ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Projectivization.independent_pair_iff_neq
@[simp]
theorem independent_pair_iff_neq (u v : ℙ K V) : Independent ![u, v] ↔ u ≠ v 
Arguments: {K : Type u_2} {V : Type u_3} [DivisionRing K] [AddCommGroup V] [Module K V] (u v : ℙ K V)
Goal: Projectivization.Independent ![u, v] ↔ u ≠ v
-- hasEigenvector_toLin_diagonal
lemma hasEigenvector_toLin_diagonal (d : n → R) (i : n) (b : Basis n R M) :
    HasEigenvector (toLin b b (diagonal d)) (d i) (b i) 
Arguments: {R : Type u_1} {n : Type u_2} {M : Type u_3} [DecidableEq n] [Fintype n] [CommRing R] [Nontrivial R] [AddCommGroup M] [Module R M] (d : n → R) (i : n) (b : Basis n R M)
Goal: Module.End.HasEigenvector ((Matrix.toLin b b) (Matrix.diagonal d)) (d i) (b i)
-- hasEigenvector_toLin'_diagonal
lemma hasEigenvector_toLin'_diagonal (d : n → R) (i : n) :
    HasEigenvector (toLin' (diagonal d)) (d i) (Pi.basisFun R n i)  
Arguments: {R : Type u_1} {n : Type u_2} [DecidableEq n] [Fintype n] [CommRing R] [Nontrivial R] (d : n → R) (i : n)
Goal: Module.End.HasEigenvector (Matrix.toLin' (Matrix.diagonal d)) (d i) ((Pi.basisFun R n) i)
-- hasEigenvalue_toLin_diagonal_iff
lemma hasEigenvalue_toLin_diagonal_iff (d : n → R) {μ : R} [NoZeroSMulDivisors R M]
    (b : Basis n R M) : HasEigenvalue (toLin b b (diagonal d)) μ ↔ ∃ i, d i = μ 
Arguments: {R : Type u_1} {n : Type u_2} {M : Type u_3} [DecidableEq n] [Fintype n] [CommRing R] [Nontrivial R] [AddCommGroup M] [Module R M] (d : n → R) {μ : R} [NoZeroSMulDivisors R M] (b : Basis n R M)
Goal: Module.End.HasEigenvalue ((Matrix.toLin b b) (Matrix.diagonal d)) μ ↔ ∃ i, d i = μ
-- hasEigenvalue_toLin'_diagonal_iff
lemma hasEigenvalue_toLin'_diagonal_iff [NoZeroDivisors R] (d : n → R) {μ : R} :
    HasEigenvalue (toLin' (diagonal d)) μ ↔ (∃ i, d i = μ) 
Arguments: {R : Type u_1} {n : Type u_2} [DecidableEq n] [Fintype n] [CommRing R] [Nontrivial R] [NoZeroDivisors R] (d : n → R) {μ : R}
Goal: Module.End.HasEigenvalue (Matrix.toLin' (Matrix.diagonal d)) μ ↔ ∃ i, d i = μ
-- spectrum_diagonal
lemma spectrum_diagonal [Field R] (d : n → R) :
    spectrum R (diagonal d) = Set.range d 
Arguments: {R : Type u_1} {n : Type u_2} [DecidableEq n] [Fintype n] [Field R] (d : n → R)
Goal: spectrum R (Matrix.diagonal d) = Set.range d
-- GradedTensorProduct.of_one
@[simp]
