/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll

! This file was ported from Lean 3 source module linear_algebra.basis.bilinear
! leanprover-community/mathlib commit 832f7b9162039c28b9361289c8681f155cae758f
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.LinearAlgebra.Basis
import Mathbin.LinearAlgebra.BilinearMap

/-!
# Lemmas about bilinear maps with a basis over each argument

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

namespace LinearMap

variable {ι₁ ι₂ : Type _}

variable {R R₂ S S₂ M N P : Type _}

variable {Mₗ Nₗ Pₗ : Type _}

variable [CommSemiring R] [CommSemiring S] [CommSemiring R₂] [CommSemiring S₂]

section AddCommMonoid

variable [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P]

variable [AddCommMonoid Mₗ] [AddCommMonoid Nₗ] [AddCommMonoid Pₗ]

variable [Module R M] [Module S N] [Module R₂ P] [Module S₂ P]

variable [Module R Mₗ] [Module R Nₗ] [Module R Pₗ]

variable [SMulCommClass S₂ R₂ P]

variable {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂}

variable (b₁ : Basis ι₁ R M) (b₂ : Basis ι₂ S N) (b₁' : Basis ι₁ R Mₗ) (b₂' : Basis ι₂ R Nₗ)

/- warning: linear_map.ext_basis -> LinearMap.ext_basis is a dubious translation:
lean 3 declaration is
  forall {ι₁ : Type.{u1}} {ι₂ : Type.{u2}} {R : Type.{u3}} {R₂ : Type.{u4}} {S : Type.{u5}} {S₂ : Type.{u6}} {M : Type.{u7}} {N : Type.{u8}} {P : Type.{u9}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : CommSemiring.{u5} S] [_inst_3 : CommSemiring.{u4} R₂] [_inst_4 : CommSemiring.{u6} S₂] [_inst_5 : AddCommMonoid.{u7} M] [_inst_6 : AddCommMonoid.{u8} N] [_inst_7 : AddCommMonoid.{u9} P] [_inst_11 : Module.{u3, u7} R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5] [_inst_12 : Module.{u5, u8} S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6] [_inst_13 : Module.{u4, u9} R₂ P (CommSemiring.toSemiring.{u4} R₂ _inst_3) _inst_7] [_inst_14 : Module.{u6, u9} S₂ P (CommSemiring.toSemiring.{u6} S₂ _inst_4) _inst_7] [_inst_18 : SMulCommClass.{u6, u4, u9} S₂ R₂ P (SMulZeroClass.toHasSmul.{u6, u9} S₂ P (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (SMulWithZero.toSmulZeroClass.{u6, u9} S₂ P (MulZeroClass.toHasZero.{u6} S₂ (MulZeroOneClass.toMulZeroClass.{u6} S₂ (MonoidWithZero.toMulZeroOneClass.{u6} S₂ (Semiring.toMonoidWithZero.{u6} S₂ (CommSemiring.toSemiring.{u6} S₂ _inst_4))))) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (MulActionWithZero.toSMulWithZero.{u6, u9} S₂ P (Semiring.toMonoidWithZero.{u6} S₂ (CommSemiring.toSemiring.{u6} S₂ _inst_4)) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (Module.toMulActionWithZero.{u6, u9} S₂ P (CommSemiring.toSemiring.{u6} S₂ _inst_4) _inst_7 _inst_14)))) (SMulZeroClass.toHasSmul.{u4, u9} R₂ P (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (SMulWithZero.toSmulZeroClass.{u4, u9} R₂ P (MulZeroClass.toHasZero.{u4} R₂ (MulZeroOneClass.toMulZeroClass.{u4} R₂ (MonoidWithZero.toMulZeroOneClass.{u4} R₂ (Semiring.toMonoidWithZero.{u4} R₂ (CommSemiring.toSemiring.{u4} R₂ _inst_3))))) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (MulActionWithZero.toSMulWithZero.{u4, u9} R₂ P (Semiring.toMonoidWithZero.{u4} R₂ (CommSemiring.toSemiring.