ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TensorProduct.lid_eq_rid
theorem lid_eq_rid : TensorProduct.lid R R = TensorProduct.rid R R 
Arguments: (R : Type u_1) [CommSemiring R]
Goal: TensorProduct.lid R R = TensorProduct.rid R R
-- TensorProduct.assoc_tmul
@[simp]
theorem assoc_tmul (m : M) (n : N) (p : P) :
    (TensorProduct.assoc R M N P) (m ⊗ₜ n ⊗ₜ p) = m ⊗ₜ (n ⊗ₜ 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] (m : M) (n : N) (p : P)
Goal: (TensorProduct.assoc R M N P) ((m ⊗ₜ[R] n) ⊗ₜ[R] p) = m ⊗ₜ[R] n ⊗ₜ[R] p
-- TensorProduct.assoc_symm_tmul
@[simp]
theorem assoc_symm_tmul (m : M) (n : N) (p : P) :
    (TensorProduct.assoc R M N P).symm (m ⊗ₜ (n ⊗ₜ p)) = m ⊗ₜ n ⊗ₜ 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] (m : M) (n : N) (p : P)
Goal: (TensorProduct.assoc R M N P).symm (m ⊗ₜ[R] n ⊗ₜ[R] p) = (m ⊗ₜ[R] n) ⊗ₜ[R] p
-- TensorProduct.map_tmul
@[simp]
theorem map_tmul (f : M →ₗ[R] P) (g : N →ₗ[R] Q) (m : M) (n : N) : map f g (m ⊗ₜ n) = f m ⊗ₜ g n 
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) (m : M) (n : N)
Goal: (TensorProduct.map f g) (m ⊗ₜ[R] n) = f m ⊗ₜ[R] g n
-- TensorProduct.map_comp_comm_eq
lemma map_comp_comm_eq (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
    map f g ∘ₗ TensorProduct.comm R N M = TensorProduct.comm R Q P ∘ₗ map g f 
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: TensorProduct.map f g ∘ₗ ↑(TensorProduct.comm R N M) = ↑(TensorProduct.comm R Q P) ∘ₗ TensorProduct.map g f
-- TensorProduct.map_comm
lemma map_comm (f : M →ₗ[R] P) (g : N →ₗ[R] Q) (x : N ⊗[R] M) :
    map f g (TensorProduct.comm R N M x) = TensorProduct.comm R Q P (map g f 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] (f : M →ₗ[R] P) (g : N →ₗ[R] Q) (x : TensorProduct R N M)
Goal: (TensorProduct.map f g) ((TensorProduct.comm R N M) x) = (TensorProduct.comm R Q P) ((TensorProduct.map g f) x)
-- TensorProduct.map_map_comp_assoc_eq
lemma map_map_comp_assoc_eq (f : M →ₗ[R] Q) (g : N →ₗ[R] S) (h : P →ₗ[R] T) :
    map f (map g h) ∘ₗ TensorProduct.assoc R M N P =
      TensorProduct.assoc R Q S T ∘ₗ map (map f g) h 
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} {T : Type u_9} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [AddCommMonoid T] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] [Module R T] (f : M →ₗ[R] Q) (g : N →ₗ[R] S) (h : P →ₗ[R] T)
Goal: TensorProduct.map f (TensorProduct.map g h) ∘ₗ ↑(TensorProduct.assoc R M N P) =
  ↑(TensorProduct.assoc R Q S T) ∘ₗ TensorProduct.map (TensorProduct.map f g) h
-- TensorProduct.map_map_assoc
lemma map_map_assoc (f : M →ₗ[R] Q) (g : N →ₗ[R] S) (h : P →ₗ[R] T) (x : (M ⊗[R] N) ⊗[R] P) :
    map f (map g h) (TensorProduct.assoc R M N P x) =
      TensorProduct.assoc R Q S T (map (map f g) h x) 
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} {T : Type u_9} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [AddCommMonoid T] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] [Module R T] (f : M →ₗ[R] Q) (g : N →ₗ[R] S) (h : P →ₗ[R] T) (x : TensorProduct R (TensorProduct R M N) P)