theorem of_one : of R 𝒜 ℬ 1 = 1 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ]
Goal: (GradedTensorProduct.of R 𝒜 ℬ) 1 = 1
-- GradedTensorProduct.of_symm_one
@[simp]
theorem of_symm_one : (of R 𝒜 ℬ).symm 1 = 1 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ]
Goal: (GradedTensorProduct.of R 𝒜 ℬ).symm 1 = 1
-- GradedTensorProduct.of_symm_of
@[simp]
theorem of_symm_of (x : A ⊗[R] B) : (of R 𝒜 ℬ).symm (of R 𝒜 ℬ x) = x 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (x : A ⊗[R] B)
Goal: (GradedTensorProduct.of R 𝒜 ℬ).symm ((GradedTensorProduct.of R 𝒜 ℬ) x) = x
-- GradedTensorProduct.symm_of_of
@[simp]
theorem symm_of_of (x : 𝒜 ᵍ⊗[R] ℬ) : of R 𝒜 ℬ ((of R 𝒜 ℬ).symm x) = x 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (x : 𝒜 ᵍ⊗[R] ℬ)
Goal: (GradedTensorProduct.of R 𝒜 ℬ) ((GradedTensorProduct.of R 𝒜 ℬ).symm x) = x
-- GradedTensorProduct.hom_ext
@[ext]
theorem hom_ext {M} [AddCommMonoid M] [Module R M] ⦃f g : 𝒜 ᵍ⊗[R] ℬ →ₗ[R] M⦄
    (h : f ∘ₗ of R 𝒜 ℬ = (g ∘ₗ of R 𝒜 ℬ : A ⊗[R] B →ₗ[R] M)) :
    f = g 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {M : Type u_5} [AddCommMonoid M] [Module R M] ⦃f g : 𝒜 ᵍ⊗[R] ℬ →ₗ[R] M⦄ (h : f ∘ₗ ↑(GradedTensorProduct.of R 𝒜 ℬ) = g ∘ₗ ↑(GradedTensorProduct.of R 𝒜 ℬ))
Goal: f = g
-- GradedTensorProduct.auxEquiv_tmul
theorem auxEquiv_tmul (a : A) (b : B) :
    auxEquiv R 𝒜 ℬ (a ᵍ⊗ₜ b) = decompose 𝒜 a ⊗ₜ decompose ℬ b 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (a : A) (b : B)
Goal: (GradedTensorProduct.auxEquiv R 𝒜 ℬ) (a ᵍ⊗ₜ[R] b) = (DirectSum.decompose 𝒜) a ⊗ₜ[R] (DirectSum.decompose ℬ) b
-- GradedTensorProduct.auxEquiv_one
theorem auxEquiv_one : auxEquiv R 𝒜 ℬ 1 = 1 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ]
Goal: (GradedTensorProduct.auxEquiv R 𝒜 ℬ) 1 = 1
-- GradedTensorProduct.auxEquiv_symm_one
theorem auxEquiv_symm_one : (auxEquiv R 𝒜 ℬ).symm 1 = 1 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ]
Goal: (GradedTensorProduct.auxEquiv R 𝒜 ℬ).symm 1 = 1
-- GradedTensorProduct.mulHom_apply
theorem mulHom_apply (x y : 𝒜 ᵍ⊗[R] ℬ) :
    mulHom 𝒜 ℬ x y
      = (auxEquiv R 𝒜 ℬ).symm (gradedMul R (𝒜 ·) (ℬ ·) (auxEquiv R 𝒜 ℬ x) (auxEquiv R 𝒜 ℬ y)) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (x y : 𝒜 ᵍ⊗[R] ℬ)
Goal: ((GradedTensorProduct.mulHom 𝒜 ℬ) x) y =
  (GradedTensorProduct.auxEquiv R 𝒜 ℬ).symm
    (((TensorProduct.gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x)) ((GradedTensorProduct.auxEquiv R 𝒜 ℬ) x))