{u4} R₂ _inst_3)) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (Module.toMulActionWithZero.{u4, u9} R₂ P (CommSemiring.toSemiring.{u4} R₂ _inst_3) _inst_7 _inst_13))))] {ρ₁₂ : RingHom.{u3, u4} R R₂ (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u4} R₂ (CommSemiring.toSemiring.{u4} R₂ _inst_3))} {σ₁₂ : RingHom.{u5, u6} S S₂ (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) (Semiring.toNonAssocSemiring.{u6} S₂ (CommSemiring.toSemiring.{u6} S₂ _inst_4))} (b₁ : Basis.{u1, u3, u7} ι₁ R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) (b₂ : Basis.{u2, u5, u8} ι₂ S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 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but is expected to have type
  forall {ι₁ : Type.{u1}} {ι₂ : Type.{u2}} {R : Type.{u9}} {R₂ : Type.{u8}} {S : Type.{u4}} {S₂ : Type.{u3}} {M : Type.{u7}} {N : Type.{u5}} {P : Type.{u6}} [_inst_1 : CommSemiring.{u9} R] [_inst_2 : CommSemiring.{u4} S] [_inst_3 : CommSemiring.{u8} R₂] [_inst_4 : CommSemiring.{u3} S₂] [_inst_5 : AddCommMonoid.{u7} M] [_inst_6 : AddCommMonoid.{u5} N] [_inst_7 : AddCommMonoid.{u6} P] [_inst_11 : Module.{u9, u7} R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5] [_inst_12 : Module.{u4, u5} S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6] [_inst_13 : Module.{u8, u6} R₂ P (CommSemiring.toSemiring.{u8} R₂ _inst_3) _inst_7] [_inst_14 : Module.{u3, u6} S₂ P (CommSemiring.toSemiring.{u3} S₂ _inst_4) _inst_7] [_inst_18 : SMulCommClass.{u3, u8, u6} S₂ R₂ P (SMulZeroClass.toSMul.{u3, u6} S₂ P (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (SMulWithZero.toSMulZeroClass.{u3, u6} S₂ P (CommMonoidWithZero.toZero.{u3} S₂ (CommSemiring.toCommMonoidWithZero.{u3} S₂ 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RingHom.{u9, u8} R R₂ (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (Semiring.toNonAssocSemiring.{u8} R₂ (CommSemiring.toSemiring.{u8} R₂ _inst_3))} {σ₁₂ : RingHom.{u4, u3} S S₂ (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} S₂ (CommSemiring.toSemiring.{u3} S₂ _inst_4))} (b₁ : Basis.{u1, u9, u7} ι₁ R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) (b₂ : Basis.{u2, u4, u5} ι₂ S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) {B : LinearMap.{u9, u8, u7, max u6 u5} R R₂ (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R₂ _inst_3) ρ₁₂ M (LinearMap.{u4, u3, u5, u6} S S₂ (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S₂ _inst_4) σ₁₂ N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S₂ N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S₂ _inst_4) _inst_6 _inst_7 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u6} S S₂ R₂ N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S₂ _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 σ₁₂ (CommSemiring.toSemiring.{u8} R₂ _inst_3) _inst_13 _inst_18)) B B')
Case conversion may be inaccurate. Consider using '#align linear_map.ext_basis LinearMap.ext_basisₓ'. -/
/-- Two bilinear maps are equal when they are equal on all basis vectors. -/
theorem ext_basis {B B' : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (h : ∀ i j, B (b₁ i) (b₂ j) = B' (b₁ i) (b₂ j)) :
    B = B' :=
  b₁.ext fun i => b₂.ext fun j => h i j
#align linear_map.ext_basis LinearMap.ext_basis

