ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ExteriorAlgebra.algebraMap_leftInverse
theorem algebraMap_leftInverse :
    Function.LeftInverse algebraMapInv (algebraMap R <| ExteriorAlgebra R M) 
Arguments: {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M]
Goal: Function.LeftInverse ⇑ExteriorAlgebra.algebraMapInv ⇑(algebraMap R (ExteriorAlgebra R M))
-- ExteriorAlgebra.algebraMap_inj
@[simp]
theorem algebraMap_inj (x y : R) :
    algebraMap R (ExteriorAlgebra R M) x = algebraMap R (ExteriorAlgebra R M) y ↔ x = y 
Arguments: {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (x y : R)
Goal: (algebraMap R (ExteriorAlgebra R M)) x = (algebraMap R (ExteriorAlgebra R M)) y ↔ x = y
-- ExteriorAlgebra.algebraMap_eq_zero_iff
@[simp]
theorem algebraMap_eq_zero_iff (x : R) : algebraMap R (ExteriorAlgebra R M) x = 0 ↔ x = 0 
Arguments: {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (x : R)
Goal: (algebraMap R (ExteriorAlgebra R M)) x = 0 ↔ x = 0
-- ExteriorAlgebra.algebraMap_eq_one_iff
@[simp]
theorem algebraMap_eq_one_iff (x : R) : algebraMap R (ExteriorAlgebra R M) x = 1 ↔ x = 1 
Arguments: {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (x : R)
Goal: (algebraMap R (ExteriorAlgebra R M)) x = 1 ↔ x = 1
-- ExteriorAlgebra.isUnit_algebraMap
theorem isUnit_algebraMap (r : R) : IsUnit (algebraMap R (ExteriorAlgebra R M) r) ↔ IsUnit r 
Arguments: {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (r : R)
Goal: IsUnit ((algebraMap R (ExteriorAlgebra R M)) r) ↔ IsUnit r
-- ExteriorAlgebra.invertibleAlgebraMapEquiv_apply_invOf
@[simps!]
def invertibleAlgebraMapEquiv (r : R) :
    Invertible (algebraMap R (ExteriorAlgebra R M) r) ≃ Invertible r 
Arguments: {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (r : R)
Goal: ∀ (x : Invertible ((algebraMap R (ExteriorAlgebra R M)) r)),
  ⅟r = ⅟(ExteriorAlgebra.algebraMapInv ((algebraMap R (ExteriorAlgebra R M)) r))
-- ExteriorAlgebra.invertibleAlgebraMapEquiv_symm_apply_invOf_toQuot
@[simps!]
def invertibleAlgebraMapEquiv (r : R) :
    Invertible (algebraMap R (ExteriorAlgebra R M) r) ≃ Invertible r 
Arguments: {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (r : R)
Goal: ∀ (x : Invertible r),
  (⅟((algebraMap R (ExteriorAlgebra R M)) r)).toQuot =
    Quot.mk (RingQuot.Rel (CliffordAlgebra.Rel 0)) ((algebraMap R (TensorAlgebra R M)) ⅟r)
-- ExteriorAlgebra.toTrivSqZeroExt_ι
@[simp]
theorem toTrivSqZeroExt_ι [Module Rᵐᵒᵖ M] [IsCentralScalar R M] (x : M) :
    toTrivSqZeroExt (ι R x) = TrivSqZeroExt.inr x 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] (x : M)
Goal: ExteriorAlgebra.toTrivSqZeroExt ((ExteriorAlgebra.ι R) x) = TrivSqZeroExt.inr x
-- ExteriorAlgebra.ι_leftInverse
theorem ι_leftInverse : Function.LeftInverse ιInv (ι R : M → ExteriorAlgebra R M) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M]
Goal: Function.LeftInverse ⇑ExteriorAlgebra.ιInv ⇑(ExteriorAlgebra.ι R)
-- ExteriorAlgebra.ι_inj
@[simp]
theorem ι_inj (x y : M) : ι R x = ι R y ↔ x = y 
Arguments: (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (x y : M)
