/-
Copyright (c) 2021 Eric Weiser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser

! This file was ported from Lean 3 source module algebra.algebra.subalgebra.pointwise
! leanprover-community/mathlib commit 932872382355f00112641d305ba0619305dc8642
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Algebra.Algebra.Operations
import Mathbin.Algebra.Algebra.Subalgebra.Basic
import Mathbin.RingTheory.Subring.Pointwise
import Mathbin.RingTheory.Adjoin.Basic

/-!
# Pointwise actions on subalgebras.

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.

If `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)
then we get an `R'` action on the collection of `R`-subalgebras.
-/

namespace Subalgebra

section Pointwise

variable {R : Type _} {A : Type _} [CommSemiring R] [Semiring A] [Algebra R A]

/- warning: subalgebra.mul_to_submodule_le -> Subalgebra.mul_toSubmodule_le is a dubious translation:
lean 3 declaration is
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(Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_3) T)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Sup.sup.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SemilatticeSup.toHasSup.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Lattice.toSemilatticeSup.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (ConditionallyCompleteLattice.toLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_3))))) S T))
but is expected to have type
  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : Semiring.{u1} A] [_inst_3 : Algebra.{u2, u1} R A _inst_1 _inst_2] (S : Subalgebra.{u2, u1} R A _inst_1 _inst_2 _inst_3) (T : Subalgebra.{u2, u1} R A _inst_1 _inst_2 _inst_3), LE.le.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u1} R A _inst_1 _inst_2 _inst_3) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A _inst_2))) (Algebra.toModule.{u2, u1} R A _inst_1 _inst_2 _inst_3)) S) (Preorder.toLE.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u1} R A _inst_1 _inst_2 _inst_3) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A _inst_2))) (Algebra.toModule.{u2, u1} R A _inst_1 _inst_2 _inst_3)) S) (PartialOrder.toPreorder.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u1} R A _inst_1 _inst_2 _inst_3) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A _inst_2))) (Algebra.toModule.{u2, u1} R A _inst_1 _inst_2 _inst_3)) S) (OmegaCompletePartialOrder.toPartialOrder.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u1} R A _inst_1 _inst_2 _inst_3) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A _inst_2))) (Algebra.toModule.{u2, u1} R A _inst_1 _inst_2 _inst_3)) S) (CompleteLattice.instOmegaCompletePartialOrder.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : 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_inst_2 _inst_3)) S) ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u1} R A _inst_1 _inst_2 _inst_3) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A _inst_2))) (Algebra.toModule.{u2, u1} R A _inst_1 _inst_2 _inst_3)) T) ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u1} R A _inst_1 _inst_2 _inst_3) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A _inst_2))) (Algebra.toModule.{u2, u1} R A _inst_1 _inst_2 _inst_3)) S) (instHMul.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u1} R A _inst_1 _inst_2 _inst_3) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_1) 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Case conversion may be inaccurate. Consider using '#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_leₓ'. -/
theorem mul_toSubmodule_le (S T : Subalgebra R A) :
    S.toSubmodule * T.toSubmodule ≤ (S ⊔ T).toSubmodule :=
  by
  rw [Submodule.mul_le]
  intro y hy z hz
  show y * z ∈ S ⊔ T
  exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)
#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le

/- warning: subalgebra.mul_self -> Subalgebra.mul_self is a dubious translation:
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R A _inst_1 _inst_2 _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A 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_inst_1 _inst_2 _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_3) S)) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) 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(Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_3) S)
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align subalgebra.mul_self Subalgebra.mul_selfₓ'. -/
/-- As submodules, subalgebras are idempotent. -/
@[simp]
theorem mul_self (S : Subalgebra R A) : S.toSubmodule * S.toSubmodule = S.toSubmodule :=
  by
  apply le_antisymm
  · refine' (mul_to_submodule_le _ _).trans_eq _
    rw [sup_idem]
  · intro x hx1
    rw [← mul_one x]
    exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)
#align subalgebra.mul_self Subalgebra.mul_self