/- warning: linear_map.sum_repr_mul_repr_mulₛₗ -> LinearMap.sum_repr_mul_repr_mulₛₗ is a dubious translation:
lean 3 declaration is
  forall {ι₁ : Type.{u1}} {ι₂ : Type.{u2}} {R : Type.{u3}} {R₂ : Type.{u4}} {S : Type.{u5}} {S₂ : Type.{u6}} {M : Type.{u7}} {N : Type.{u8}} {P : Type.{u9}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : CommSemiring.{u5} S] [_inst_3 : CommSemiring.{u4} R₂] [_inst_4 : CommSemiring.{u6} S₂] [_inst_5 : AddCommMonoid.{u7} M] [_inst_6 : AddCommMonoid.{u8} N] [_inst_7 : AddCommMonoid.{u9} P] [_inst_11 : Module.{u3, u7} R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5] [_inst_12 : Module.{u5, u8} S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6] [_inst_13 : Module.{u4, u9} R₂ P (CommSemiring.toSemiring.{u4} R₂ _inst_3) _inst_7] [_inst_14 : Module.{u6, u9} S₂ P (CommSemiring.toSemiring.{u6} S₂ _inst_4) _inst_7] [_inst_18 : SMulCommClass.{u6, u4, u9} S₂ R₂ P (SMulZeroClass.toHasSmul.{u6, u9} S₂ P (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (SMulWithZero.toSmulZeroClass.{u6, u9} S₂ P (MulZeroClass.toHasZero.{u6} S₂ 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but is expected to have type
  forall {ι₁ : Type.{u2}} {ι₂ : Type.{u1}} {R : Type.{u9}} {R₂ : Type.{u8}} {S : Type.{u4}} {S₂ : Type.{u3}} {M : Type.{u7}} {N : Type.{u5}} {P : Type.{u6}} [_inst_1 : CommSemiring.{u9} R] [_inst_2 : CommSemiring.{u4} S] [_inst_3 : CommSemiring.{u8} R₂] [_inst_4 : CommSemiring.{u3} S₂] [_inst_5 : AddCommMonoid.{u7} M] [_inst_6 : AddCommMonoid.{u5} N] [_inst_7 : AddCommMonoid.{u6} P] [_inst_11 : Module.{u9, u7} R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5] [_inst_12 : Module.{u4, u5} S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6] [_inst_13 : Module.{u8, u6} R₂ P (CommSemiring.toSemiring.{u8} R₂ _inst_3) _inst_7] [_inst_14 : Module.{u3, u6} S₂ P (CommSemiring.toSemiring.{u3} S₂ _inst_4) _inst_7] [_inst_18 : SMulCommClass.{u3, u8, u6} S₂ R₂ P (SMulZeroClass.toSMul.{u3, u6} S₂ P (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (SMulWithZero.toSMulZeroClass.{u3, u6} S₂ P (CommMonoidWithZero.toZero.{u3} S₂ (CommSemiring.toCommMonoidWithZero.{u3} S₂ 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(CommSemiring.toSemiring.{u9} R _inst_1)))))) (Module.toDistribMulAction.{u9, max u2 u9} R (Finsupp.{u2, u9} ι₁ R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (CommSemiring.toSemiring.{u9} R _inst_1) (Finsupp.addCommMonoid.{u2, u9} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) (Finsupp.module.{u2, u9, u9} ι₁ R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))))) (DistribMulActionHomClass.toSMulHomClass.{max (max u2 u9) u7, u9, u7, max u2 u9} (LinearEquiv.{u9, u9, u7, max u9 u2} R R (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u9} R _inst_1) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) M (Finsupp.{u2, u9} ι₁ R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) _inst_5 (Finsupp.addCommMonoid.{u2, u9} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} ι₁ R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) R M (Finsupp.{u2, u9} ι₁ R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (MonoidWithZero.toMonoid.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (AddCommMonoid.toAddMonoid.{u7} M _inst_5) (AddCommMonoid.toAddMonoid.{max u2 u9} (Finsupp.{u2, u9} ι₁ R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u9} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))))) (Module.toDistribMulAction.{u9, u7} R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) (Module.toDistribMulAction.{u9, max u2 u9} R (Finsupp.{u2, u9} ι₁ R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R 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(Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) M (Finsupp.{u2, u9} ι₁ R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) _inst_5 (Finsupp.addCommMonoid.{u2, u9} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} ι₁ R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 (Finsupp.addCommMonoid.{u2, u9} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} ι₁ R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (SemilinearEquivClass.instSemilinearMapClass.{u9, u9, u7, max u2 u9, max (max u2 u9) u7} R R M (Finsupp.{u2, u9} ι₁ R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (LinearEquiv.{u9, u9, u7, max u9 u2} R R (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u9} R _inst_1) 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(RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))))) (Basis.repr.{u2, u9, u7} ι₁ R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11 b₁) x) (fun (i : ι₁) (xi : R) => Finsupp.sum.{u1, u4, u6} ι₂ S P (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) _inst_7 (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u5, max (succ u1) (succ u4)} (LinearEquiv.{u4, u4, u5, max u4 u1} S S (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u4} S _inst_2) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) N (Finsupp.{u1, u4} ι₂ S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) _inst_6 (Finsupp.addCommMonoid.{u1, u4} ι₂ S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) _inst_12 (Finsupp.module.{u1, u4, u4} ι₂ S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : N) => Finsupp.{u1, u4} ι₂ S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) _x) 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Case conversion may be inaccurate. Consider using '#align linear_map.sum_repr_mul_repr_mulₛₗ LinearMap.sum_repr_mul_repr_mulₛₗₓ'. -/
/-- Write out `B x y` as a sum over `B (b i) (b j)` if `b` is a basis.