Goal: (TensorProduct.map f (TensorProduct.map g h)) ((TensorProduct.assoc R M N P) x) =
  (TensorProduct.assoc R Q S T) ((TensorProduct.map (TensorProduct.map f g) h) x)
-- TensorProduct.map_map_comp_assoc_symm_eq
lemma map_map_comp_assoc_symm_eq (f : M →ₗ[R] Q) (g : N →ₗ[R] S) (h : P →ₗ[R] T) :
    map (map f g) h ∘ₗ (TensorProduct.assoc R M N P).symm =
      (TensorProduct.assoc R Q S T).symm ∘ₗ map f (map g h) 
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} {T : Type u_9} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [AddCommMonoid T] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] [Module R T] (f : M →ₗ[R] Q) (g : N →ₗ[R] S) (h : P →ₗ[R] T)
Goal: TensorProduct.map (TensorProduct.map f g) h ∘ₗ ↑(TensorProduct.assoc R M N P).symm =
  ↑(TensorProduct.assoc R Q S T).symm ∘ₗ TensorProduct.map f (TensorProduct.map g h)
-- TensorProduct.map_map_assoc_symm
lemma map_map_assoc_symm (f : M →ₗ[R] Q) (g : N →ₗ[R] S) (h : P →ₗ[R] T) (x : M ⊗[R] (N ⊗[R] P)) :
    map (map f g) h ((TensorProduct.assoc R M N P).symm x) =
      (TensorProduct.assoc R Q S T).symm (map f (map g h) x) 
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} {T : Type u_9} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [AddCommMonoid T] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] [Module R T] (f : M →ₗ[R] Q) (g : N →ₗ[R] S) (h : P →ₗ[R] T) (x : TensorProduct R M (TensorProduct R N P))
Goal: (TensorProduct.map (TensorProduct.map f g) h) ((TensorProduct.assoc R M N P).symm x) =
  (TensorProduct.assoc R Q S T).symm ((TensorProduct.map f (TensorProduct.map g h)) x)
-- TensorProduct.map_range_eq_span_tmul
theorem map_range_eq_span_tmul (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
    range (map f g) = Submodule.span R { t | ∃ m n, f m ⊗ₜ g n = t } 
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.range (TensorProduct.map f g) = Submodule.span R {t | ∃ m n, f m ⊗ₜ[R] g n = t}
-- TensorProduct.range_mapIncl
lemma range_mapIncl (p : Submodule R P) (q : Submodule R Q) :
    LinearMap.range (mapIncl p q) = Submodule.span R (Set.image2 (· ⊗ₜ ·) p q) 
Arguments: {R : Type u_1} [CommSemiring R] {P : Type u_6} {Q : Type u_7} [AddCommMonoid P] [AddCommMonoid Q] [Module R P] [Module R Q] (p : Submodule R P) (q : Submodule R Q)
Goal: LinearMap.range (TensorProduct.mapIncl p q) = Submodule.span R (Set.image2 (fun x x_1 => x ⊗ₜ[R] x_1) ↑p ↑q)
-- TensorProduct.map₂_eq_range_lift_comp_mapIncl
theorem map₂_eq_range_lift_comp_mapIncl (f : P →ₗ[R] Q →ₗ[R] M)
    (p : Submodule R P) (q : Submodule R Q) :
    Submodule.map₂ f p q = LinearMap.range (lift f ∘ₗ mapIncl p q) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_4} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R P] [Module R Q] (f : P →ₗ[R] Q →ₗ[R] M) (p : Submodule R P) (q : Submodule R Q)
Goal: Submodule.map₂ f p q = LinearMap.range (TensorProduct.lift f ∘ₗ TensorProduct.mapIncl p q)