/- warning: subalgebra.mul_to_submodule -> Subalgebra.mul_toSubmodule is a dubious translation:
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(SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6) A (Subalgebra.setLike.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_5)))) (Algebra.toModule.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_5)))) (Algebra.toModule.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_5)))) (Algebra.toModule.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_5)))) (Algebra.toModule.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_5)))) (Algebra.toModule.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_5)))) (Algebra.toModule.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6) (Sup.sup.{u2} (Subalgebra.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6) (SemilatticeSup.toHasSup.{u2} (Subalgebra.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6) (Lattice.toSemilatticeSup.{u2} (Subalgebra.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6) (ConditionallyCompleteLattice.toLattice.{u2} (Subalgebra.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Subalgebra.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_4 (CommSemiring.toSemiring.{u2} A _inst_5) _inst_6))))) S T))
but is expected to have type
  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_4 : CommSemiring.{u2} R] [_inst_5 : CommSemiring.{u1} A] [_inst_6 : Algebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5)] (S : Subalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6) (T : Subalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6), Eq.{succ u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) S) (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A 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(CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) S) (instHMul.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) S) (Submodule.mul.{u2, u1} R _inst_4 A (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) (FunLike.coe.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} (Subalgebra.{u2, u1} R 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R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6) => LE.le.{u1} (Subalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6) (SetLike.instPartialOrder.{u1, u1} (Subalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) => LE.le.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A _inst_5)))) (Algebra.toModule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u2, u1} R A _inst_4 (CommSemiring.toSemiring.{u1} A _inst_5) _inst_6)) S) 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_inst_6))))) S T))
Case conversion may be inaccurate. Consider using '#align subalgebra.mul_to_submodule Subalgebra.mul_toSubmoduleₓ'. -/
/-- When `A` is commutative, `subalgebra.mul_to_submodule_le` is strict. -/
theorem mul_toSubmodule {R : Type _} {A : Type _} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : S.toSubmodule * T.toSubmodule = (S ⊔ T).toSubmodule :=
  by
  refine' le_antisymm (mul_to_submodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
  · cases' hx with hxS hxT
    · rw [← mul_one x]
      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)
    · rw [← one_mul x]
      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT
  · rw [← one_mul (algebraMap _ _ _)]
    exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebra_map_mem _ _)
  have := Submodule.mul_mem_mul hx hy
  rwa [mul_assoc, mul_comm _ T.to_submodule, ← mul_assoc _ _ S.to_submodule, mul_self,
    mul_comm T.to_submodule, ← mul_assoc, mul_self] at this
#align subalgebra.mul_to_submodule Subalgebra.mul_toSubmodule

variable {R' : Type _} [Semiring R'] [MulSemiringAction R' A] [SMulCommClass R' R A]

/- warning: subalgebra.pointwise_mul_action -> Subalgebra.pointwiseMulAction is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] {R' : Type.{u3}} [_inst_4 : Semiring.{u3} R'] [_inst_5 : MulSemiringAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2] [_inst_6 : SMulCommClass.{u3, u1, u2} R' R A (SMulZeroClass.toHasSmul.{u3, u2} R' A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (DistribSMul.toSmulZeroClass.{u3, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2 _inst_5)))) (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))], MulAction.{u3, u2} R' (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4))
but is expected to have type
  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] {R' : Type.{u3}} [_inst_4 : Semiring.{u3} R'] [_inst_5 : MulSemiringAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2] [_inst_6 : SMulCommClass.{u3, u1, u2} R' R A (SMulZeroClass.toSMul.{u3, u2} R' A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (DistribSMul.toSMulZeroClass.{u3, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2 _inst_5)))) (Algebra.toSMul.{u1, u2} R A _inst_1 _inst_2 _inst_3)], MulAction.{u3, u2} R' (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4))
Case conversion may be inaccurate. Consider using '#align subalgebra.pointwise_mul_action Subalgebra.pointwiseMulActionₓ'. -/
/-- The action on a subalgebra corresponding to applying the action to every element.