Version for semi-bilinear maps, see `sum_repr_mul_repr_mul` for the bilinear version. -/
theorem sum_repr_mul_repr_mulₛₗ {B : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (x y) :
    ((b₁.repr x).Sum fun i xi => (b₂.repr y).Sum fun j yj => ρ₁₂ xi • σ₁₂ yj • B (b₁ i) (b₂ j)) =
      B x y :=
  by
  conv_rhs => rw [← b₁.total_repr x, ← b₂.total_repr y]
  simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smulₛₗ₂,
    LinearMap.map_smulₛₗ]
#align linear_map.sum_repr_mul_repr_mulₛₗ LinearMap.sum_repr_mul_repr_mulₛₗ

/- warning: linear_map.sum_repr_mul_repr_mul -> LinearMap.sum_repr_mul_repr_mul is a dubious translation:
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_inst_8 (LinearMap.addCommMonoid.{u3, u3, u5, u6} R R Nₗ Pₗ (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) _inst_15 (LinearMap.module.{u3, u3, u3, u5, u6} R R R Nₗ Pₗ (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_17 (smulCommClass_self.{u3, u6} R Pₗ (CommSemiring.toCommMonoid.{u3} R _inst_1) (MulActionWithZero.toMulAction.{u3, u6} R Pₗ (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} Pₗ (AddMonoid.toAddZeroClass.{u6} Pₗ (AddCommMonoid.toAddMonoid.{u6} Pₗ _inst_10))) (Module.toMulActionWithZero.{u3, u6} R Pₗ (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17))))) => Mₗ -> (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) Nₗ Pₗ _inst_9 _inst_10 _inst_16 _inst_17)) (LinearMap.hasCoeToFun.{u3, u3, u4, max u5 u6} R R Mₗ (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) Nₗ Pₗ _inst_9 _inst_10 _inst_16 _inst_17) (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8 (LinearMap.addCommMonoid.{u3, u3, u5, u6} R R Nₗ Pₗ (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) _inst_15 (LinearMap.module.{u3, u3, u3, u5, u6} R R R Nₗ Pₗ (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_17 (smulCommClass_self.{u3, u6} R Pₗ (CommSemiring.toCommMonoid.{u3} R _inst_1) (MulActionWithZero.toMulAction.{u3, u6} R Pₗ (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} Pₗ (AddMonoid.toAddZeroClass.{u6} Pₗ (AddCommMonoid.toAddMonoid.{u6} Pₗ _inst_10))) (Module.toMulActionWithZero.{u3, u6} R Pₗ (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17)))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) B x) y)
but is expected to have type
  forall {ι₁ : Type.{u2}} {ι₂ : Type.{u1}} {R : Type.{u6}} {Mₗ : Type.{u5}} {Nₗ : Type.{u3}} {Pₗ : Type.{u4}} [_inst_1 : CommSemiring.{u6} R] [_inst_8 : AddCommMonoid.{u5} Mₗ] [_inst_9 : AddCommMonoid.{u3} Nₗ] [_inst_10 : AddCommMonoid.{u4} Pₗ] [_inst_15 : Module.{u6, u5} R Mₗ (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8] [_inst_16 : Module.{u6, u3} R Nₗ (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9] [_inst_17 : Module.{u6, u4} R Pₗ (CommSemiring.toSemiring.{u6} R _inst_1) _inst_10] (b₁' : Basis.{u2, u6, u5} ι₁ R Mₗ (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15) (b₂' : Basis.{u1, u6, u3} ι₂ R Nₗ (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) {B : LinearMap.{u6, u6, u5, max u4 u3} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) Mₗ (LinearMap.{u6, u6, u3, u4} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) Nₗ Pₗ _inst_9 _inst_10 _inst_16 _inst_17) _inst_8 (LinearMap.addCommMonoid.{u6, u6, u3, u4} R R Nₗ Pₗ (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_15 (LinearMap.instModuleLinearMapAddCommMonoid.{u6, u6, u6, u3, u4} R R R Nₗ Pₗ (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_17 (smulCommClass_self.{u6, u4} R Pₗ (CommSemiring.toCommMonoid.{u6} R _inst_1) (MulActionWithZero.toMulAction.{u6, u4} R Pₗ (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (AddMonoid.toZero.{u4} Pₗ (AddCommMonoid.toAddMonoid.{u4} Pₗ _inst_10)) (Module.toMulActionWithZero.{u6, u4} R Pₗ (CommSemiring.toSemiring.{u6} R _inst_1) _inst_10 _inst_17))))} (x : Mₗ) (y : Nₗ), Eq.{succ u4} Pₗ (Finsupp.sum.{u2, u6, u4} ι₁ R Pₗ (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) _inst_10 (FunLike.coe.{max (max (succ u2) (succ u6)) (succ u5), succ u5, max (succ u2) (succ u6)} (LinearEquiv.