-- TensorProduct.map_comp
theorem map_comp (f₂ : P →ₗ[R] P') (f₁ : M →ₗ[R] P) (g₂ : Q →ₗ[R] Q') (g₁ : N →ₗ[R] Q) :
    map (f₂.comp f₁) (g₂.comp g₁) = (map f₂ g₂).comp (map f₁ g₁) 
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] {P' : Type u_10} {Q' : Type u_11} [AddCommMonoid P'] [Module R P'] [AddCommMonoid Q'] [Module R Q'] (f₂ : P →ₗ[R] P') (f₁ : M →ₗ[R] P) (g₂ : Q →ₗ[R] Q') (g₁ : N →ₗ[R] Q)
Goal: TensorProduct.map (f₂ ∘ₗ f₁) (g₂ ∘ₗ g₁) = TensorProduct.map f₂ g₂ ∘ₗ TensorProduct.map f₁ g₁
-- TensorProduct.range_mapIncl_mono
lemma range_mapIncl_mono {p p' : Submodule R P} {q q' : Submodule R Q} (hp : p ≤ p') (hq : q ≤ q') :
    LinearMap.range (mapIncl p q) ≤ LinearMap.range (mapIncl p' q') 
Arguments: {R : Type u_1} [CommSemiring R] {P : Type u_6} {Q : Type u_7} [AddCommMonoid P] [AddCommMonoid Q] [Module R P] [Module R Q] {p p' : Submodule R P} {q q' : Submodule R Q} (hp : p ≤ p') (hq : q ≤ q')
Goal: LinearMap.range (TensorProduct.mapIncl p q) ≤ LinearMap.range (TensorProduct.mapIncl p' q')
-- TensorProduct.lift_comp_map
theorem lift_comp_map (i : P →ₗ[R] Q →ₗ[R] Q') (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
    (lift i).comp (map f g) = lift ((i.comp f).compl₂ 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] {Q' : Type u_11} [AddCommMonoid Q'] [Module R Q'] (i : P →ₗ[R] Q →ₗ[R] Q') (f : M →ₗ[R] P) (g : N →ₗ[R] Q)
Goal: TensorProduct.lift i ∘ₗ TensorProduct.map f g = TensorProduct.lift ((i ∘ₗ f).compl₂ g)
-- TensorProduct.map_id
@[simp]
theorem map_id : map (id : M →ₗ[R] M) (id : N →ₗ[R] N) = .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: TensorProduct.map LinearMap.id LinearMap.id = LinearMap.id
-- TensorProduct.map_one
@[simp]
theorem map_one : map (1 : M →ₗ[R] M) (1 : N →ₗ[R] N) = 1 
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: TensorProduct.map 1 1 = 1
-- TensorProduct.map_mul
theorem map_mul (f₁ f₂ : M →ₗ[R] M) (g₁ g₂ : N →ₗ[R] N) :
    map (f₁ * f₂) (g₁ * g₂) = map f₁ g₁ * map f₂ g₂ 
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] (f₁ f₂ : M →ₗ[R] M) (g₁ g₂ : N →ₗ[R] N)
Goal: TensorProduct.map (f₁ * f₂) (g₁ * g₂) = TensorProduct.map f₁ g₁ * TensorProduct.map f₂ g₂
-- TensorProduct.map_pow
@[simp]
protected theorem map_pow (f : M →ₗ[R] M) (g : N →ₗ[R] N) (n : ℕ) :
    map f g ^ n = map (f ^ n) (g ^ 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) (g : N →ₗ[R] N) (n : ℕ)
Goal: TensorProduct.map f g ^ n = TensorProduct.map (f ^ n) (g ^ n)
-- TensorProduct.map_add_left
theorem map_add_left (f₁ f₂ : M →ₗ[R] P) (g : N →ₗ[R] Q) :
    map (f₁ + f₂) g = map f₁ g + map f₂ 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] (f₁ f₂ : M →ₗ[R] P) (g : N →ₗ[R] Q)
Goal: TensorProduct.map (f₁ + f₂) g = TensorProduct.map f₁ g + TensorProduct.map f₂ g
-- TensorProduct.map_add_right
theorem map_add_right (f : M →ₗ[R] P) (g₁ g₂ : N →ₗ[R] Q) :
    map f (g₁ + g₂) = map f g₁ + map f g₂ 
