ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Projectivization.submodule_injective
theorem submodule_injective :
    Function.Injective (Projectivization.submodule : ℙ K V → Submodule K V) 
Arguments: {K : Type u_1} {V : Type u_2} [DivisionRing K] [AddCommGroup V] [Module K V]
Goal: Function.Injective Projectivization.submodule
-- Projectivization.submodule_mk''
@[simp]
theorem submodule_mk'' (H : Submodule K V) (h : finrank K H = 1) : (mk'' H h).submodule = H 
Arguments: {K : Type u_1} {V : Type u_2} [DivisionRing K] [AddCommGroup V] [Module K V] (H : Submodule K V) (h : FiniteDimensional.finrank K ↥H = 1)
Goal: (Projectivization.mk'' H h).submodule = H
-- Projectivization.mk''_submodule
@[simp]
theorem mk''_submodule (v : ℙ K V) : mk'' v.submodule v.finrank_submodule = v 
Arguments: {K : Type u_1} {V : Type u_2} [DivisionRing K] [AddCommGroup V] [Module K V] (v : Projectivization K V)
Goal: Projectivization.mk'' v.submodule ⋯ = v
-- Projectivization.map_mk
theorem map_mk {σ : K →+* L} (f : V →ₛₗ[σ] W) (hf : Function.Injective f) (v : V) (hv : v ≠ 0) :
    map f hf (mk K v hv) = mk L (f v) (map_zero f ▸ hf.ne hv) 
Arguments: {K : Type u_1} {V : Type u_2} [DivisionRing K] [AddCommGroup V] [Module K V] {L : Type u_3} {W : Type u_4} [DivisionRing L] [AddCommGroup W] [Module L W] {σ : K →+* L} (f : V →ₛₗ[σ] W) (hf : Function.Injective ⇑f) (v : V) (hv : v ≠ 0)
Goal: Projectivization.map f hf (Projectivization.mk K v hv) = Projectivization.mk L (f v) ⋯
-- Projectivization.map_injective
theorem map_injective {σ : K →+* L} {τ : L →+* K} [RingHomInvPair σ τ] (f : V →ₛₗ[σ] W)
    (hf : Function.Injective f) : Function.Injective (map f hf) 
Arguments: {K : Type u_1} {V : Type u_2} [DivisionRing K] [AddCommGroup V] [Module K V] {L : Type u_3} {W : Type u_4} [DivisionRing L] [AddCommGroup W] [Module L W] {σ : K →+* L} {τ : L →+* K} [RingHomInvPair σ τ] (f : V →ₛₗ[σ] W) (hf : Function.Injective ⇑f)
Goal: Function.Injective (Projectivization.map f hf)
-- Projectivization.map_id
@[simp]
theorem map_id : map (LinearMap.id : V →ₗ[K] V) (LinearEquiv.refl K V).injective = id 
Arguments: {K : Type u_1} {V : Type u_2} [DivisionRing K] [AddCommGroup V] [Module K V]
Goal: Projectivization.map LinearMap.id ⋯ = id
-- Projectivization.map_comp
theorem map_comp {F U : Type*} [Field F] [AddCommGroup U] [Module F U] {σ : K →+* L} {τ : L →+* F}
    {γ : K →+* F} [RingHomCompTriple σ τ γ] (f : V →ₛₗ[σ] W) (hf : Function.Injective f)
    (g : W →ₛₗ[τ] U) (hg : Function.Injective g) :
    map (g.comp f) (hg.comp hf) = map g hg ∘ map f hf 
Arguments: {K : Type u_1} {V : Type u_2} [DivisionRing K] [AddCommGroup V] [Module K V] {L : Type u_3} {W : Type u_4} [DivisionRing L] [AddCommGroup W] [Module L W] {F : Type u_5} {U : Type u_6} [Field F] [AddCommGroup U] [Module F U] {σ : K →+* L} {τ : L →+* F} {γ : K →+* F} [RingHomCompTriple σ τ γ] (f : V →ₛₗ[σ] W) (hf : Function.Injective ⇑f) (g : W →ₛₗ[τ] U) (hg : Function.Injective ⇑g)
Goal: Projectivization.map (g.comp f) ⋯ = Projectivization.map g hg ∘ Projectivization.map f hf
-- Module.End.apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub
lemma apply_eq_of_mem_genEigenspace_of_comm_of_isSemisimple_of_isNilpotent_sub