      ((GradedTensorProduct.auxEquiv R 𝒜 ℬ) y))
-- GradedTensorProduct.mul_def
theorem mul_def (x y : 𝒜 ᵍ⊗[R] ℬ) : x * y = mulHom 𝒜 ℬ x y 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (x y : 𝒜 ᵍ⊗[R] ℬ)
Goal: x * y = ((GradedTensorProduct.mulHom 𝒜 ℬ) x) y
-- GradedTensorProduct.auxEquiv_mul
theorem auxEquiv_mul (x y : 𝒜 ᵍ⊗[R] ℬ) :
    auxEquiv R 𝒜 ℬ (x * y) = gradedMul R (𝒜 ·) (ℬ ·) (auxEquiv R 𝒜 ℬ x) (auxEquiv R 𝒜 ℬ y) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (x y : 𝒜 ᵍ⊗[R] ℬ)
Goal: (GradedTensorProduct.auxEquiv R 𝒜 ℬ) (x * y) =
  ((TensorProduct.gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x)) ((GradedTensorProduct.auxEquiv R 𝒜 ℬ) x))
    ((GradedTensorProduct.auxEquiv R 𝒜 ℬ) y)
-- GradedTensorProduct.tmul_coe_mul_coe_tmul
theorem tmul_coe_mul_coe_tmul {j₁ i₂ : ι} (a₁ : A) (b₁ : ℬ j₁) (a₂ : 𝒜 i₂) (b₂ : B) :
    (a₁ ᵍ⊗ₜ[R] (b₁ : B) * (a₂ : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) =
      (-1 : ℤˣ)^(j₁ * i₂) • ((a₁ * a₂ : A) ᵍ⊗ₜ (b₁ * b₂ : B)) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {j₁ i₂ : ι} (a₁ : A) (b₁ : ↥(ℬ j₁)) (a₂ : ↥(𝒜 i₂)) (b₂ : B)
Goal: a₁ ᵍ⊗ₜ[R] ↑b₁ * ↑a₂ ᵍ⊗ₜ[R] b₂ = (-1) ^ (j₁ * i₂) • (a₁ * ↑a₂) ᵍ⊗ₜ[R] (↑b₁ * b₂)
-- GradedTensorProduct.tmul_zero_coe_mul_coe_tmul
theorem tmul_zero_coe_mul_coe_tmul {i₂ : ι} (a₁ : A) (b₁ : ℬ 0) (a₂ : 𝒜 i₂) (b₂ : B) :
    (a₁ ᵍ⊗ₜ[R] (b₁ : B) * (a₂ : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) =
      ((a₁ * a₂ : A) ᵍ⊗ₜ (b₁ * b₂ : B)) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {i₂ : ι} (a₁ : A) (b₁ : ↥(ℬ 0)) (a₂ : ↥(𝒜 i₂)) (b₂ : B)
Goal: a₁ ᵍ⊗ₜ[R] ↑b₁ * ↑a₂ ᵍ⊗ₜ[R] b₂ = (a₁ * ↑a₂) ᵍ⊗ₜ[R] (↑b₁ * b₂)
-- GradedTensorProduct.tmul_coe_mul_zero_coe_tmul
theorem tmul_coe_mul_zero_coe_tmul {j₁ : ι} (a₁ : A) (b₁ : ℬ j₁) (a₂ : 𝒜 0) (b₂ : B) :
    (a₁ ᵍ⊗ₜ[R] (b₁ : B) * (a₂ : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) =
      ((a₁ * a₂ : A) ᵍ⊗ₜ (b₁ * b₂ : B)) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {j₁ : ι} (a₁ : A) (b₁ : ↥(ℬ j₁)) (a₂ : ↥(𝒜 0)) (b₂ : B)
Goal: a₁ ᵍ⊗ₜ[R] ↑b₁ * ↑a₂ ᵍ⊗ₜ[R] b₂ = (a₁ * ↑a₂) ᵍ⊗ₜ[R] (↑b₁ * b₂)
-- GradedTensorProduct.tmul_one_mul_coe_tmul
theorem tmul_one_mul_coe_tmul {i₂ : ι} (a₁ : A) (a₂ : 𝒜 i₂) (b₂ : B) :
    (a₁ ᵍ⊗ₜ[R] (1 : B) * (a₂ : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) = (a₁ * a₂ : A) ᵍ⊗ₜ (b₂ : B) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {i₂ : ι} (a₁ : A) (a₂ : ↥(𝒜 i₂)) (b₂ : B)
