ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearMap.comm_comp_rTensor_comp_comm_eq
lemma comm_comp_rTensor_comp_comm_eq (g : N →ₗ[R] P) :
    TensorProduct.comm R P Q ∘ₗ rTensor Q g ∘ₗ TensorProduct.comm R Q N =
      lTensor Q g 
Arguments: {R : Type u_1} [CommSemiring R] {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] (g : N →ₗ[R] P)
Goal: ↑(TensorProduct.comm R P Q) ∘ₗ LinearMap.rTensor Q g ∘ₗ ↑(TensorProduct.comm R Q N) = LinearMap.lTensor Q g
-- LinearMap.comm_comp_lTensor_comp_comm_eq
lemma comm_comp_lTensor_comp_comm_eq (g : N →ₗ[R] P) :
    TensorProduct.comm R Q P ∘ₗ lTensor Q g ∘ₗ TensorProduct.comm R N Q =
      rTensor Q g 
Arguments: {R : Type u_1} [CommSemiring R] {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] (g : N →ₗ[R] P)
Goal: ↑(TensorProduct.comm R Q P) ∘ₗ LinearMap.lTensor Q g ∘ₗ ↑(TensorProduct.comm R N Q) = LinearMap.rTensor Q g
-- LinearMap.lTensor_inj_iff_rTensor_inj
theorem lTensor_inj_iff_rTensor_inj :
    Function.Injective (lTensor M f) ↔ Function.Injective (rTensor M f) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P)
Goal: Function.Injective ⇑(LinearMap.lTensor M f) ↔ Function.Injective ⇑(LinearMap.rTensor M f)
-- LinearMap.lTensor_surj_iff_rTensor_surj
theorem lTensor_surj_iff_rTensor_surj :
    Function.Surjective (lTensor M f) ↔ Function.Surjective (rTensor M f) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P)
Goal: Function.Surjective ⇑(LinearMap.lTensor M f) ↔ Function.Surjective ⇑(LinearMap.rTensor M f)
-- LinearMap.lTensor_bij_iff_rTensor_bij
theorem lTensor_bij_iff_rTensor_bij :
    Function.Bijective (lTensor M f) ↔ Function.Bijective (rTensor M f) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P)
Goal: Function.Bijective ⇑(LinearMap.lTensor M f) ↔ Function.Bijective ⇑(LinearMap.rTensor M f)
-- LinearMap.coe_lTensorHom
@[simp]
theorem coe_lTensorHom : (lTensorHom M : (N →ₗ[R] P) → M ⊗[R] N →ₗ[R] M ⊗[R] P) = lTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P]
Goal: ⇑(LinearMap.lTensorHom M) = LinearMap.lTensor M
-- LinearMap.coe_rTensorHom
@[simp]
theorem coe_rTensorHom : (rTensorHom M : (N →ₗ[R] P) → N ⊗[R] M →ₗ[R] P ⊗[R] M) = rTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P]
Goal: ⇑(LinearMap.rTensorHom M) = LinearMap.rTensor M
-- LinearMap.lTensor_add
@[simp]
theorem lTensor_add (f g : N →ₗ[R] P) : (f + g).lTensor M = f.lTensor M + g.lTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f g : N →ₗ[R] P)
Goal: LinearMap.lTensor M (f + g) = LinearMap.lTensor M f + LinearMap.lTensor M g
-- LinearMap.rTensor_add
@[simp]
theorem rTensor_add (f g : N →ₗ[R] P) : (f + g).rTensor M = f.rTensor M + g.rTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f g : N →ₗ[R] P)
Goal: LinearMap.rTensor M (f + g) = LinearMap.rTensor M f + LinearMap.rTensor M g
-- LinearMap.lTensor_zero
@[simp]
theorem lTensor_zero : lTensor M (0 : N →ₗ[R] P) = 0 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P]
Goal: LinearMap.lTensor M 0 = 0
-- LinearMap.rTensor_zero
@[simp]
theorem rTensor_zero : rTensor M (0 : N →ₗ[R] P) = 0 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P]
Goal: LinearMap.rTensor M 0 = 0
-- LinearMap.lTensor_smul
@[simp]
theorem lTensor_smul (r : R) (f : N →ₗ[R] P) : (r • f).lTensor M = r • f.lTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (r : R) (f : N →ₗ[R] P)