    {μ : R} {k : ℕ} {m : M} (hm : m ∈ f.genEigenspace μ k)
    (hfg : Commute f g) (hss : g.IsSemisimple) (hnil : IsNilpotent (f - g)) :
    g m = μ • m 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {f g : Module.End R M} {μ : R} {k : ℕ} {m : M} (hm : m ∈ (f.genEigenspace μ) k) (hfg : Commute f g) (hss : g.IsSemisimple) (hnil : IsNilpotent (f - g))
Goal: g m = μ • m
-- Module.End.IsSemisimple.genEigenspace_eq_eigenspace
lemma IsSemisimple.genEigenspace_eq_eigenspace
    (hf : f.IsSemisimple) (μ : R) {k : ℕ} (hk : 0 < k) :
    f.genEigenspace μ k = f.eigenspace μ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {f : Module.End R M} (hf : f.IsSemisimple) (μ : R) {k : ℕ} (hk : 0 < k)
Goal: (f.genEigenspace μ) k = f.eigenspace μ
-- Module.End.IsSemisimple.maxGenEigenspace_eq_eigenspace
lemma IsSemisimple.maxGenEigenspace_eq_eigenspace
    (hf : f.IsSemisimple) (μ : R) :
    f.maxGenEigenspace μ = f.eigenspace μ 
Arguments: {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {f : Module.End R M} (hf : f.IsSemisimple) (μ : R)
Goal: f.maxGenEigenspace μ = f.eigenspace μ
-- TensorAlgebra.ι_def
irreducible_def ι : M →ₗ[R] TensorAlgebra R M 
Arguments: (R : Type u_3) [CommSemiring R] {M : Type u_4} [AddCommMonoid M] [Module R M]
Goal: TensorAlgebra.ι R =
  { toFun := fun m => (RingQuot.mkAlgHom R (TensorAlgebra.Rel R M)) (FreeAlgebra.ι R m), map_add' := ⋯, map_smul' := ⋯ }
-- TensorAlgebra.ringQuot_mkAlgHom_freeAlgebra_ι_eq_ι
theorem ringQuot_mkAlgHom_freeAlgebra_ι_eq_ι (m : M) :
    RingQuot.mkAlgHom R (Rel R M) (FreeAlgebra.ι R m) = ι R m 
Arguments: (R : Type u_1) [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (m : M)
Goal: (RingQuot.mkAlgHom R (TensorAlgebra.Rel R M)) (FreeAlgebra.ι R m) = (TensorAlgebra.ι R) m
-- TensorAlgebra.lift_symm_apply
@[simps symm_apply]
def lift {A : Type*} [Semiring A] [Algebra R A] : (M →ₗ[R] A) ≃ (TensorAlgebra R M →ₐ[R] A) 
Arguments: (R : Type u_1) [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {A : Type u_3} [Semiring A] [Algebra R A] (F : TensorAlgebra R M →ₐ[R] A)
Goal: (TensorAlgebra.lift R).symm F = F.toLinearMap ∘ₗ TensorAlgebra.ι R
-- TensorAlgebra.ι_comp_lift
@[simp]
theorem ι_comp_lift {A : Type*} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) :
    (lift R f).toLinearMap.comp (ι R) = f 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {A : Type u_3} [Semiring A] [Algebra R A] (f : M →ₗ[R] A)
Goal: ((TensorAlgebra.lift R) f).toLinearMap ∘ₗ TensorAlgebra.ι R = f
-- TensorAlgebra.lift_ι_apply
@[simp]
theorem lift_ι_apply {A : Type*} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (x) :
    lift R f (ι R x) = f x 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {A : Type u_3} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (x : M)
Goal: ((TensorAlgebra.lift R) f) ((TensorAlgebra.ι R) x) = f x
-- TensorAlgebra.lift_unique
@[simp]
theorem lift_unique {A : Type*} [Semiring A] [Algebra R A] (f : M →ₗ[R] A)
    (g : TensorAlgebra R M →ₐ[R] A) : g.toLinearMap.comp (ι R) = f ↔ g = lift R f 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {A : Type u_3} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (g : TensorAlgebra R M →ₐ[R] A)
Goal: g.toLinearMap ∘ₗ TensorAlgebra.ι R = f ↔ g = (TensorAlgebra.lift R) f