Goal: a₁ ᵍ⊗ₜ[R] 1 * ↑a₂ ᵍ⊗ₜ[R] b₂ = (a₁ * ↑a₂) ᵍ⊗ₜ[R] b₂
-- GradedTensorProduct.tmul_coe_mul_one_tmul
theorem tmul_coe_mul_one_tmul {j₁ : ι} (a₁ : A) (b₁ : ℬ j₁) (b₂ : B) :
    (a₁ ᵍ⊗ₜ[R] (b₁ : B) * (1 : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) = (a₁ : A) ᵍ⊗ₜ (b₁ * b₂ : B) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {j₁ : ι} (a₁ : A) (b₁ : ↥(ℬ j₁)) (b₂ : B)
Goal: a₁ ᵍ⊗ₜ[R] ↑b₁ * 1 ᵍ⊗ₜ[R] b₂ = a₁ ᵍ⊗ₜ[R] (↑b₁ * b₂)
-- GradedTensorProduct.tmul_one_mul_one_tmul
theorem tmul_one_mul_one_tmul (a₁ : A) (b₂ : B) :
    (a₁ ᵍ⊗ₜ[R] (1 : B) * (1 : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) = (a₁ : A) ᵍ⊗ₜ (b₂ : B) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (a₁ : A) (b₂ : B)
Goal: a₁ ᵍ⊗ₜ[R] 1 * 1 ᵍ⊗ₜ[R] b₂ = a₁ ᵍ⊗ₜ[R] b₂
-- GradedTensorProduct.includeLeftRingHom_apply
@[simps]
def includeLeftRingHom : A →+* 𝒜 ᵍ⊗[R] ℬ 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (a : A)
Goal: (GradedTensorProduct.includeLeftRingHom 𝒜 ℬ) a = a ᵍ⊗ₜ[R] 1
-- GradedTensorProduct.algebraMap_def
lemma algebraMap_def (r : R) : algebraMap R (𝒜 ᵍ⊗[R] ℬ) r = algebraMap R A r ᵍ⊗ₜ[R] 1 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (r : R)
Goal: (algebraMap R (𝒜 ᵍ⊗[R] ℬ)) r = (algebraMap R A) r ᵍ⊗ₜ[R] 1
-- GradedTensorProduct.tmul_algebraMap_mul_coe_tmul
theorem tmul_algebraMap_mul_coe_tmul {i₂ : ι} (a₁ : A) (r : R) (a₂ : 𝒜 i₂) (b₂ : B) :
    (a₁ ᵍ⊗ₜ[R] algebraMap R B r * (a₂ : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ)
      = (a₁ * a₂ : A) ᵍ⊗ₜ (algebraMap R B r * b₂ : B) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {i₂ : ι} (a₁ : A) (r : R) (a₂ : ↥(𝒜 i₂)) (b₂ : B)
Goal: a₁ ᵍ⊗ₜ[R] (algebraMap R B) r * ↑a₂ ᵍ⊗ₜ[R] b₂ = (a₁ * ↑a₂) ᵍ⊗ₜ[R] ((algebraMap R B) r * b₂)
-- GradedTensorProduct.tmul_coe_mul_algebraMap_tmul
theorem tmul_coe_mul_algebraMap_tmul {j₁ : ι} (a₁ : A) (b₁ : ℬ j₁) (r : R) (b₂ : B) :
    (a₁ ᵍ⊗ₜ[R] (b₁ : B) * algebraMap R A r ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ)
      = (a₁ * algebraMap R A r : A) ᵍ⊗ₜ (b₁ * b₂ : B) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {j₁ : ι} (a₁ : A) (b₁ : ↥(ℬ j₁)) (r : R) (b₂ : B)
Goal: a₁ ᵍ⊗ₜ[R] ↑b₁ * (algebraMap R A) r ᵍ⊗ₜ[R] b₂ = (a₁ * (algebraMap R A) r) ᵍ⊗ₜ[R] (↑b₁ * b₂)
-- GradedTensorProduct.includeLeft_apply
@[simps!]