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) (g₁ g₂ : N →ₗ[R] Q)
Goal: TensorProduct.map f (g₁ + g₂) = TensorProduct.map f g₁ + TensorProduct.map f g₂
-- TensorProduct.map_smul_left
theorem map_smul_left (r : R) (f : M →ₗ[R] P) (g : N →ₗ[R] Q) : map (r • f) g = r • 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] (r : R) (f : M →ₗ[R] P) (g : N →ₗ[R] Q)
Goal: TensorProduct.map (r • f) g = r • TensorProduct.map f g
-- TensorProduct.map_smul_right
theorem map_smul_right (r : R) (f : M →ₗ[R] P) (g : N →ₗ[R] Q) : map f (r • g) = r • 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] (r : R) (f : M →ₗ[R] P) (g : N →ₗ[R] Q)
Goal: TensorProduct.map f (r • g) = r • TensorProduct.map f g
-- TensorProduct.mapBilinear_apply
@[simp]
theorem mapBilinear_apply (f : M →ₗ[R] P) (g : N →ₗ[R] Q) : mapBilinear R M N P Q f g = 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: ((TensorProduct.mapBilinear R M N P Q) f) g = TensorProduct.map f g
-- TensorProduct.lTensorHomToHomLTensor_apply
@[simp]
theorem lTensorHomToHomLTensor_apply (p : P) (f : M →ₗ[R] Q) (m : M) :
    lTensorHomToHomLTensor R M P Q (p ⊗ₜ f) m = p ⊗ₜ f m 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_4} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R P] [Module R Q] (p : P) (f : M →ₗ[R] Q) (m : M)
Goal: ((TensorProduct.lTensorHomToHomLTensor R M P Q) (p ⊗ₜ[R] f)) m = p ⊗ₜ[R] f m
-- TensorProduct.rTensorHomToHomRTensor_apply
@[simp]
theorem rTensorHomToHomRTensor_apply (f : M →ₗ[R] P) (q : Q) (m : M) :
    rTensorHomToHomRTensor R M P Q (f ⊗ₜ q) m = f m ⊗ₜ q 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_4} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R P] [Module R Q] (f : M →ₗ[R] P) (q : Q) (m : M)
Goal: ((TensorProduct.rTensorHomToHomRTensor R M P Q) (f ⊗ₜ[R] q)) m = f m ⊗ₜ[R] q
-- TensorProduct.homTensorHomMap_apply
@[simp]
theorem homTensorHomMap_apply (f : M →ₗ[R] P) (g : N →ₗ[R] Q) :
    homTensorHomMap R M N P Q (f ⊗ₜ g) = 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: (TensorProduct.homTensorHomMap R M N P Q) (f ⊗ₜ[R] g) = TensorProduct.map f g
-- TensorProduct.map₂_apply_tmul
@[simp]
theorem map₂_apply_tmul (f : M →ₗ[R] P →ₗ[R] Q) (g : N →ₗ[R] S →ₗ[R] T) (m : M) (n : N) :
    map₂ f g (m ⊗ₜ n) = map (f m) (g n) 
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} {T : Type u_9} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [AddCommMonoid T] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] [Module R T] (f : M →ₗ[R] P →ₗ[R] Q) (g : N →ₗ[R] S →ₗ[R] T) (m : M) (n : N)
Goal: (TensorProduct.map₂ f g) (m ⊗ₜ[R] n) = TensorProduct.map (f m) (g n)
-- TensorProduct.map_zero_left
@[simp]
theorem map_zero_left (g : N →ₗ[R] Q) : map (0 : M →ₗ[R] P) g = 0 
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 : N →ₗ[R] Q)
Goal: TensorProduct.map 0 g = 0
-- TensorProduct.map_zero_right
@[simp]
theorem map_zero_right (f : M →ₗ[R] P) : map f (0 : N →ₗ[R] Q) = 0 
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)