Goal: LinearMap.lTensor M (r • f) = r • LinearMap.lTensor M f
-- LinearMap.rTensor_smul
@[simp]
theorem rTensor_smul (r : R) (f : N →ₗ[R] P) : (r • f).rTensor M = r • f.rTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (r : R) (f : N →ₗ[R] P)
Goal: LinearMap.rTensor M (r • f) = r • LinearMap.rTensor M f
-- LinearMap.lTensor_comp
theorem lTensor_comp : (g.comp f).lTensor M = (g.lTensor M).comp (f.lTensor M) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (g : P →ₗ[R] Q) (f : N →ₗ[R] P)
Goal: LinearMap.lTensor M (g ∘ₗ f) = LinearMap.lTensor M g ∘ₗ LinearMap.lTensor M f
-- LinearMap.lTensor_comp_apply
theorem lTensor_comp_apply (x : M ⊗[R] N) :
    (g.comp f).lTensor M x = (g.lTensor M) ((f.lTensor M) x) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (g : P →ₗ[R] Q) (f : N →ₗ[R] P) (x : TensorProduct R M N)
Goal: (LinearMap.lTensor M (g ∘ₗ f)) x = (LinearMap.lTensor M g) ((LinearMap.lTensor M f) x)
-- LinearMap.rTensor_comp
theorem rTensor_comp : (g.comp f).rTensor M = (g.rTensor M).comp (f.rTensor M) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (g : P →ₗ[R] Q) (f : N →ₗ[R] P)
Goal: LinearMap.rTensor M (g ∘ₗ f) = LinearMap.rTensor M g ∘ₗ LinearMap.rTensor M f
-- LinearMap.rTensor_comp_apply
theorem rTensor_comp_apply (x : N ⊗[R] M) :
    (g.comp f).rTensor M x = (g.rTensor M) ((f.rTensor M) x) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (g : P →ₗ[R] Q) (f : N →ₗ[R] P) (x : TensorProduct R N M)
Goal: (LinearMap.rTensor M (g ∘ₗ f)) x = (LinearMap.rTensor M g) ((LinearMap.rTensor M f) x)
-- LinearMap.lTensor_mul
theorem lTensor_mul (f g : Module.End R N) : (f * g).lTensor M = f.lTensor M * g.lTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f g : Module.End R N)
Goal: LinearMap.lTensor M (f * g) = LinearMap.lTensor M f * LinearMap.lTensor M g
-- LinearMap.rTensor_mul
theorem rTensor_mul (f g : Module.End R N) : (f * g).rTensor M = f.rTensor M * g.rTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f g : Module.End R N)
Goal: LinearMap.rTensor M (f * g) = LinearMap.rTensor M f * LinearMap.rTensor M g
-- LinearMap.lTensor_id
@[simp]
theorem lTensor_id : (id : N →ₗ[R] N).lTensor M = id 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
Goal: LinearMap.lTensor M LinearMap.id = LinearMap.id
-- LinearMap.lTensor_id_apply
theorem lTensor_id_apply (x : M ⊗[R] N) : (LinearMap.id : N →ₗ[R] N).lTensor M x = x 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x : TensorProduct R M N)
Goal: (LinearMap.lTensor M LinearMap.id) x = x
-- LinearMap.rTensor_id
@[simp]
theorem rTensor_id : (id : N →ₗ[R] N).rTensor M = id 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
Goal: LinearMap.rTensor M LinearMap.id = LinearMap.id
-- LinearMap.rTensor_id_apply
theorem rTensor_id_apply (x : N ⊗[R] M) : (LinearMap.id : N →ₗ[R] N).rTensor M x = x 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x : TensorProduct R N M)
Goal: (LinearMap.rTensor M LinearMap.id) x = x
-- LinearMap.lTensor_smul_action
@[simp]
theorem lTensor_smul_action (r : R) :
    (DistribMulAction.toLinearMap R N r).lTensor M =
      DistribMulAction.toLinearMap R (M ⊗[R] N) r 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (r : R)
Goal: LinearMap.lTensor M (DistribMulAction.toLinearMap R N r) = DistribMulAction.toLinearMap R (TensorProduct R M N) r
-- LinearMap.rTensor_smul_action
@[simp]