-- TensorAlgebra.lift_comp_ι
@[simp]
theorem lift_comp_ι {A : Type*} [Semiring A] [Algebra R A] (g : TensorAlgebra R M →ₐ[R] A) :
    lift R (g.toLinearMap.comp (ι R)) = g 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {A : Type u_3} [Semiring A] [Algebra R A] (g : TensorAlgebra R M →ₐ[R] A)
Goal: (TensorAlgebra.lift R) (g.toLinearMap ∘ₗ TensorAlgebra.ι R) = g
-- TensorAlgebra.hom_ext
@[ext]
theorem hom_ext {A : Type*} [Semiring A] [Algebra R A] {f g : TensorAlgebra R M →ₐ[R] A}
    (w : f.toLinearMap.comp (ι R) = g.toLinearMap.comp (ι R)) : f = g 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {A : Type u_3} [Semiring A] [Algebra R A] {f g : TensorAlgebra R M →ₐ[R] A} (w : f.toLinearMap ∘ₗ TensorAlgebra.ι R = g.toLinearMap ∘ₗ TensorAlgebra.ι R)
Goal: f = g
-- TensorAlgebra.induction
@[elab_as_elim]
theorem induction {C : TensorAlgebra R M → Prop}
    (algebraMap : ∀ r, C (algebraMap R (TensorAlgebra R M) r)) (ι : ∀ x, C (ι R x))
    (mul : ∀ a b, C a → C b → C (a * b)) (add : ∀ a b, C a → C b → C (a + b))
    (a : TensorAlgebra R M) : C a 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {C : TensorAlgebra R M → Prop} (algebraMap : ∀ (r : R), C ((algebraMap R (TensorAlgebra R M)) r)) (ι : ∀ (x : M), C ((TensorAlgebra.ι R) x)) (mul : ∀ (a b : TensorAlgebra R M), C a → C b → C (a * b)) (add : ∀ (a b : TensorAlgebra R M), C a → C b → C (a + b)) (a : TensorAlgebra R M)
Goal: C a
-- TensorAlgebra.algebraMap_leftInverse
theorem algebraMap_leftInverse :
    Function.LeftInverse algebraMapInv (algebraMap R <| TensorAlgebra R M) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_2) [AddCommMonoid M] [Module R M]
Goal: Function.LeftInverse ⇑TensorAlgebra.algebraMapInv ⇑(algebraMap R (TensorAlgebra R M))
-- TensorAlgebra.algebraMap_inj
@[simp]
theorem algebraMap_inj (x y : R) :
    algebraMap R (TensorAlgebra R M) x = algebraMap R (TensorAlgebra R M) y ↔ x = y 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] (x y : R)
Goal: (algebraMap R (TensorAlgebra R M)) x = (algebraMap R (TensorAlgebra R M)) y ↔ x = y
-- TensorAlgebra.algebraMap_eq_zero_iff
@[simp]
theorem algebraMap_eq_zero_iff (x : R) : algebraMap R (TensorAlgebra R M) x = 0 ↔ x = 0 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] (x : R)
Goal: (algebraMap R (TensorAlgebra R M)) x = 0 ↔ x = 0
-- TensorAlgebra.algebraMap_eq_one_iff
@[simp]
theorem algebraMap_eq_one_iff (x : R) : algebraMap R (TensorAlgebra R M) x = 1 ↔ x = 1 
Arguments: {R : Type u_1} [CommSemiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] (x : R)
Goal: (algebraMap R (TensorAlgebra R M)) x = 1 ↔ x = 1
-- TensorAlgebra.toTrivSqZeroExt_ι
@[simp]
theorem toTrivSqZeroExt_ι (x : M) [Module Rᵐᵒᵖ M] [IsCentralScalar R M] :
    toTrivSqZeroExt (ι R x) = TrivSqZeroExt.inr x 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (x : M) [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
Goal: TensorAlgebra.toTrivSqZeroExt ((TensorAlgebra.ι R) x) = TrivSqZeroExt.inr x
-- TensorAlgebra.ι_leftInverse
theorem ι_leftInverse : Function.LeftInverse ιInv (ι R : M → TensorAlgebra R M) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M]
Goal: Function.LeftInverse ⇑TensorAlgebra.ιInv ⇑(TensorAlgebra.ι R)
-- TensorAlgebra.ι_inj
@[simp]