Goal: (ExteriorAlgebra.ι R) x = (ExteriorAlgebra.ι R) y ↔ x = y
-- ExteriorAlgebra.ι_eq_zero_iff
@[simp]
theorem ι_eq_zero_iff (x : M) : ι R x = 0 ↔ x = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (x : M)
Goal: (ExteriorAlgebra.ι R) x = 0 ↔ x = 0
-- ExteriorAlgebra.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 u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (x : M) (r : R)
Goal: (ExteriorAlgebra.ι R) x = (algebraMap R (ExteriorAlgebra R M)) r ↔ x = 0 ∧ r = 0
-- ExteriorAlgebra.ι_ne_one
@[simp]
theorem ι_ne_one [Nontrivial R] (x : M) : ι R x ≠ 1 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] [Nontrivial R] (x : M)
Goal: (ExteriorAlgebra.ι R) x ≠ 1
-- ExteriorAlgebra.ι_range_disjoint_one
theorem ι_range_disjoint_one :
    Disjoint (LinearMap.range (ι R : M →ₗ[R] ExteriorAlgebra R M))
      (1 : Submodule R (ExteriorAlgebra R M)) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M]
Goal: Disjoint (LinearMap.range (ExteriorAlgebra.ι R)) 1
-- ExteriorAlgebra.ι_add_mul_swap
@[simp]
theorem ι_add_mul_swap (x y : M) : ι R x * ι R y + ι R y * ι R x = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (x y : M)
Goal: (ExteriorAlgebra.ι R) x * (ExteriorAlgebra.ι R) y + (ExteriorAlgebra.ι R) y * (ExteriorAlgebra.ι R) x = 0
-- ExteriorAlgebra.ι_mul_prod_list
theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
    (ι R <| f i) * (List.ofFn fun i => ι R <| f i).prod = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {n : ℕ} (f : Fin n → M) (i : Fin n)
Goal: (ExteriorAlgebra.ι R) (f i) * (List.ofFn fun i => (ExteriorAlgebra.ι R) (f i)).prod = 0
-- ExteriorAlgebra.ιMulti_apply
theorem ιMulti_apply {n : ℕ} (v : Fin n → M) : ιMulti R n v = (List.ofFn fun i => ι R (v i)).prod 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {n : ℕ} (v : Fin n → M)
Goal: (ExteriorAlgebra.ιMulti R n) v = (List.ofFn fun i => (ExteriorAlgebra.ι R) (v i)).prod
-- ExteriorAlgebra.ιMulti_zero_apply
@[simp]
theorem ιMulti_zero_apply (v : Fin 0 → M) : ιMulti R 0 v = 1 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (v : Fin 0 → M)
Goal: (ExteriorAlgebra.ιMulti R 0) v = 1
-- ExteriorAlgebra.ιMulti_succ_apply
@[simp]
theorem ιMulti_succ_apply {n : ℕ} (v : Fin n.succ → M) :
    ιMulti R _ v = ι R (v 0) * ιMulti R _ (Matrix.vecTail v) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {n : ℕ} (v : Fin n.succ → M)
Goal: (ExteriorAlgebra.ιMulti R n.succ) v = (ExteriorAlgebra.ι R) (v 0) * (ExteriorAlgebra.ιMulti R n) (Matrix.vecTail v)
-- ExteriorAlgebra.ιMulti_succ_curryLeft
theorem ιMulti_succ_curryLeft {n : ℕ} (m : M) :
    (ιMulti R n.succ).curryLeft m = (LinearMap.mulLeft R (ι R m)).compAlternatingMap (ιMulti R n) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {n : ℕ} (m : M)
Goal: (ExteriorAlgebra.ιMulti R n.succ).curryLeft m =
  (LinearMap.mulLeft R ((ExteriorAlgebra.ι R) m)).compAlternatingMap (ExteriorAlgebra.ιMulti R n)
-- ExteriorAlgebra.ιMulti_range
lemma ιMulti_range (n : ℕ) :
    Set.range (ιMulti R n (M := M)) ⊆ ↑(⋀[R]^n M) 
Arguments: (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (n : ℕ)