theorem rTensor_smul_action (r : R) :
    (DistribMulAction.toLinearMap R N r).rTensor M =
      DistribMulAction.toLinearMap R (N ⊗[R] M) r 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (r : R)
Goal: LinearMap.rTensor M (DistribMulAction.toLinearMap R N r) = DistribMulAction.toLinearMap R (TensorProduct R N M) r
-- LinearMap.lid_comp_rTensor
theorem lid_comp_rTensor (f : N →ₗ[R] R) :
    (TensorProduct.lid R M).comp (rTensor M f) = lift ((lsmul R M).comp f) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : N →ₗ[R] R)
Goal: ↑(TensorProduct.lid R M) ∘ₗ LinearMap.rTensor M f = TensorProduct.lift (LinearMap.lsmul R M ∘ₗ f)
-- LinearMap.lTensor_comp_rTensor
@[simp]
theorem lTensor_comp_rTensor (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
    (g.lTensor P).comp (f.rTensor N) = map f g 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (f : M →ₗ[R] P) (g : N →ₗ[R] Q)
Goal: LinearMap.lTensor P g ∘ₗ LinearMap.rTensor N f = TensorProduct.map f g
-- LinearMap.rTensor_comp_lTensor
@[simp]
theorem rTensor_comp_lTensor (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
    (f.rTensor Q).comp (g.lTensor M) = map f g 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (f : M →ₗ[R] P) (g : N →ₗ[R] Q)
Goal: LinearMap.rTensor Q f ∘ₗ LinearMap.lTensor M g = TensorProduct.map f g
-- LinearMap.map_comp_rTensor
@[simp]
theorem map_comp_rTensor (f : M →ₗ[R] P) (g : N →ₗ[R] Q) (f' : S →ₗ[R] M) :
    (map f g).comp (f'.rTensor _) = map (f.comp f') g 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} {S : Type u_8} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] (f : M →ₗ[R] P) (g : N →ₗ[R] Q) (f' : S →ₗ[R] M)
Goal: TensorProduct.map f g ∘ₗ LinearMap.rTensor N f' = TensorProduct.map (f ∘ₗ f') g
-- LinearMap.map_comp_lTensor
@[simp]
theorem map_comp_lTensor (f : M →ₗ[R] P) (g : N →ₗ[R] Q) (g' : S →ₗ[R] N) :
    (map f g).comp (g'.lTensor _) = map f (g.comp g') 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} {S : Type u_8} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] (f : M →ₗ[R] P) (g : N →ₗ[R] Q) (g' : S →ₗ[R] N)
Goal: TensorProduct.map f g ∘ₗ LinearMap.lTensor M g' = TensorProduct.map f (g ∘ₗ g')
-- LinearMap.rTensor_comp_map
@[simp]
theorem rTensor_comp_map (f' : P →ₗ[R] S) (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
    (f'.rTensor _).comp (map f g) = map (f'.comp f) g 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} {S : Type u_8} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] (f' : P →ₗ[R] S) (f : M →ₗ[R] P) (g : N →ₗ[R] Q)
Goal: LinearMap.rTensor Q f' ∘ₗ TensorProduct.map f g = TensorProduct.map (f' ∘ₗ f) g
-- LinearMap.lTensor_comp_map
@[simp]
theorem lTensor_comp_map (g' : Q →ₗ[R] S) (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
    (g'.lTensor _).comp (map f g) = map f (g'.comp g) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} {S : Type u_8} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] (g' : Q →ₗ[R] S) (f : M →ₗ[R] P) (g : N →ₗ[R] Q)
Goal: LinearMap.lTensor P g' ∘ₗ TensorProduct.map f g = TensorProduct.map f (g' ∘ₗ g)
-- LinearMap.rTensor_pow
@[simp]
theorem rTensor_pow (f : M →ₗ[R] M) (n : ℕ) : f.rTensor N ^ n = (f ^ n).rTensor N 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : M →ₗ[R] M) (n : ℕ)
Goal: LinearMap.rTensor N f ^ n = LinearMap.rTensor N (f ^ n)
-- LinearMap.lTensor_pow
@[simp]