theorem ι_inj (x y : M) : ι R x = ι R y ↔ x = y 
Arguments: (R : Type u_1) [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (x y : M)
Goal: (TensorAlgebra.ι R) x = (TensorAlgebra.ι R) y ↔ x = y
-- TensorAlgebra.ι_eq_zero_iff
@[simp]
theorem ι_eq_zero_iff (x : M) : ι R x = 0 ↔ x = 0 
Arguments: (R : Type u_1) [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (x : M)
Goal: (TensorAlgebra.ι R) x = 0 ↔ x = 0
-- TensorAlgebra.ι_eq_algebraMap_iff
@[simp]
theorem ι_eq_algebraMap_iff (x : M) (r : R) : ι R x = algebraMap R _ r ↔ x = 0 ∧ r = 0 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (x : M) (r : R)
Goal: (TensorAlgebra.ι R) x = (algebraMap R (TensorAlgebra R M)) r ↔ x = 0 ∧ r = 0
-- TensorAlgebra.ι_ne_one
@[simp]
theorem ι_ne_one [Nontrivial R] (x : M) : ι R x ≠ 1 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] [Nontrivial R] (x : M)
Goal: (TensorAlgebra.ι R) x ≠ 1
-- TensorAlgebra.ι_range_disjoint_one
theorem ι_range_disjoint_one :
    Disjoint (LinearMap.range (ι R : M →ₗ[R] TensorAlgebra R M))
      (1 : Submodule R (TensorAlgebra R M)) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M]
Goal: Disjoint (LinearMap.range (TensorAlgebra.ι R)) 1
-- TensorAlgebra.tprod_apply
@[simp]
theorem tprod_apply {n : ℕ} (x : Fin n → M) : tprod R M n x = (List.ofFn fun i => ι R (x i)).prod 
Arguments: (R : Type u_1) [CommSemiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] {n : ℕ} (x : Fin n → M)
Goal: (TensorAlgebra.tprod R M n) x = (List.ofFn fun i => (TensorAlgebra.ι R) (x i)).prod
-- FreeAlgebra.toTensor_ι
@[simp]
theorem toTensor_ι (m : M) : FreeAlgebra.toTensor (FreeAlgebra.ι R m) = TensorAlgebra.ι R m 
Arguments: {R : Type u_1} [CommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (m : M)
Goal: FreeAlgebra.toTensor (FreeAlgebra.ι R m) = (TensorAlgebra.ι R) m
-- LinearMap.ext_basis
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' 
Arguments: {ι₁ : Type u_1} {ι₂ : Type u_2} {R : Type u_3} {R₂ : Type u_4} {S : Type u_5} {S₂ : Type u_6} {M : Type u_7} {N : Type u_8} {P : Type u_9} [Semiring R] [Semiring S] [Semiring R₂] [Semiring S₂] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (b₁ : Basis ι₁ R M) (b₂ : Basis ι₂ S N) {B B' : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (h : ∀ (i : ι₁) (j : ι₂), (B (b₁ i)) (b₂ j) = (B' (b₁ i)) (b₂ j))
Goal: B = B'
-- LinearMap.sum_repr_mul_repr_mulₛₗ
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 
Arguments: {ι₁ : Type u_1} {ι₂ : Type u_2} {R : Type u_3} {R₂ : Type u_4} {S : Type u_5} {S₂ : Type u_6} {M : Type u_7} {N : Type u_8} {P : Type u_9} [Semiring R] [Semiring S] [Semiring R₂] [Semiring S₂] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (b₁ : Basis ι₁ R M) (b₂ : Basis ι₂ S N) {B : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (x : M) (y : N)
Goal: ((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
-- LinearMap.sum_repr_mul_repr_mul
theorem sum_repr_mul_repr_mul {B : Mₗ →ₗ[Rₗ] Nₗ →ₗ[Rₗ] 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 
Arguments: {ι₁ : Type u_1} {ι₂ : Type u_2} {Rₗ : Type u_10} {Mₗ : Type u_11} {Nₗ : Type u_12} {Pₗ : Type u_13} [CommSemiring Rₗ] [AddCommMonoid Mₗ] [AddCommMonoid Nₗ] [AddCommMonoid Pₗ] [Module Rₗ Mₗ] [Module Rₗ Nₗ] [Module Rₗ Pₗ] (b₁' : Basis ι₁ Rₗ Mₗ) (b₂' : Basis ι₂ Rₗ Nₗ) {B : Mₗ →ₗ[Rₗ] Nₗ →ₗ[Rₗ] Pₗ} (x : Mₗ) (y : Nₗ)