Goal: Set.range ⇑(ExteriorAlgebra.ιMulti R n) ⊆ ↑(⋀[R]^n M)
-- ExteriorAlgebra.ιMulti_span_fixedDegree
lemma ιMulti_span_fixedDegree (n : ℕ) :
    Submodule.span R (Set.range (ιMulti R n)) = ⋀[R]^n M 
Arguments: (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (n : ℕ)
Goal: Submodule.span R (Set.range ⇑(ExteriorAlgebra.ιMulti R n)) = ⋀[R]^n M
-- ExteriorAlgebra.map_comp_ι
@[simp]
theorem map_comp_ι (f : M →ₗ[R] N) : (map f).toLinearMap ∘ₗ ι R = ι R ∘ₗ f 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N)
Goal: (ExteriorAlgebra.map f).toLinearMap ∘ₗ ExteriorAlgebra.ι R = ExteriorAlgebra.ι R ∘ₗ f
-- ExteriorAlgebra.map_apply_ι
@[simp]
theorem map_apply_ι (f : M →ₗ[R] N) (m : M) : map f (ι R m) = ι R (f m) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) (m : M)
Goal: (ExteriorAlgebra.map f) ((ExteriorAlgebra.ι R) m) = (ExteriorAlgebra.ι R) (f m)
-- ExteriorAlgebra.map_apply_ιMulti
@[simp]
theorem map_apply_ιMulti {n : ℕ} (f : M →ₗ[R] N) (m : Fin n → M) :
    map f (ιMulti R n m) = ιMulti R n (f ∘ m) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {n : ℕ} (f : M →ₗ[R] N) (m : Fin n → M)
Goal: (ExteriorAlgebra.map f) ((ExteriorAlgebra.ιMulti R n) m) = (ExteriorAlgebra.ιMulti R n) (⇑f ∘ m)
-- ExteriorAlgebra.map_comp_ιMulti
@[simp]
theorem map_comp_ιMulti {n : ℕ} (f : M →ₗ[R] N) :
    (map f).toLinearMap.compAlternatingMap (ιMulti R n (M := M)) =
    (ιMulti R n (M := N)).compLinearMap f 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {n : ℕ} (f : M →ₗ[R] N)
Goal: (ExteriorAlgebra.map f).toLinearMap.compAlternatingMap (ExteriorAlgebra.ιMulti R n) =
  (ExteriorAlgebra.ιMulti R n).compLinearMap f
-- ExteriorAlgebra.map_id
@[simp]
theorem map_id :
    map LinearMap.id = AlgHom.id R (ExteriorAlgebra R M) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M]
Goal: ExteriorAlgebra.map LinearMap.id = AlgHom.id R (ExteriorAlgebra R M)
-- ExteriorAlgebra.map_comp_map
@[simp]
theorem map_comp_map (f : M →ₗ[R] N) (g : N →ₗ[R] N') :
    AlgHom.comp (map g) (map f) = map (LinearMap.comp g f) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} {N' : Type u5} [AddCommGroup N] [Module R N] [AddCommGroup N'] [Module R N'] (f : M →ₗ[R] N) (g : N →ₗ[R] N')
Goal: (ExteriorAlgebra.map g).comp (ExteriorAlgebra.map f) = ExteriorAlgebra.map (g ∘ₗ f)
-- ExteriorAlgebra.ι_range_map_map
@[simp]
theorem ι_range_map_map (f : M →ₗ[R] N) :
    Submodule.map (AlgHom.toLinearMap (map f)) (LinearMap.range (ι R (M := M))) =
    Submodule.map (ι R) (LinearMap.range f) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N)
Goal: Submodule.map (ExteriorAlgebra.map f).toLinearMap (LinearMap.range (ExteriorAlgebra.ι R)) =
  Submodule.map (ExteriorAlgebra.ι R) (LinearMap.range f)
-- ExteriorAlgebra.toTrivSqZeroExt_comp_map
theorem toTrivSqZeroExt_comp_map [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [Module Rᵐᵒᵖ N]
    [IsCentralScalar R N] (f : M →ₗ[R] N) :
    toTrivSqZeroExt.comp (map f) = (TrivSqZeroExt.map f).comp toTrivSqZeroExt 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [Module Rᵐᵒᵖ N] [IsCentralScalar R N] (f : M →ₗ[R] N)