This is available as an instance in the `pointwise` locale. -/
protected def pointwiseMulAction : MulAction R' (Subalgebra R A)
    where
  smul a S := S.map (MulSemiringAction.toAlgHom _ _ a)
  one_smul S := (congr_arg (fun f => S.map f) (AlgHom.ext <| one_smul R')).trans S.map_id
  mul_smul a₁ a₂ S :=
    (congr_arg (fun f => S.map f) (AlgHom.ext <| mul_smul _ _)).trans (S.map_map _ _).symm
#align subalgebra.pointwise_mul_action Subalgebra.pointwiseMulAction

scoped[Pointwise] attribute [instance] Subalgebra.pointwiseMulAction

open Pointwise

/- warning: subalgebra.coe_pointwise_smul -> Subalgebra.coe_pointwise_smul is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] {R' : Type.{u3}} [_inst_4 : Semiring.{u3} R'] [_inst_5 : MulSemiringAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2] [_inst_6 : SMulCommClass.{u3, u1, u2} R' R A (SMulZeroClass.toHasSmul.{u3, u2} R' A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (DistribSMul.toSmulZeroClass.{u3, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2 _inst_5)))) (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))] (m : R') (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3), Eq.{succ u2} (Set.{u2} A) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) (SMul.smul.{u3, u2} R' (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (MulAction.toHasSmul.{u3, u2} R' (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (Subalgebra.pointwiseMulAction.{u1, u2, u3} R A _inst_1 _inst_2 _inst_3 R' _inst_4 _inst_5 _inst_6)) m S)) (SMul.smul.{u3, u2} R' (Set.{u2} A) (Set.smulSet.{u3, u2} R' A (SMulZeroClass.toHasSmul.{u3, u2} R' A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (DistribSMul.toSmulZeroClass.{u3, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2 _inst_5))))) m ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) S))
but is expected to have type
  forall {R : Type.{u3}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u3, u2} R A _inst_1 _inst_2] {R' : Type.{u1}} [_inst_4 : Semiring.{u1} R'] [_inst_5 : MulSemiringAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2] [_inst_6 : SMulCommClass.{u1, u3, u2} R' R A (SMulZeroClass.toSMul.{u1, u2} R' A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2 _inst_5)))) (Algebra.toSMul.{u3, u2} R A _inst_1 _inst_2 _inst_3)] (m : R') (S : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3), Eq.{succ u2} (Set.{u2} A) (SetLike.coe.{u2, u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (HSMul.hSMul.{u1, u2, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (instHSMul.{u1, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MulAction.toSMul.{u1, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (Subalgebra.pointwiseMulAction.{u3, u2, u1} R A _inst_1 _inst_2 _inst_3 R' _inst_4 _inst_5 _inst_6))) m S)) (HSMul.hSMul.{u1, u2, u2} R' (Set.{u2} A) (Set.{u2} A) (instHSMul.{u1, u2} R' (Set.{u2} A) (Set.smulSet.{u1, u2} R' A (SMulZeroClass.toSMul.{u1, u2} R' A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2 _inst_5)))))) m (SetLike.coe.{u2, u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) S))
Case conversion may be inaccurate. Consider using '#align subalgebra.coe_pointwise_smul Subalgebra.coe_pointwise_smulₓ'. -/
@[simp]
theorem coe_pointwise_smul (m : R') (S : Subalgebra R A) : ↑(m • S) = m • (S : Set A) :=
  rfl
#align subalgebra.coe_pointwise_smul Subalgebra.coe_pointwise_smul