Goal: ((b₁'.repr x).sum fun i xi => (b₂'.repr y).sum fun j yj => xi • yj • (B (b₁' i)) (b₂' j)) = (B x) y
-- Matrix.det_apply
theorem det_apply (M : Matrix n n R) : M.det = ∑ σ : Perm n, Equiv.Perm.sign σ • ∏ i, M (σ i) i 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (M : Matrix n n R)
Goal: M.det = ∑ σ : Equiv.Perm n, Equiv.Perm.sign σ • ∏ i : n, M (σ i) i
-- Matrix.det_apply'
theorem det_apply' (M : Matrix n n R) : M.det = ∑ σ : Perm n, ε σ * ∏ i, M (σ i) i 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (M : Matrix n n R)
Goal: M.det = ∑ σ : Equiv.Perm n, ↑↑(Equiv.Perm.sign σ) * ∏ i : n, M (σ i) i
-- Matrix.det_diagonal
@[simp]
theorem det_diagonal {d : n → R} : det (diagonal d) = ∏ i, d i 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {d : n → R}
Goal: (Matrix.diagonal d).det = ∏ i : n, d i
-- Matrix.det_zero
theorem det_zero (_ : Nonempty n) : det (0 : Matrix n n R) = 0 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R]
Goal: Nonempty n → Matrix.det 0 = 0
-- Matrix.det_one
@[simp]
theorem det_one : det (1 : Matrix n n R) = 1 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R]
Goal: Matrix.det 1 = 1
-- Matrix.det_isEmpty
theorem det_isEmpty [IsEmpty n] {A : Matrix n n R} : det A = 1 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [IsEmpty n] {A : Matrix n n R}
Goal: A.det = 1
-- Matrix.coe_det_isEmpty
@[simp]
theorem coe_det_isEmpty [IsEmpty n] : (det : Matrix n n R → R) = Function.const _ 1 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [IsEmpty n]
Goal: Matrix.det = Function.const (Matrix n n R) 1
-- Matrix.det_eq_one_of_card_eq_zero
theorem det_eq_one_of_card_eq_zero {A : Matrix n n R} (h : Fintype.card n = 0) : det A = 1 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {A : Matrix n n R} (h : Fintype.card n = 0)
Goal: A.det = 1
-- Matrix.det_unique
@[simp]
theorem det_unique {n : Type*} [Unique n] [DecidableEq n] [Fintype n] (A : Matrix n n R) :
    det A = A default default 
Arguments: {R : Type v} [CommRing R] {n : Type u_3} [Unique n] [DecidableEq n] [Fintype n] (A : Matrix n n R)
Goal: A.det = A default default
-- Matrix.det_eq_elem_of_subsingleton
theorem det_eq_elem_of_subsingleton [Subsingleton n] (A : Matrix n n R) (k : n) :
    det A = A k k 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [Subsingleton n] (A : Matrix n n R) (k : n)
Goal: A.det = A k k
-- Matrix.det_eq_elem_of_card_eq_one
theorem det_eq_elem_of_card_eq_one {A : Matrix n n R} (h : Fintype.card n = 1) (k : n) :
    det A = A k k 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {A : Matrix n n R} (h : Fintype.card n = 1) (k : n)
Goal: A.det = A k k
-- Matrix.det_mul_aux
theorem det_mul_aux {M N : Matrix n n R} {p : n → n} (H : ¬Bijective p) :
    (∑ σ : Perm n, ε σ * ∏ x, M (σ x) (p x) * N (p x) x) = 0 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {M N : Matrix n n R} {p : n → n} (H : ¬Function.Bijective p)
Goal: ∑ σ : Equiv.Perm n, ↑↑(Equiv.Perm.sign σ) * ∏ x : n, M (σ x) (p x) * N (p x) x = 0
-- Matrix.det_mul
@[simp]
theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (M N : Matrix n n R)
Goal: (M * N).det = M.det * N.det