Goal: ExteriorAlgebra.toTrivSqZeroExt.comp (ExteriorAlgebra.map f) =
  (TrivSqZeroExt.map f).comp ExteriorAlgebra.toTrivSqZeroExt
-- ExteriorAlgebra.ιInv_comp_map
theorem ιInv_comp_map (f : M →ₗ[R] N) :
    ιInv.comp (map f).toLinearMap = f.comp ιInv 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N)
Goal: ExteriorAlgebra.ιInv ∘ₗ (ExteriorAlgebra.map f).toLinearMap = f ∘ₗ ExteriorAlgebra.ιInv
-- ExteriorAlgebra.leftInverse_map_iff
@[simp]
lemma leftInverse_map_iff {f : M →ₗ[R] N} {g : N →ₗ[R] M} :
    LeftInverse (map g) (map f) ↔ LeftInverse g f 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N} {g : N →ₗ[R] M}
Goal: Function.LeftInverse ⇑(ExteriorAlgebra.map g) ⇑(ExteriorAlgebra.map f) ↔ Function.LeftInverse ⇑g ⇑f
-- ExteriorAlgebra.map_injective
lemma map_injective {f : M →ₗ[R] N} (hf : ∃ (g : N →ₗ[R] M), g.comp f = LinearMap.id) :
    Function.Injective (map f) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N} (hf : ∃ g, g ∘ₗ f = LinearMap.id)
Goal: Function.Injective ⇑(ExteriorAlgebra.map f)
-- ExteriorAlgebra.map_surjective_iff
@[simp]
lemma map_surjective_iff {f : M →ₗ[R] N} :
    Function.Surjective (map f) ↔ Function.Surjective f 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N}
Goal: Function.Surjective ⇑(ExteriorAlgebra.map f) ↔ Function.Surjective ⇑f
-- ExteriorAlgebra.map_injective_field
lemma map_injective_field {f : E →ₗ[K] F} (hf : LinearMap.ker f = ⊥) :
    Function.Injective (map f) 
Arguments: {K : Type u_2} {E : Type u_3} {F : Type u_4} [Field K] [AddCommGroup E] [Module K E] [AddCommGroup F] [Module K F] {f : E →ₗ[K] F} (hf : LinearMap.ker f = ⊥)
Goal: Function.Injective ⇑(ExteriorAlgebra.map f)
-- TensorAlgebra.toExterior_ι
@[simp]
theorem toExterior_ι (m : M) :
    TensorAlgebra.toExterior (TensorAlgebra.ι R m) = ExteriorAlgebra.ι R m 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (m : M)
Goal: TensorAlgebra.toExterior ((TensorAlgebra.ι R) m) = (ExteriorAlgebra.ι R) m
-- mk_eq_mk_of_basis
theorem mk_eq_mk_of_basis (v : Basis ι R M) (v' : Basis ι' R M) :
    Cardinal.lift.{w'} #ι = Cardinal.lift.{w} #ι' 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {ι : Type w} {ι' : Type w'} [InvariantBasisNumber R] (v : Basis ι R M) (v' : Basis ι' R M)
Goal: Cardinal.lift.{w', w} #ι = Cardinal.lift.{w, w'} #ι'
-- mk_eq_mk_of_basis'
theorem mk_eq_mk_of_basis' {ι' : Type w} (v : Basis ι R M) (v' : Basis ι' R M) : #ι = #ι' 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {ι : Type w} [InvariantBasisNumber R] {ι' : Type w} (v : Basis ι R M) (v' : Basis ι' R M)
Goal: #ι = #ι'
-- Basis.le_span''
theorem Basis.le_span'' {ι : Type*} [Fintype ι] (b : Basis ι R M) {w : Set M} [Fintype w]
    (s : span R w = ⊤) : Fintype.card ι ≤ Fintype.card w 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [RankCondition R] {ι : Type u_1} [Fintype ι] (b : Basis ι R M) {w : Set M} [Fintype ↑w] (s : Submodule.span R w = ⊤)
Goal: Fintype.card ι ≤ Fintype.card ↑w
-- basis_le_span'
theorem basis_le_span' {ι : Type*} (b : Basis ι R M) {w : Set M} [Fintype w] (s : span R w = ⊤) :