/- warning: subalgebra.pointwise_smul_to_subsemiring -> Subalgebra.pointwise_smul_toSubsemiring is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] {R' : Type.{u3}} [_inst_4 : Semiring.{u3} R'] [_inst_5 : MulSemiringAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2] [_inst_6 : SMulCommClass.{u3, u1, u2} R' R A (SMulZeroClass.toHasSmul.{u3, u2} R' A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (DistribSMul.toSmulZeroClass.{u3, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2 _inst_5)))) (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))] (m : R') (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3), Eq.{succ u2} (Subsemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Subalgebra.toSubsemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 (SMul.smul.{u3, u2} R' (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (MulAction.toHasSmul.{u3, u2} R' (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (Subalgebra.pointwiseMulAction.{u1, u2, u3} R A _inst_1 _inst_2 _inst_3 R' _inst_4 _inst_5 _inst_6)) m S)) (SMul.smul.{u3, u2} R' (Subsemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (MulAction.toHasSmul.{u3, u2} R' (Subsemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (Subsemiring.pointwiseMulAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2 _inst_5)) m (Subalgebra.toSubsemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 S))
but is expected to have type
  forall {R : Type.{u3}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u3, u2} R A _inst_1 _inst_2] {R' : Type.{u1}} [_inst_4 : Semiring.{u1} R'] [_inst_5 : MulSemiringAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2] [_inst_6 : SMulCommClass.{u1, u3, u2} R' R A (SMulZeroClass.toSMul.{u1, u2} R' A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2 _inst_5)))) (Algebra.toSMul.{u3, u2} R A _inst_1 _inst_2 _inst_3)] (m : R') (S : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3), Eq.{succ u2} (Subsemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Subalgebra.toSubsemiring.{u3, u2} R A _inst_1 _inst_2 _inst_3 (HSMul.hSMul.{u1, u2, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (instHSMul.{u1, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MulAction.toSMul.{u1, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (Subalgebra.pointwiseMulAction.{u3, u2, u1} R A _inst_1 _inst_2 _inst_3 R' _inst_4 _inst_5 _inst_6))) m S)) (HSMul.hSMul.{u1, u2, u2} R' (Subsemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Subsemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (instHSMul.{u1, u2} R' (Subsemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (MulAction.toSMul.{u1, u2} R' (Subsemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (Subsemiring.pointwiseMulAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2 _inst_5))) m (Subalgebra.toSubsemiring.{u3, u2} R A _inst_1 _inst_2 _inst_3 S))
Case conversion may be inaccurate. Consider using '#align subalgebra.pointwise_smul_to_subsemiring Subalgebra.pointwise_smul_toSubsemiringₓ'. -/
@[simp]
theorem pointwise_smul_toSubsemiring (m : R') (S : Subalgebra R A) :
    (m • S).toSubsemiring = m • S.toSubsemiring :=
  rfl
#align subalgebra.pointwise_smul_to_subsemiring Subalgebra.pointwise_smul_toSubsemiring