def includeLeft : A →ₐ[R] 𝒜 ᵍ⊗[R] ℬ 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (a : A)
Goal: (GradedTensorProduct.includeLeft 𝒜 ℬ) a = a ᵍ⊗ₜ[R] 1
-- GradedTensorProduct.includeRight_apply
@[simps!]
def includeRight : B →ₐ[R] (𝒜 ᵍ⊗[R] ℬ) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (a : B)
Goal: (GradedTensorProduct.includeRight 𝒜 ℬ) a = 1 ᵍ⊗ₜ[R] a
-- GradedTensorProduct.algebraMap_def'
lemma algebraMap_def' (r : R) : algebraMap R (𝒜 ᵍ⊗[R] ℬ) r = 1 ᵍ⊗ₜ[R] algebraMap R B r 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (r : R)
Goal: (algebraMap R (𝒜 ᵍ⊗[R] ℬ)) r = 1 ᵍ⊗ₜ[R] (algebraMap R B) r
-- GradedTensorProduct.lift_tmul
@[simp]
theorem lift_tmul (f : A →ₐ[R] C) (g : B →ₐ[R] C)
    (h_anti_commutes : ∀ ⦃i j⦄ (a : 𝒜 i) (b : ℬ j), f a * g b = (-1 : ℤˣ)^(j * i) • (g b * f a))
    (a : A) (b : B) :
    lift 𝒜 ℬ f g h_anti_commutes (a ᵍ⊗ₜ b) = f a * g b 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {C : Type u_5} [Ring C] [Algebra R C] (f : A →ₐ[R] C) (g : B →ₐ[R] C) (h_anti_commutes : ∀ ⦃i j : ι⦄ (a : ↥(𝒜 i)) (b : ↥(ℬ j)), f ↑a * g ↑b = (-1) ^ (j * i) • (g ↑b * f ↑a)) (a : A) (b : B)
Goal: (GradedTensorProduct.lift 𝒜 ℬ f g h_anti_commutes) (a ᵍ⊗ₜ[R] b) = f a * g b
-- GradedTensorProduct.algHom_ext
@[ext]
lemma algHom_ext ⦃f g : (𝒜 ᵍ⊗[R] ℬ) →ₐ[R] C⦄
    (ha : f.comp (includeLeft 𝒜 ℬ) = g.comp (includeLeft 𝒜 ℬ))
    (hb : f.comp (includeRight 𝒜 ℬ) = g.comp (includeRight 𝒜 ℬ)) : f = g 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {C : Type u_5} [Ring C] [Algebra R C] ⦃f g : 𝒜 ᵍ⊗[R] ℬ →ₐ[R] C⦄ (ha : f.comp (GradedTensorProduct.includeLeft 𝒜 ℬ) = g.comp (GradedTensorProduct.includeLeft 𝒜 ℬ)) (hb : f.comp (GradedTensorProduct.includeRight 𝒜 ℬ) = g.comp (GradedTensorProduct.includeRight 𝒜 ℬ))
Goal: f = g
-- GradedTensorProduct.auxEquiv_comm
lemma auxEquiv_comm (x : 𝒜 ᵍ⊗[R] ℬ) :
    auxEquiv R ℬ 𝒜 (comm 𝒜 ℬ x) = gradedComm R (𝒜 ·) (ℬ ·) (auxEquiv R 𝒜 ℬ x) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] (x : 𝒜 ᵍ⊗[R] ℬ)
Goal: (GradedTensorProduct.auxEquiv R ℬ 𝒜) ((GradedTensorProduct.comm 𝒜 ℬ) x) =
  (TensorProduct.gradedComm R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x)) ((GradedTensorProduct.auxEquiv R 𝒜 ℬ) x)
-- GradedTensorProduct.comm_coe_tmul_coe
@[simp] lemma comm_coe_tmul_coe {i j : ι} (a : 𝒜 i) (b : ℬ j) :
    comm 𝒜 ℬ (a ᵍ⊗ₜ b) = (-1 : ℤˣ)^(j * i) • (b ᵍ⊗ₜ a : ℬ ᵍ⊗[R] 𝒜) 
Arguments: {R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] {i j : ι} (a : ↥(𝒜 i)) (b : ↥(ℬ j))
Goal: (GradedTensorProduct.comm 𝒜 ℬ) (↑a ᵍ⊗ₜ[R] ↑b) = (-1) ^ (j * i) • ↑b ᵍ⊗ₜ[R] ↑a
-- Projectivization.Subspace.mem_add'
@[ext]
structure Subspace where
    carrier : Set (ℙ K V)
    mem_add' (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) (hvw : v + w ≠ 0) :
    mk K v hv ∈ carrier → mk K w hw ∈ carrier → mk K (v + w) hvw ∈ carrier
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] (self : Projectivization.Subspace K V) (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) (hvw : v + w ≠ 0)