{u6, u6, u5, max u6 u2} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) Mₗ (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_8 (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} ι₁ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) Mₗ (fun (_x : Mₗ) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Mₗ) => Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _x) (SMulHomClass.toFunLike.{max (max u2 u6) u5, u6, u5, max u2 u6} (LinearEquiv.{u6, u6, u5, max u6 u2} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) Mₗ (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_8 (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} ι₁ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) R Mₗ (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (SMulZeroClass.toSMul.{u6, u5} R Mₗ (AddMonoid.toZero.{u5} Mₗ (AddCommMonoid.toAddMonoid.{u5} Mₗ _inst_8)) (DistribSMul.toSMulZeroClass.{u6, u5} R Mₗ (AddMonoid.toAddZeroClass.{u5} Mₗ (AddCommMonoid.toAddMonoid.{u5} Mₗ _inst_8)) (DistribMulAction.toDistribSMul.{u6, u5} R Mₗ (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{u5} Mₗ _inst_8) (Module.toDistribMulAction.{u6, u5} R Mₗ (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15)))) (SMulZeroClass.toSMul.{u6, max u2 u6} R (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddMonoid.toZero.{max u2 u6} (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddCommMonoid.toAddMonoid.{max u2 u6} (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribSMul.toSMulZeroClass.{u6, max u2 u6} R (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddMonoid.toAddZeroClass.{max u2 u6} (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddCommMonoid.toAddMonoid.{max u2 u6} (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribMulAction.toDistribSMul.{u6, max u2 u6} R (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{max u2 u6} (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))))) (Module.toDistribMulAction.{u6, max u2 u6} R (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) (Finsupp.module.{u2, u6, u6} ι₁ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribMulActionHomClass.toSMulHomClass.{max (max u2 u6) u5, u6, u5, max u2 u6} (LinearEquiv.{u6, u6, u5, max u6 u2} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) Mₗ (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_8 (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} ι₁ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) R Mₗ (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{u5} Mₗ _inst_8) (AddCommMonoid.toAddMonoid.{max u2 u6} (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))))) (Module.toDistribMulAction.{u6, u5} R Mₗ (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15) (Module.toDistribMulAction.{u6, max u2 u6} R (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) (Finsupp.module.{u2, u6, u6} ι₁ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (SemilinearMapClass.distribMulActionHomClass.{u6, u5, max u2 u6, max (max u2 u6) u5} R Mₗ (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (LinearEquiv.{u6, u6, u5, max u6 u2} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) Mₗ (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_8 (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} ι₁ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} ι₁ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (SemilinearEquivClass.instSemilinearMapClass.{u6, u6, u5, max u2 u6, max (max u2 u6) u5} R R Mₗ (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (LinearEquiv.{u6, u6, u5, max u6 u2} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) Mₗ (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_8 (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} ι₁ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} ι₁ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (LinearEquiv.instSemilinearEquivClassLinearEquiv.{u6, u6, u5, max u2 u6} R R Mₗ (Finsupp.{u2, u6} ι₁ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 (Finsupp.addCommMonoid.