Goal: TensorProduct.map f 0 = 0
-- TensorProduct.congr_tmul
@[simp]
theorem congr_tmul (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q) (m : M) (n : N) :
    congr f g (m ⊗ₜ n) = f m ⊗ₜ g n 
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) (m : M) (n : N)
Goal: (TensorProduct.congr f g) (m ⊗ₜ[R] n) = f m ⊗ₜ[R] g n
-- TensorProduct.congr_symm_tmul
@[simp]
theorem congr_symm_tmul (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q) (p : P) (q : Q) :
    (congr f g).symm (p ⊗ₜ q) = f.symm p ⊗ₜ g.symm q 
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) (p : P) (q : Q)
Goal: (TensorProduct.congr f g).symm (p ⊗ₜ[R] q) = f.symm p ⊗ₜ[R] g.symm q
-- TensorProduct.congr_symm
theorem congr_symm (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q) : (congr f g).symm = congr f.symm g.symm 
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: (TensorProduct.congr f g).symm = TensorProduct.congr f.symm g.symm
-- TensorProduct.congr_refl_refl
@[simp] theorem congr_refl_refl : congr (.refl R M) (.refl R N) = .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: TensorProduct.congr (LinearEquiv.refl R M) (LinearEquiv.refl R N) = LinearEquiv.refl R (TensorProduct R M N)
-- TensorProduct.congr_trans
theorem congr_trans (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q) (f' : P ≃ₗ[R] S) (g' : Q ≃ₗ[R] T) :
    congr (f ≪≫ₗ f') (g ≪≫ₗ g') = congr f g ≪≫ₗ 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} {S : Type u_8} {T : Type u_9} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [AddCommMonoid T] [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] [Module R T] (f : M ≃ₗ[R] P) (g : N ≃ₗ[R] Q) (f' : P ≃ₗ[R] S) (g' : Q ≃ₗ[R] T)
Goal: TensorProduct.congr (f ≪≫ₗ f') (g ≪≫ₗ g') = TensorProduct.congr f g ≪≫ₗ TensorProduct.congr f' g'
-- TensorProduct.congr_mul
theorem congr_mul (f : M ≃ₗ[R] M) (g : N ≃ₗ[R] N) (f' : M ≃ₗ[R] M) (g' : N ≃ₗ[R] N) :
    congr (f * f') (g * g') = congr f g * congr f' g' 
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) (g : N ≃ₗ[R] N) (f' : M ≃ₗ[R] M) (g' : N ≃ₗ[R] N)
Goal: TensorProduct.congr (f * f') (g * g') = TensorProduct.congr f g * TensorProduct.congr f' g'
-- TensorProduct.congr_pow
@[simp] theorem congr_pow (f : M ≃ₗ[R] M) (g : N ≃ₗ[R] N) (n : ℕ) :
    congr f g ^ n = congr (f ^ n) (g ^ 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) (g : N ≃ₗ[R] N) (n : ℕ)
Goal: TensorProduct.congr f g ^ n = TensorProduct.congr (f ^ n) (g ^ n)
-- TensorProduct.congr_zpow
@[simp] theorem congr_zpow (f : M ≃ₗ[R] M) (g : N ≃ₗ[R] N) (n : ℤ) :
    congr f g ^ n = congr (f ^ n) (g ^ 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) (g : N ≃ₗ[R] N) (n : ℤ)
Goal: TensorProduct.congr f g ^ n = TensorProduct.congr (f ^ n) (g ^ n)
-- TensorProduct.leftComm_tmul
@[simp]
theorem leftComm_tmul (m : M) (n : N) (p : P) : leftComm R M N P (m ⊗ₜ (n ⊗ₜ p)) = n ⊗ₜ (m ⊗ₜ 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] (m : M) (n : N) (p : P)