/- warning: subalgebra.pointwise_smul_to_submodule -> Subalgebra.pointwise_smul_toSubmodule is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] {R' : Type.{u3}} [_inst_4 : Semiring.{u3} R'] [_inst_5 : MulSemiringAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2] [_inst_6 : SMulCommClass.{u3, u1, u2} R' R A (SMulZeroClass.toHasSmul.{u3, u2} R' A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (DistribSMul.toSmulZeroClass.{u3, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' 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u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SMul.smul.{u3, u2} R' (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (MulAction.toHasSmul.{u3, u2} R' (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (Subalgebra.pointwiseMulAction.{u1, u2, u3} R A _inst_1 _inst_2 _inst_3 R' _inst_4 _inst_5 _inst_6)) m S)) (SMul.smul.{u3, u2} R' (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (SMulZeroClass.toHasSmul.{u3, u2} R' (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (AddZeroClass.toHasZero.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (AddMonoid.toAddZeroClass.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (AddCommMonoid.toAddMonoid.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.pointwiseAddCommMonoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))))) (DistribSMul.toSmulZeroClass.{u3, u2} R' (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (AddMonoid.toAddZeroClass.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (AddCommMonoid.toAddMonoid.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.pointwiseAddCommMonoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) (DistribMulAction.toDistribSMul.{u3, u2} R' (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (AddCommMonoid.toAddMonoid.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A 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(Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))))) => (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_3) S))
but is expected to have type
  forall {R : Type.{u3}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u3, u2} R A _inst_1 _inst_2] {R' : Type.{u1}} [_inst_4 : Semiring.{u1} R'] [_inst_5 : MulSemiringAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2] [_inst_6 : SMulCommClass.{u1, u3, u2} R' R A (SMulZeroClass.toSMul.{u1, u2} R' A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2 _inst_5)))) (Algebra.toSMul.{u3, u2} R A _inst_1 _inst_2 _inst_3)] (m : R') (S : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3), Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (HSMul.hSMul.{u1, u2, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (instHSMul.{u1, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MulAction.toSMul.{u1, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (Subalgebra.pointwiseMulAction.{u3, u2, u1} R A _inst_1 _inst_2 _inst_3 R' _inst_4 _inst_5 _inst_6))) m S)) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3))) (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (fun (_x : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A 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_inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) => LE.le.{u2} (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (HSMul.hSMul.{u1, u2, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (instHSMul.{u1, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MulAction.toSMul.{u1, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (Subalgebra.pointwiseMulAction.{u3, u2, u1} R A _inst_1 _inst_2 _inst_3 R' _inst_4 _inst_5 _inst_6))) m S)) (HSMul.hSMul.{u1, u2, u2} R' ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (instHSMul.{u1, u2} R' ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (SMulZeroClass.toSMul.{u1, u2} R' ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (AddMonoidWithOne.toAddMonoid.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (NonAssocSemiring.toAddCommMonoidWithOne.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (Semiring.toNonAssocSemiring.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (IdemSemiring.toSemiring.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (Submodule.idemSemiring.{u3, u2} R _inst_1 A _inst_2 _inst_3))))))) (DistribMulAction.toDistribSMul.{u1, u2} R' ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S) (NonAssocSemiring.toAddCommMonoidWithOne.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A 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(Submodule.pointwiseDistribMulAction.{u1, u3, u2} R' R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (MulSemiringAction.toDistribMulAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2 _inst_5) _inst_6))))) m (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3))) (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) 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Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) => LE.le.{u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) => LE.le.{u2} (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u3, u2} R A (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u3, u2} R A _inst_1 _inst_2 _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u3, u2} R A _inst_1 _inst_2 _inst_3)) S))
Case conversion may be inaccurate. Consider using '#align subalgebra.pointwise_smul_to_submodule Subalgebra.pointwise_smul_toSubmoduleₓ'. -/
@[simp]
theorem pointwise_smul_toSubmodule (m : R') (S : Subalgebra R A) :
    (m • S).toSubmodule = m • S.toSubmodule :=
  rfl
#align subalgebra.pointwise_smul_to_submodule Subalgebra.pointwise_smul_toSubmodule