-- Matrix.coe_detMonoidHom
@[simp]
theorem coe_detMonoidHom : (detMonoidHom : Matrix n n R → R) = det 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R]
Goal: ⇑Matrix.detMonoidHom = Matrix.det
-- Matrix.det_mul_comm
theorem det_mul_comm (M N : Matrix m m R) : det (M * N) = det (N * M) 
Arguments: {m : Type u_1} [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] (M N : Matrix m m R)
Goal: (M * N).det = (N * M).det
-- Matrix.det_mul_left_comm
theorem det_mul_left_comm (M N P : Matrix m m R) : det (M * (N * P)) = det (N * (M * P)) 
Arguments: {m : Type u_1} [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] (M N P : Matrix m m R)
Goal: (M * (N * P)).det = (N * (M * P)).det
-- Matrix.det_mul_right_comm
theorem det_mul_right_comm (M N P : Matrix m m R) : det (M * N * P) = det (M * P * N) 
Arguments: {m : Type u_1} [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] (M N P : Matrix m m R)
Goal: (M * N * P).det = (M * P * N).det
-- Matrix.det_units_conj
theorem det_units_conj (M : (Matrix m m R)ˣ) (N : Matrix m m R) :
    det ((M : Matrix _ _ _) * N * (↑M⁻¹ : Matrix _ _ _)) = det N 
Arguments: {m : Type u_1} [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] (M : (Matrix m m R)ˣ) (N : Matrix m m R)
Goal: (↑M * N * ↑M⁻¹).det = N.det
-- Matrix.det_units_conj'
theorem det_units_conj' (M : (Matrix m m R)ˣ) (N : Matrix m m R) :
    det ((↑M⁻¹ : Matrix _ _ _) * N * (↑M : Matrix _ _ _)) = det N 
Arguments: {m : Type u_1} [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] (M : (Matrix m m R)ˣ) (N : Matrix m m R)
Goal: (↑M⁻¹ * N * ↑M).det = N.det
-- Matrix.det_transpose
@[simp]
theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (M : Matrix n n R)
Goal: M.transpose.det = M.det
-- Matrix.det_permute
theorem det_permute (σ : Perm n) (M : Matrix n n R) :
    (M.submatrix σ id).det = Perm.sign σ * M.det 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (σ : Equiv.Perm n) (M : Matrix n n R)
Goal: (M.submatrix (⇑σ) id).det = ↑↑(Equiv.Perm.sign σ) * M.det
-- Matrix.det_permute'
theorem det_permute' (σ : Perm n) (M : Matrix n n R) :
    (M.submatrix id σ).det = Perm.sign σ * M.det 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (σ : Equiv.Perm n) (M : Matrix n n R)
Goal: (M.submatrix id ⇑σ).det = ↑↑(Equiv.Perm.sign σ) * M.det
-- Matrix.det_submatrix_equiv_self
@[simp]
theorem det_submatrix_equiv_self (e : n ≃ m) (A : Matrix m m R) :
    det (A.submatrix e e) = det A 
Arguments: {m : Type u_1} {n : Type u_2} [DecidableEq n] [Fintype n] [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] (e : n ≃ m) (A : Matrix m m R)
Goal: (A.submatrix ⇑e ⇑e).det = A.det
-- Matrix.det_reindex_self
theorem det_reindex_self (e : m ≃ n) (A : Matrix m m R) : det (reindex e e A) = det A 
Arguments: {m : Type u_1} {n : Type u_2} [DecidableEq n] [Fintype n] [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] (e : m ≃ n) (A : Matrix m m R)
Goal: ((Matrix.reindex e e) A).det = A.det
-- Matrix.det_smul
theorem det_smul (A : Matrix n n R) (c : R) : det (c • A) = c ^ Fintype.card n * det A 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix n n R) (c : R)
Goal: (c • A).det = c ^ Fintype.card n * A.det
-- Matrix.det_smul_of_tower
@[simp]