    #ι ≤ Fintype.card w 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [RankCondition R] {ι : Type u_1} (b : Basis ι R M) {w : Set M} [Fintype ↑w] (s : Submodule.span R w = ⊤)
Goal: #ι ≤ ↑(Fintype.card ↑w)
-- Basis.le_span
theorem Basis.le_span {J : Set M} (v : Basis ι R M) (hJ : span R J = ⊤) : #(range v) ≤ #J 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {ι : Type w} [RankCondition R] {J : Set M} (v : Basis ι R M) (hJ : Submodule.span R J = ⊤)
Goal: #↑(Set.range ⇑v) ≤ #↑J
-- linearIndependent_le_span_aux'
theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
    (i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
    Fintype.card ι ≤ Fintype.card w 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type u_1} [Fintype ι] (v : ι → M) (i : LinearIndependent R v) (w : Set M) [Fintype ↑w] (s : Set.range v ≤ ↑(Submodule.span R w))
Goal: Fintype.card ι ≤ Fintype.card ↑w
-- LinearIndependent.finite_of_le_span_finite
lemma LinearIndependent.finite_of_le_span_finite {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
    (w : Set M) [Finite w] (s : range v ≤ span R w) : Finite ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type u_1} (v : ι → M) (i : LinearIndependent R v) (w : Set M) [Finite ↑w] (s : Set.range v ≤ ↑(Submodule.span R w))
Goal: Finite ι
-- linearIndependent_le_span'
theorem linearIndependent_le_span' {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
    [Fintype w] (s : range v ≤ span R w) : #ι ≤ Fintype.card w 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type u_1} (v : ι → M) (i : LinearIndependent R v) (w : Set M) [Fintype ↑w] (s : Set.range v ≤ ↑(Submodule.span R w))
Goal: #ι ≤ ↑(Fintype.card ↑w)
-- linearIndependent_le_span
theorem linearIndependent_le_span {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
    [Fintype w] (s : span R w = ⊤) : #ι ≤ Fintype.card w 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type u_1} (v : ι → M) (i : LinearIndependent R v) (w : Set M) [Fintype ↑w] (s : Submodule.span R w = ⊤)
Goal: #ι ≤ ↑(Fintype.card ↑w)
-- linearIndependent_le_span_finset
theorem linearIndependent_le_span_finset {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
    (w : Finset M) (s : span R (w : Set M) = ⊤) : #ι ≤ w.card 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type u_1} (v : ι → M) (i : LinearIndependent R v) (w : Finset M) (s : Submodule.span R ↑w = ⊤)
Goal: #ι ≤ ↑w.card
-- linearIndependent_le_infinite_basis
theorem linearIndependent_le_infinite_basis {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w}
    (v : κ → M) (i : LinearIndependent R v) : #κ ≤ #ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w} (v : κ → M) (i : LinearIndependent R v)
Goal: #κ ≤ #ι
-- linearIndependent_le_basis
theorem linearIndependent_le_basis {ι : Type w} (b : Basis ι R M) {κ : Type w} (v : κ → M)
    (i : LinearIndependent R v) : #κ ≤ #ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type w} (b : Basis ι R M) {κ : Type w} (v : κ → M) (i : LinearIndependent R v)
Goal: #κ ≤ #ι
-- Basis.card_le_card_of_linearIndependent_aux
theorem Basis.card_le_card_of_linearIndependent_aux {R : Type*} [Ring R] [StrongRankCondition R]