Goal: Projectivization.mk K v hv ∈ self.carrier →
  Projectivization.mk K w hw ∈ self.carrier → Projectivization.mk K (v + w) hvw ∈ self.carrier
-- Projectivization.Subspace.mem_carrier_iff
@[simp]
theorem mem_carrier_iff (A : Subspace K V) (x : ℙ K V) : x ∈ A.carrier ↔ x ∈ A 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] (A : Projectivization.Subspace K V) (x : Projectivization K V)
Goal: x ∈ A.carrier ↔ x ∈ A
-- Projectivization.Subspace.mem_add
theorem mem_add (T : Subspace K V) (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) (hvw : v + w ≠ 0) :
    Projectivization.mk K v hv ∈ T →
      Projectivization.mk K w hw ∈ T → Projectivization.mk K (v + w) hvw ∈ T 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] (T : Projectivization.Subspace K V) (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) (hvw : v + w ≠ 0)
Goal: Projectivization.mk K v hv ∈ T → Projectivization.mk K w hw ∈ T → Projectivization.mk K (v + w) hvw ∈ T
-- Projectivization.Subspace.subset_span
theorem subset_span (S : Set (ℙ K V)) : S ⊆ span S 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] (S : Set (Projectivization K V))
Goal: S ⊆ ↑(Projectivization.Subspace.span S)
-- Projectivization.Subspace.span_coe
@[simp]
theorem span_coe (W : Subspace K V) : span ↑W = W 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] (W : Projectivization.Subspace K V)
Goal: Projectivization.Subspace.span ↑W = W
-- Projectivization.Subspace.span_empty
@[simp]
theorem span_empty : span (∅ : Set (ℙ K V)) = ⊥ 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V]
Goal: Projectivization.Subspace.span ∅ = ⊥
-- Projectivization.Subspace.span_univ
@[simp]
theorem span_univ : span (Set.univ : Set (ℙ K V)) = ⊤ 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V]
Goal: Projectivization.Subspace.span Set.univ = ⊤
-- Projectivization.Subspace.span_le_subspace_iff
theorem span_le_subspace_iff {S : Set (ℙ K V)} {W : Subspace K V} : span S ≤ W ↔ S ⊆ W 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {S : Set (Projectivization K V)} {W : Projectivization.Subspace K V}
Goal: Projectivization.Subspace.span S ≤ W ↔ S ⊆ ↑W
-- Projectivization.Subspace.monotone_span
@[mono]
theorem monotone_span : Monotone (span : Set (ℙ K V) → Subspace K V) 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V]
Goal: Monotone Projectivization.Subspace.span
-- Projectivization.Subspace.subset_span_trans
theorem subset_span_trans {S T U : Set (ℙ K V)} (hST : S ⊆ span T) (hTU : T ⊆ span U) :
    S ⊆ span U 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {S T U : Set (Projectivization K V)} (hST : S ⊆ ↑(Projectivization.Subspace.span T)) (hTU : T ⊆ ↑(Projectivization.Subspace.span U))
Goal: S ⊆ ↑(Projectivization.Subspace.span U)
-- Projectivization.Subspace.span_union
theorem span_union (S T : Set (ℙ K V)) : span (S ∪ T) = span S ⊔ span T 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] (S T : Set (Projectivization K V))