theorem lTensor_pow (f : N →ₗ[R] N) (n : ℕ) : f.lTensor M ^ n = (f ^ n).lTensor M 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : N →ₗ[R] N) (n : ℕ)
Goal: LinearMap.lTensor M f ^ n = LinearMap.lTensor M (f ^ n)
-- LinearEquiv.coe_lTensor
@[simp] theorem coe_lTensor : lTensor M f = (f : N →ₗ[R] P).lTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P)
Goal: ↑(LinearEquiv.lTensor M f) = LinearMap.lTensor M ↑f
-- LinearEquiv.coe_lTensor_symm
@[simp] theorem coe_lTensor_symm : (lTensor M f).symm = (f.symm : P →ₗ[R] N).lTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P)
Goal: ↑(LinearEquiv.lTensor M f).symm = LinearMap.lTensor M ↑f.symm
-- LinearEquiv.coe_rTensor
@[simp] theorem coe_rTensor : rTensor M f = (f : N →ₗ[R] P).rTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P)
Goal: ↑(LinearEquiv.rTensor M f) = LinearMap.rTensor M ↑f
-- LinearEquiv.coe_rTensor_symm
@[simp] theorem coe_rTensor_symm : (rTensor M f).symm = (f.symm : P →ₗ[R] N).rTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P)
Goal: ↑(LinearEquiv.rTensor M f).symm = LinearMap.rTensor M ↑f.symm
-- LinearEquiv.lTensor_tmul
@[simp] theorem lTensor_tmul : f.lTensor M (m ⊗ₜ n) = m ⊗ₜ f n 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P) (m : M) (n : N)
Goal: (LinearEquiv.lTensor M f) (m ⊗ₜ[R] n) = m ⊗ₜ[R] f n
-- LinearEquiv.lTensor_symm_tmul
@[simp] theorem lTensor_symm_tmul : (f.lTensor M).symm (m ⊗ₜ p) = m ⊗ₜ f.symm p 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P) (m : M) (p : P)
Goal: (LinearEquiv.lTensor M f).symm (m ⊗ₜ[R] p) = m ⊗ₜ[R] f.symm p
-- LinearEquiv.rTensor_tmul
@[simp] theorem rTensor_tmul : f.rTensor M (n ⊗ₜ m) = f n ⊗ₜ m 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P) (m : M) (n : N)
Goal: (LinearEquiv.rTensor M f) (n ⊗ₜ[R] m) = f n ⊗ₜ[R] m
-- LinearEquiv.rTensor_symm_tmul
@[simp] theorem rTensor_symm_tmul : (f.rTensor M).symm (p ⊗ₜ m) = f.symm p ⊗ₜ m 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P) (m : M) (p : P)
Goal: (LinearEquiv.rTensor M f).symm (p ⊗ₜ[R] m) = f.symm p ⊗ₜ[R] m
-- LinearEquiv.comm_trans_rTensor_trans_comm_eq
lemma comm_trans_rTensor_trans_comm_eq (g : N ≃ₗ[R] P) :
    TensorProduct.comm R Q N ≪≫ₗ rTensor Q g ≪≫ₗ TensorProduct.comm R P Q = lTensor Q g 
Arguments: {R : Type u_1} [CommSemiring R] {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] (g : N ≃ₗ[R] P)
Goal: TensorProduct.comm R Q N ≪≫ₗ LinearEquiv.rTensor Q g ≪≫ₗ TensorProduct.comm R P Q = LinearEquiv.lTensor Q g
-- LinearEquiv.comm_trans_lTensor_trans_comm_eq
lemma comm_trans_lTensor_trans_comm_eq (g : N ≃ₗ[R] P) :
    TensorProduct.comm R N Q ≪≫ₗ lTensor Q g ≪≫ₗ TensorProduct.comm R Q P = rTensor Q g 
Arguments: {R : Type u_1} [CommSemiring R] {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] (g : N ≃ₗ[R] P)
Goal: TensorProduct.comm R N Q ≪≫ₗ LinearEquiv.lTensor Q g ≪≫ₗ TensorProduct.comm R Q P = LinearEquiv.rTensor Q g
-- LinearEquiv.lTensor_trans
theorem lTensor_trans : (f ≪≫ₗ g).lTensor M = f.lTensor M ≪≫ₗ g.lTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (g : P ≃ₗ[R] Q) (f : N ≃ₗ[R] P)
Goal: LinearEquiv.lTensor M (f ≪≫ₗ g) = LinearEquiv.lTensor M f ≪≫ₗ LinearEquiv.lTensor M g
-- LinearEquiv.lTensor_trans_apply
theorem lTensor_trans_apply : (f ≪≫ₗ g).lTensor M x = g.lTensor M (f.lTensor M x) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (g : P ≃ₗ[R] Q) (f : N ≃ₗ[R] P) (x : TensorProduct R M N)