/- warning: subalgebra.pointwise_smul_to_subring -> Subalgebra.pointwise_smul_toSubring is a dubious translation:
lean 3 declaration is
  forall {R' : Type.{u1}} {R : Type.{u2}} {A : Type.{u3}} [_inst_7 : Semiring.{u1} R'] [_inst_8 : CommRing.{u2} R] [_inst_9 : Ring.{u3} A] [_inst_10 : MulSemiringAction.{u1, u3} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_7)) (Ring.toSemiring.{u3} A _inst_9)] [_inst_11 : Algebra.{u2, u3} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u3} A _inst_9)] [_inst_12 : SMulCommClass.{u1, u2, u3} R' R A (SMulZeroClass.toHasSmul.{u1, u3} R' A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddMonoidWithOne.toAddMonoid.{u3} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u3} A (NonAssocSemiring.toAddCommMonoidWithOne.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_9))))))) (DistribSMul.toSmulZeroClass.{u1, u3} R' A (AddMonoid.toAddZeroClass.{u3} A (AddMonoidWithOne.toAddMonoid.{u3} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u3} A (NonAssocSemiring.toAddCommMonoidWithOne.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_9)))))) (DistribMulAction.toDistribSMul.{u1, u3} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_7)) (AddMonoidWithOne.toAddMonoid.{u3} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u3} A (NonAssocSemiring.toAddCommMonoidWithOne.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_9))))) (MulSemiringAction.toDistribMulAction.{u1, u3} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_7)) (Ring.toSemiring.{u3} A _inst_9) _inst_10)))) (SMulZeroClass.toHasSmul.{u2, u3} R A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_9))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} R A (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_8)))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_9))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R A (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_8))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_9))))))) (Module.toMulActionWithZero.{u2, u3} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_8)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_9)))) (Algebra.toModule.{u2, u3} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u3} A _inst_9) _inst_11)))))] (m : R') (S : Subalgebra.{u2, u3} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u3} A _inst_9) _inst_11), Eq.{succ u3} (Subring.{u3} A _inst_9) (Subalgebra.toSubring.{u2, u3} R A _inst_8 _inst_9 _inst_11 (SMul.smul.{u1, u3} R' (Subalgebra.{u2, u3} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u3} A _inst_9) _inst_11) (MulAction.toHasSmul.{u1, u3} R' (Subalgebra.{u2, u3} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u3} A _inst_9) _inst_11) (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_7)) (Subalgebra.pointwiseMulAction.{u2, u3, u1} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u3} A _inst_9) _inst_11 R' _inst_7 _inst_10 _inst_12)) m S)) (SMul.smul.{u1, u3} R' (Subring.{u3} A _inst_9) (MulAction.toHasSmul.{u1, u3} R' (Subring.{u3} A _inst_9) (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_7)) (Subring.pointwiseMulAction.{u1, u3} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_7)) _inst_9 _inst_10)) m (Subalgebra.toSubring.{u2, u3} R A _inst_8 _inst_9 _inst_11 S))
but is expected to have type
  forall {R' : Type.{u3}} {R : Type.{u2}} {A : Type.{u1}} [_inst_7 : Semiring.{u3} R'] [_inst_8 : CommRing.{u2} R] [_inst_9 : Ring.{u1} A] [_inst_10 : MulSemiringAction.{u3, u1} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_7)) (Ring.toSemiring.{u1} A _inst_9)] [_inst_11 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u1} A _inst_9)] [_inst_12 : SMulCommClass.{u3, u2, u1} R' R A (SMulZeroClass.toSMul.{u3, u1} R' A (MonoidWithZero.toZero.{u1} A (Semiring.toMonoidWithZero.{u1} A (Ring.toSemiring.{u1} A _inst_9))) (DistribSMul.toSMulZeroClass.{u3, u1} R' A (AddMonoid.toAddZeroClass.{u1} A (AddMonoidWithOne.toAddMonoid.{u1} A (AddGroupWithOne.toAddMonoidWithOne.{u1} A (Ring.toAddGroupWithOne.{u1} A _inst_9)))) (DistribMulAction.toDistribSMul.{u3, u1} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_7)) (AddMonoidWithOne.toAddMonoid.{u1} A (AddGroupWithOne.toAddMonoidWithOne.{u1} A (Ring.toAddGroupWithOne.{u1} A _inst_9))) (MulSemiringAction.toDistribMulAction.{u3, u1} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_7)) (Ring.toSemiring.{u1} A _inst_9) _inst_10)))) (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u1} A _inst_9) _inst_11)] (m : R') (S : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u1} A _inst_9) _inst_11), Eq.{succ u1} (Subring.{u1} A _inst_9) (Subalgebra.toSubring.{u2, u1} R A _inst_8 _inst_9 _inst_11 (HSMul.hSMul.{u3, u1, u1} R' (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u1} A _inst_9) _inst_11) (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u1} A _inst_9) _inst_11) (instHSMul.{u3, u1} R' (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u1} A _inst_9) _inst_11) (MulAction.toSMul.{u3, u1} R' (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u1} A _inst_9) _inst_11) (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_7)) (Subalgebra.pointwiseMulAction.{u2, u1, u3} R A (CommRing.toCommSemiring.{u2} R _inst_8) (Ring.toSemiring.{u1} A _inst_9) _inst_11 R' _inst_7 _inst_10 _inst_12))) m S)) (HSMul.hSMul.{u3, u1, u1} R' (Subring.{u1} A _inst_9) (Subring.{u1} A _inst_9) (instHSMul.{u3, u1} R' (Subring.{u1} A _inst_9) (MulAction.toSMul.{u3, u1} R' (Subring.{u1} A _inst_9) (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_7)) (Subring.pointwiseMulAction.{u3, u1} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_7)) _inst_9 _inst_10))) m (Subalgebra.toSubring.{u2, u1} R A _inst_8 _inst_9 _inst_11 S))
Case conversion may be inaccurate. Consider using '#align subalgebra.pointwise_smul_to_subring Subalgebra.pointwise_smul_toSubringₓ'. -/
@[simp]
theorem pointwise_smul_toSubring {R' R A : Type _} [Semiring R'] [CommRing R] [Ring A]
    [MulSemiringAction R' A] [Algebra R A] [SMulCommClass R' R A] (m : R') (S : Subalgebra R A) :
    (m • S).toSubring = m • S.toSubring :=
  rfl
#align subalgebra.pointwise_smul_to_subring Subalgebra.pointwise_smul_toSubring