    (n : ℕ) {m : ℕ} (v : Fin m → Fin n → R) : LinearIndependent R v → m ≤ n 
Arguments: {R : Type u_1} [Ring R] [StrongRankCondition R] (n : ℕ) {m : ℕ} (v : Fin m → Fin n → R)
Goal: LinearIndependent R v → m ≤ n
-- maximal_linearIndependent_eq_infinite_basis
theorem maximal_linearIndependent_eq_infinite_basis {ι : Type w} (b : Basis ι R M) [Infinite ι]
    {κ : Type w} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal) : #κ = #ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal)
Goal: #κ = #ι
-- Basis.mk_eq_rank''
theorem Basis.mk_eq_rank'' {ι : Type v} (v : Basis ι R M) : #ι = Module.rank R M 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type v} (v : Basis ι R M)
Goal: #ι = Module.rank R M
-- Basis.mk_range_eq_rank
theorem Basis.mk_range_eq_rank (v : Basis ι R M) : #(range v) = Module.rank R M 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {ι : Type w} [StrongRankCondition R] (v : Basis ι R M)
Goal: #↑(Set.range ⇑v) = Module.rank R M
-- rank_eq_card_basis
theorem rank_eq_card_basis {ι : Type w} [Fintype ι] (h : Basis ι R M) :
    Module.rank R M = Fintype.card ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type w} [Fintype ι] (h : Basis ι R M)
Goal: Module.rank R M = ↑(Fintype.card ι)
-- Basis.card_le_card_of_linearIndependent
theorem Basis.card_le_card_of_linearIndependent {ι : Type*} [Fintype ι] (b : Basis ι R M)
    {ι' : Type*} [Fintype ι'] {v : ι' → M} (hv : LinearIndependent R v) :
    Fintype.card ι' ≤ Fintype.card ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type u_1} [Fintype ι] (b : Basis ι R M) {ι' : Type u_2} [Fintype ι'] {v : ι' → M} (hv : LinearIndependent R v)
Goal: Fintype.card ι' ≤ Fintype.card ι
-- Basis.card_le_card_of_submodule
theorem Basis.card_le_card_of_submodule (N : Submodule R M) [Fintype ι] (b : Basis ι R M)
    [Fintype ι'] (b' : Basis ι' R N) : Fintype.card ι' ≤ Fintype.card ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {ι : Type w} {ι' : Type w'} [StrongRankCondition R] (N : Submodule R M) [Fintype ι] (b : Basis ι R M) [Fintype ι'] (b' : Basis ι' R ↥N)
Goal: Fintype.card ι' ≤ Fintype.card ι
-- Basis.card_le_card_of_le
theorem Basis.card_le_card_of_le {N O : Submodule R M} (hNO : N ≤ O) [Fintype ι] (b : Basis ι R O)
    [Fintype ι'] (b' : Basis ι' R N) : Fintype.card ι' ≤ Fintype.card ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {ι : Type w} {ι' : Type w'} [StrongRankCondition R] {N O : Submodule R M} (hNO : N ≤ O) [Fintype ι] (b : Basis ι R ↥O) [Fintype ι'] (b' : Basis ι' R ↥N)
Goal: Fintype.card ι' ≤ Fintype.card ι
-- Basis.mk_eq_rank
theorem Basis.mk_eq_rank (v : Basis ι R M) :
    Cardinal.lift.{v} #ι = Cardinal.lift.{w} (Module.rank R M) 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {ι : Type w} [StrongRankCondition R] (v : Basis ι R M)
Goal: Cardinal.lift.{v, w} #ι = Cardinal.lift.{w, v} (Module.rank R M)
-- Basis.mk_eq_rank'
theorem Basis.mk_eq_rank'.{m} (v : Basis ι R M) :
    Cardinal.lift.{max v m} #ι = Cardinal.lift.{max w m} (Module.rank R M) 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {ι : Type w} [StrongRankCondition R] (v : Basis ι R M)