{u2, u6} ι₁ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} ι₁ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (Basis.repr.{u2, u6, u5} ι₁ R Mₗ (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15 b₁') x) (fun (i : ι₁) (xi : R) => Finsupp.sum.{u1, u6, u4} ι₂ R Pₗ (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) _inst_10 (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u3, max (succ u1) (succ u6)} (LinearEquiv.{u6, u6, u3, max u6 u1} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) Nₗ (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_9 (Finsupp.addCommMonoid.{u1, u6} ι₂ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} ι₂ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) Nₗ (fun (_x : Nₗ) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Nₗ) => Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _x) (SMulHomClass.toFunLike.{max (max u1 u6) u3, u6, u3, max u1 u6} (LinearEquiv.{u6, u6, u3, max u6 u1} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) Nₗ (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_9 (Finsupp.addCommMonoid.{u1, u6} ι₂ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} ι₂ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) R Nₗ (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (SMulZeroClass.toSMul.{u6, u3} R Nₗ (AddMonoid.toZero.{u3} Nₗ (AddCommMonoid.toAddMonoid.{u3} Nₗ _inst_9)) (DistribSMul.toSMulZeroClass.{u6, u3} R Nₗ (AddMonoid.toAddZeroClass.{u3} Nₗ (AddCommMonoid.toAddMonoid.{u3} Nₗ _inst_9)) (DistribMulAction.toDistribSMul.{u6, u3} R Nₗ (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{u3} Nₗ _inst_9) (Module.toDistribMulAction.{u6, u3} R Nₗ (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16)))) (SMulZeroClass.toSMul.{u6, max u1 u6} R (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddMonoid.toZero.{max u1 u6} (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddCommMonoid.toAddMonoid.{max u1 u6} (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u1, u6} ι₂ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribSMul.toSMulZeroClass.{u6, max u1 u6} R (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddMonoid.toAddZeroClass.{max u1 u6} (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddCommMonoid.toAddMonoid.{max u1 u6} (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u1, u6} ι₂ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribMulAction.toDistribSMul.{u6, max u1 u6} R (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{max u1 u6} (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u1, u6} ι₂ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))))) (Module.toDistribMulAction.{u6, max u1 u6} R (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (Finsupp.addCommMonoid.{u1, u6} ι₂ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) (Finsupp.module.{u1, u6, u6} ι₂ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribMulActionHomClass.toSMulHomClass.{max (max u1 u6) u3, u6, u3, max u1 u6} (LinearEquiv.{u6, u6, u3, max u6 u1} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) Nₗ (Finsupp.{u1, u6} ι₂ R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_9 (Finsupp.addCommMonoid.{u1, u6} ι₂ R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} ι₂ R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R 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B x) y)
Case conversion may be inaccurate. Consider using '#align linear_map.sum_repr_mul_repr_mul LinearMap.sum_repr_mul_repr_mulₓ'. -/
/-- Write out `B x y` as a sum over `B (b i) (b j)` if `b` is a basis.

Version for bilinear maps, see `sum_repr_mul_repr_mulₛₗ` for the semi-bilinear version. -/
theorem sum_repr_mul_repr_mul {B : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ} (x y) :
    ((b₁'.repr x).Sum fun i xi => (b₂'.repr y).Sum fun j yj => xi • yj • B (b₁' i) (b₂' j)) =
      B x y :=
  by
  conv_rhs => rw [← b₁'.total_repr x, ← b₂'.total_repr y]
  simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smul₂,
    LinearMap.map_smul]
#align linear_map.sum_repr_mul_repr_mul LinearMap.sum_repr_mul_repr_mul

end AddCommMonoid

end LinearMap