/- warning: subalgebra.smul_mem_pointwise_smul -> Subalgebra.smul_mem_pointwise_smul is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] {R' : Type.{u3}} [_inst_4 : Semiring.{u3} R'] [_inst_5 : MulSemiringAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2] [_inst_6 : SMulCommClass.{u3, u1, u2} R' R A (SMulZeroClass.toHasSmul.{u3, u2} R' A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (DistribSMul.toSmulZeroClass.{u3, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2 _inst_5)))) (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))] (m : R') (r : A) (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3), (Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3)) r S) -> (Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (SMul.smul.{u3, u2} R' A (SMulZeroClass.toHasSmul.{u3, u2} R' A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))))) (DistribSMul.toSmulZeroClass.{u3, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u3, u2} R' A (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) _inst_2 _inst_5)))) m r) (SMul.smul.{u3, u2} R' (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (MulAction.toHasSmul.{u3, u2} R' (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u3} R' (Semiring.toMonoidWithZero.{u3} R' _inst_4)) (Subalgebra.pointwiseMulAction.{u1, u2, u3} R A _inst_1 _inst_2 _inst_3 R' _inst_4 _inst_5 _inst_6)) m S))
but is expected to have type
  forall {R : Type.{u3}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u3, u2} R A _inst_1 _inst_2] {R' : Type.{u1}} [_inst_4 : Semiring.{u1} R'] [_inst_5 : MulSemiringAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2] [_inst_6 : SMulCommClass.{u1, u3, u2} R' R A (SMulZeroClass.toSMul.{u1, u2} R' A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2 _inst_5)))) (Algebra.toSMul.{u3, u2} R A _inst_1 _inst_2 _inst_3)] (m : R') (r : A) (S : Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3), (Membership.mem.{u2, u2} A (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.instMembership.{u2, u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3)) r S) -> (Membership.mem.{u2, u2} A (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.instMembership.{u2, u2} (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3)) (HSMul.hSMul.{u1, u2, u2} R' A A (instHSMul.{u1, u2} R' A (SMulZeroClass.toSMul.{u1, u2} R' A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R' A (AddMonoid.toAddZeroClass.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))))) (DistribMulAction.toDistribSMul.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (AddMonoidWithOne.toAddMonoid.{u2} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} A (NonAssocSemiring.toAddCommMonoidWithOne.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (MulSemiringAction.toDistribMulAction.{u1, u2} R' A (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) _inst_2 _inst_5))))) m r) (HSMul.hSMul.{u1, u2, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (instHSMul.{u1, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MulAction.toSMul.{u1, u2} R' (Subalgebra.{u3, u2} R A _inst_1 _inst_2 _inst_3) (MonoidWithZero.toMonoid.{u1} R' (Semiring.toMonoidWithZero.{u1} R' _inst_4)) (Subalgebra.pointwiseMulAction.{u3, u2, u1} R A _inst_1 _inst_2 _inst_3 R' _inst_4 _inst_5 _inst_6))) m S))
Case conversion may be inaccurate. Consider using '#align subalgebra.smul_mem_pointwise_smul Subalgebra.smul_mem_pointwise_smulₓ'. -/
theorem smul_mem_pointwise_smul (m : R') (r : A) (S : Subalgebra R A) : r ∈ S → m • r ∈ m • S :=
  (Set.smul_mem_smul_set : _ → _ ∈ m • (S : Set A))
#align subalgebra.smul_mem_pointwise_smul Subalgebra.smul_mem_pointwise_smul

end Pointwise

end Subalgebra