Goal: Cardinal.lift.{max v m, w} #ι = Cardinal.lift.{max w m, v} (Module.rank R M)
-- rank_span
theorem rank_span {v : ι → M} (hv : LinearIndependent R v) :
    Module.rank R ↑(span R (range v)) = #(range v) 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {ι : Type w} [StrongRankCondition R] {v : ι → M} (hv : LinearIndependent R v)
Goal: Module.rank R ↥(Submodule.span R (Set.range v)) = #↑(Set.range v)
-- rank_span_set
theorem rank_span_set {s : Set M} (hs : LinearIndependent R (fun x => x : s → M)) :
    Module.rank R ↑(span R s) = #s 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {s : Set M} (hs : LinearIndependent R fun (x : ↑s) => ↑x)
Goal: Module.rank R ↥(Submodule.span R s) = #↑s
-- Ideal.rank_eq
theorem Ideal.rank_eq {R S : Type*} [CommRing R] [StrongRankCondition R] [Ring S] [IsDomain S]
    [Algebra R S] {n m : Type*} [Fintype n] [Fintype m] (b : Basis n R S) {I : Ideal S}
    (hI : I ≠ ⊥) (c : Basis m R I) : Fintype.card m = Fintype.card n 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [StrongRankCondition R] [Ring S] [IsDomain S] [Algebra R S] {n : Type u_3} {m : Type u_4} [Fintype n] [Fintype m] (b : Basis n R S) {I : Ideal S} (hI : I ≠ ⊥) (c : Basis m R ↥I)
Goal: Fintype.card m = Fintype.card n
-- finrank_eq_nat_card_basis
theorem finrank_eq_nat_card_basis (h : Basis ι R M) :
    finrank R M = Nat.card ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {ι : Type w} [StrongRankCondition R] (h : Basis ι R M)
Goal: FiniteDimensional.finrank R M = Nat.card ι
-- FiniteDimensional.finrank_eq_card_basis
theorem finrank_eq_card_basis {ι : Type w} [Fintype ι] (h : Basis ι R M) :
    finrank R M = Fintype.card ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type w} [Fintype ι] (h : Basis ι R M)
Goal: FiniteDimensional.finrank R M = Fintype.card ι
-- Module.mk_finrank_eq_card_basis
theorem _root_.Module.mk_finrank_eq_card_basis [Module.Finite R M]
    {ι : Type w} (h : Basis ι R M) : (finrank R M : Cardinal.{w}) = #ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] [Module.Finite R M] {ι : Type w} (h : Basis ι R M)
Goal: ↑(FiniteDimensional.finrank R M) = #ι
-- FiniteDimensional.finrank_eq_card_finset_basis
theorem finrank_eq_card_finset_basis {ι : Type w} {b : Finset ι} (h : Basis b R M) :
    finrank R M = Finset.card b 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] {ι : Type w} {b : Finset ι} (h : Basis { x // x ∈ b } R M)
Goal: FiniteDimensional.finrank R M = b.card
-- rank_self
@[simp]
theorem rank_self : Module.rank R R = 1 
Arguments: (R : Type u) [Ring R] [StrongRankCondition R]
Goal: Module.rank R R = 1
-- FiniteDimensional.finrank_self
@[simp]
theorem FiniteDimensional.finrank_self : finrank R R = 1 
Arguments: (R : Type u) [Ring R] [StrongRankCondition R]
Goal: FiniteDimensional.finrank R R = 1
-- rank_lt_aleph0
theorem rank_lt_aleph0 [Module.Finite R M] : Module.rank R M < ℵ₀ 
Arguments: (R : Type u) (M : Type v) [Ring R] [AddCommGroup M] [Module R M] [StrongRankCondition R] [Module.Finite R M]
Goal: Module.rank R M < ℵ₀