Goal: Projectivization.Subspace.span (S ∪ T) = Projectivization.Subspace.span S ⊔ Projectivization.Subspace.span T
-- Projectivization.Subspace.span_iUnion
theorem span_iUnion {ι} (s : ι → Set (ℙ K V)) : span (⋃ i, s i) = ⨆ i, span (s i) 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {ι : Sort u_3} (s : ι → Set (Projectivization K V))
Goal: Projectivization.Subspace.span (⋃ i, s i) = ⨆ i, Projectivization.Subspace.span (s i)
-- Projectivization.Subspace.sup_span
theorem sup_span {S : Set (ℙ K V)} {W : Subspace K V} : W ⊔ span S = span (W ∪ S) 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {S : Set (Projectivization K V)} {W : Projectivization.Subspace K V}
Goal: W ⊔ Projectivization.Subspace.span S = Projectivization.Subspace.span (↑W ∪ S)
-- Projectivization.Subspace.span_sup
theorem span_sup {S : Set (ℙ K V)} {W : Subspace K V} : span S ⊔ W = span (S ∪ W) 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {S : Set (Projectivization K V)} {W : Projectivization.Subspace K V}
Goal: Projectivization.Subspace.span S ⊔ W = Projectivization.Subspace.span (S ∪ ↑W)
-- Projectivization.Subspace.mem_span
theorem mem_span {S : Set (ℙ K V)} (u : ℙ K V) :
    u ∈ span S ↔ ∀ W : Subspace K V, S ⊆ W → u ∈ W 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {S : Set (Projectivization K V)} (u : Projectivization K V)
Goal: u ∈ Projectivization.Subspace.span S ↔ ∀ (W : Projectivization.Subspace K V), S ⊆ ↑W → u ∈ W
-- Projectivization.Subspace.span_eq_sInf
theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {S : Set (Projectivization K V)}
Goal: Projectivization.Subspace.span S = sInf {W | S ⊆ ↑W}
-- Projectivization.Subspace.span_eq_of_le
theorem span_eq_of_le {S : Set (ℙ K V)} {W : Subspace K V} (hS : S ⊆ W) (hW : W ≤ span S) :
    span S = W 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {S : Set (Projectivization K V)} {W : Projectivization.Subspace K V} (hS : S ⊆ ↑W) (hW : W ≤ Projectivization.Subspace.span S)
Goal: Projectivization.Subspace.span S = W
-- Projectivization.Subspace.span_eq_span_iff
theorem span_eq_span_iff {S T : Set (ℙ K V)} : span S = span T ↔ S ⊆ span T ∧ T ⊆ span S 
Arguments: {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {S T : Set (Projectivization K V)}
Goal: Projectivization.Subspace.span S = Projectivization.Subspace.span T ↔
  S ⊆ ↑(Projectivization.Subspace.span T) ∧ T ⊆ ↑(Projectivization.Subspace.span S)
-- LinearIndependent.sum_elim_of_quotient
theorem LinearIndependent.sum_elim_of_quotient
    {M' : Submodule R M} {ι₁ ι₂} {f : ι₁ → M'} (hf : LinearIndependent R f) (g : ι₂ → M)
    (hg : LinearIndependent R (Submodule.Quotient.mk (p := M') ∘ g)) :
      LinearIndependent R (Sum.elim (f · : ι₁ → M) g) 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {M' : Submodule R M} {ι₁ : Type u_2} {ι₂ : Type u_3} {f : ι₁ → ↥M'} (hf : LinearIndependent R f) (g : ι₂ → M) (hg : LinearIndependent R (Submodule.Quotient.mk ∘ g))
Goal: LinearIndependent R (Sum.elim (fun x => ↑(f x)) g)