Goal: (LinearEquiv.lTensor M (f ≪≫ₗ g)) x = (LinearEquiv.lTensor M g) ((LinearEquiv.lTensor M f) x)
-- LinearEquiv.rTensor_trans
theorem rTensor_trans : (f ≪≫ₗ g).rTensor M = f.rTensor M ≪≫ₗ g.rTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (g : P ≃ₗ[R] Q) (f : N ≃ₗ[R] P)
Goal: LinearEquiv.rTensor M (f ≪≫ₗ g) = LinearEquiv.rTensor M f ≪≫ₗ LinearEquiv.rTensor M g
-- LinearEquiv.rTensor_trans_apply
theorem rTensor_trans_apply : (f ≪≫ₗ g).rTensor M y = g.rTensor M (f.rTensor M y) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (g : P ≃ₗ[R] Q) (f : N ≃ₗ[R] P) (y : TensorProduct R N M)
Goal: (LinearEquiv.rTensor M (f ≪≫ₗ g)) y = (LinearEquiv.rTensor M g) ((LinearEquiv.rTensor M f) y)
-- LinearEquiv.lTensor_mul
theorem lTensor_mul (f g : N ≃ₗ[R] N) : (f * g).lTensor M = f.lTensor M * g.lTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f g : N ≃ₗ[R] N)
Goal: LinearEquiv.lTensor M (f * g) = LinearEquiv.lTensor M f * LinearEquiv.lTensor M g
-- LinearEquiv.rTensor_mul
theorem rTensor_mul (f g : N ≃ₗ[R] N) : (f * g).rTensor M = f.rTensor M * g.rTensor M 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f g : N ≃ₗ[R] N)
Goal: LinearEquiv.rTensor M (f * g) = LinearEquiv.rTensor M f * LinearEquiv.rTensor M g
-- LinearEquiv.lTensor_refl
@[simp] theorem lTensor_refl : (refl R N).lTensor M = refl R _ 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
Goal: LinearEquiv.lTensor M (LinearEquiv.refl R N) = LinearEquiv.refl R (TensorProduct R M N)
-- LinearEquiv.lTensor_refl_apply
theorem lTensor_refl_apply : (refl R N).lTensor M x = x 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x : TensorProduct R M N)
Goal: (LinearEquiv.lTensor M (LinearEquiv.refl R N)) x = x
-- LinearEquiv.rTensor_refl
@[simp] theorem rTensor_refl : (refl R N).rTensor M = refl R _ 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
Goal: LinearEquiv.rTensor M (LinearEquiv.refl R N) = LinearEquiv.refl R (TensorProduct R N M)
-- LinearEquiv.rTensor_refl_apply
theorem rTensor_refl_apply : (refl R N).rTensor M y = y 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (y : TensorProduct R N M)
Goal: (LinearEquiv.rTensor M (LinearEquiv.refl R N)) y = y
-- LinearEquiv.rTensor_trans_lTensor
@[simp] theorem rTensor_trans_lTensor (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q) :
    f.rTensor N ≪≫ₗ g.lTensor P = TensorProduct.congr f g 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q)
Goal: LinearEquiv.rTensor N f ≪≫ₗ LinearEquiv.lTensor P g = TensorProduct.congr f g
-- LinearEquiv.lTensor_trans_rTensor
@[simp] theorem lTensor_trans_rTensor (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q) :
    g.lTensor M ≪≫ₗ f.rTensor Q = TensorProduct.congr f g 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q)
Goal: LinearEquiv.lTensor M g ≪≫ₗ LinearEquiv.rTensor Q f = TensorProduct.congr f g
-- LinearEquiv.rTensor_trans_congr
@[simp] theorem rTensor_trans_congr (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q) (f' : S ≃ₗ[R] M) :
    f'.rTensor _ ≪≫ₗ TensorProduct.congr f g = TensorProduct.congr (f' ≪≫ₗ f) g 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} {S : Type u_8} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q) (f' : S ≃ₗ[R] M)
Goal: LinearEquiv.rTensor N f' ≪≫ₗ TensorProduct.congr f g = TensorProduct.congr (f' ≪≫ₗ f) g