Goal: (TensorProduct.leftComm R M N P) (m ⊗ₜ[R] n ⊗ₜ[R] p) = n ⊗ₜ[R] m ⊗ₜ[R] p
-- TensorProduct.leftComm_symm_tmul
@[simp]
theorem leftComm_symm_tmul (m : M) (n : N) (p : P) :
    (leftComm R M N P).symm (n ⊗ₜ (m ⊗ₜ p)) = m ⊗ₜ (n ⊗ₜ 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] (m : M) (n : N) (p : P)
Goal: (TensorProduct.leftComm R M N P).symm (n ⊗ₜ[R] m ⊗ₜ[R] p) = m ⊗ₜ[R] n ⊗ₜ[R] p
-- TensorProduct.tensorTensorTensorComm_tmul
@[simp]
theorem tensorTensorTensorComm_tmul (m : M) (n : N) (p : P) (q : Q) :
    tensorTensorTensorComm R M N P Q (m ⊗ₜ n ⊗ₜ (p ⊗ₜ q)) = m ⊗ₜ p ⊗ₜ (n ⊗ₜ q) 
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] (m : M) (n : N) (p : P) (q : Q)
Goal: (TensorProduct.tensorTensorTensorComm R M N P Q) ((m ⊗ₜ[R] n) ⊗ₜ[R] p ⊗ₜ[R] q) = (m ⊗ₜ[R] p) ⊗ₜ[R] n ⊗ₜ[R] q
-- TensorProduct.tensorTensorTensorComm_symm
@[simp]
theorem tensorTensorTensorComm_symm :
    (tensorTensorTensorComm R M N P Q).symm = tensorTensorTensorComm R M P N Q 
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]
Goal: (TensorProduct.tensorTensorTensorComm R M N P Q).symm = TensorProduct.tensorTensorTensorComm R M P N Q
-- TensorProduct.tensorTensorTensorAssoc_tmul
@[simp]
theorem tensorTensorTensorAssoc_tmul (m : M) (n : N) (p : P) (q : Q) :
    tensorTensorTensorAssoc R M N P Q (m ⊗ₜ n ⊗ₜ (p ⊗ₜ q)) = m ⊗ₜ (n ⊗ₜ p) ⊗ₜ q 
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] (m : M) (n : N) (p : P) (q : Q)
Goal: (TensorProduct.tensorTensorTensorAssoc R M N P Q) ((m ⊗ₜ[R] n) ⊗ₜ[R] p ⊗ₜ[R] q) = (m ⊗ₜ[R] n ⊗ₜ[R] p) ⊗ₜ[R] q
-- TensorProduct.tensorTensorTensorAssoc_symm_tmul
@[simp]
theorem tensorTensorTensorAssoc_symm_tmul (m : M) (n : N) (p : P) (q : Q) :
    (tensorTensorTensorAssoc R M N P Q).symm (m ⊗ₜ (n ⊗ₜ p) ⊗ₜ q) = m ⊗ₜ n ⊗ₜ (p ⊗ₜ q) 
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] (m : M) (n : N) (p : P) (q : Q)
Goal: (TensorProduct.tensorTensorTensorAssoc R M N P Q).symm ((m ⊗ₜ[R] n ⊗ₜ[R] p) ⊗ₜ[R] q) = (m ⊗ₜ[R] n) ⊗ₜ[R] p ⊗ₜ[R] q
-- LinearMap.lTensor_tmul
@[simp]
theorem lTensor_tmul (m : M) (n : N) : 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: (LinearMap.lTensor M f) (m ⊗ₜ[R] n) = m ⊗ₜ[R] f n
-- LinearMap.rTensor_tmul
@[simp]
theorem rTensor_tmul (m : M) (n : N) : 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: (LinearMap.rTensor M f) (n ⊗ₜ[R] m) = f n ⊗ₜ[R] m
-- LinearMap.lTensor_comp_mk
@[simp]
theorem lTensor_comp_mk (m : M) :
    f.lTensor M ∘ₗ TensorProduct.mk R M N m = TensorProduct.mk R M P m ∘ₗ 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) (m : M)
Goal: LinearMap.lTensor M f ∘ₗ (TensorProduct.mk R M N) m = (TensorProduct.mk R M P) m ∘ₗ f
-- LinearMap.rTensor_comp_flip_mk
@[simp]
theorem rTensor_comp_flip_mk (m : M) :
    f.rTensor M ∘ₗ (TensorProduct.mk R N M).flip m = (TensorProduct.mk R P M).flip m ∘ₗ 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) (m : M)
Goal: LinearMap.rTensor M f ∘ₗ (TensorProduct.mk R N M).flip m = (TensorProduct.mk R P M).flip m ∘ₗ f