theorem det_smul_of_tower {α} [Monoid α] [DistribMulAction α R] [IsScalarTower α R R]
    [SMulCommClass α R R] (c : α) (A : Matrix n n R) :
    det (c • A) = c ^ Fintype.card n • det A 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {α : Type u_3} [Monoid α] [DistribMulAction α R] [IsScalarTower α R R] [SMulCommClass α R R] (c : α) (A : Matrix n n R)
Goal: (c • A).det = c ^ Fintype.card n • A.det
-- Matrix.det_neg
theorem det_neg (A : Matrix n n R) : det (-A) = (-1) ^ Fintype.card n * det A 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix n n R)
Goal: (-A).det = (-1) ^ Fintype.card n * A.det
-- Matrix.det_neg_eq_smul
theorem det_neg_eq_smul (A : Matrix n n R) :
    det (-A) = (-1 : Units ℤ) ^ Fintype.card n • det A 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix n n R)
Goal: (-A).det = (-1) ^ Fintype.card n • A.det
-- Matrix.det_mul_row
theorem det_mul_row (v : n → R) (A : Matrix n n R) :
    det (of fun i j => v j * A i j) = (∏ i, v i) * det A 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (v : n → R) (A : Matrix n n R)
Goal: (Matrix.of fun i j => v j * A i j).det = (∏ i : n, v i) * A.det
-- Matrix.det_mul_column
theorem det_mul_column (v : n → R) (A : Matrix n n R) :
    det (of fun i j => v i * A i j) = (∏ i, v i) * det A 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (v : n → R) (A : Matrix n n R)
Goal: (Matrix.of fun i j => v i * A i j).det = (∏ i : n, v i) * A.det
-- Matrix.det_pow
@[simp]
theorem det_pow (M : Matrix m m R) (n : ℕ) : det (M ^ n) = det M ^ n 
Arguments: {m : Type u_1} [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] (M : Matrix m m R) (n : ℕ)
Goal: (M ^ n).det = M.det ^ n
-- RingHom.map_det
theorem _root_.RingHom.map_det (f : R →+* S) (M : Matrix n n R) :
    f M.det = Matrix.det (f.mapMatrix M) 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type w} [CommRing S] (f : R →+* S) (M : Matrix n n R)
Goal: f M.det = (f.mapMatrix M).det
-- RingEquiv.map_det
theorem _root_.RingEquiv.map_det (f : R ≃+* S) (M : Matrix n n R) :
    f M.det = Matrix.det (f.mapMatrix M) 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type w} [CommRing S] (f : R ≃+* S) (M : Matrix n n R)
Goal: f M.det = (f.mapMatrix M).det
-- AlgHom.map_det
theorem _root_.AlgHom.map_det [Algebra R S] {T : Type z} [CommRing T] [Algebra R T] (f : S →ₐ[R] T)
    (M : Matrix n n S) : f M.det = Matrix.det (f.mapMatrix M) 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type w} [CommRing S] [Algebra R S] {T : Type z} [CommRing T] [Algebra R T] (f : S →ₐ[R] T) (M : Matrix n n S)
Goal: f M.det = (f.mapMatrix M).det
-- AlgEquiv.map_det
theorem _root_.AlgEquiv.map_det [Algebra R S] {T : Type z} [CommRing T] [Algebra R T]
    (f : S ≃ₐ[R] T) (M : Matrix n n S) : f M.det = Matrix.det (f.mapMatrix M) 
Arguments: {n : Type u_2} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type w} [CommRing S] [Algebra R S] {T : Type z} [CommRing T] [Algebra R T] (f : S ≃ₐ[R] T) (M : Matrix n n S)
Goal: f M.det = (f.mapMatrix M).det
-- Matrix.det_conjTranspose
@[simp]
theorem det_conjTranspose [StarRing R] (M : Matrix m m R) : det Mᴴ = star (det M) 
Arguments: {m : Type u_1} [DecidableEq m] [Fintype m] {R : Type v} [CommRing R] [StarRing R] (M : Matrix m m R)
Goal: M.conjTranspose.det = star M.det