ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Basis.constr_apply_fintype
@[simp]
theorem Basis.constr_apply_fintype [Fintype ι] (b : Basis ι R M) (f : ι → M') (x : M) :
    (constr (M' := M') b S f : M → M') x = ∑ i, b.equivFun x i • f i 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] [Fintype ι] (b : Basis ι R M) (f : ι → M') (x : M)
Goal: ((b.constr S) f) x = ∑ i : ι, b.equivFun x i • f i
-- Basis.mem_submodule_iff'
theorem Basis.mem_submodule_iff' [Fintype ι] {P : Submodule R M} (b : Basis ι R P) {x : M} :
    x ∈ P ↔ ∃ c : ι → R, x = ∑ i, c i • (b i : M) 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι] {P : Submodule R M} (b : Basis ι R ↥P) {x : M}
Goal: x ∈ P ↔ ∃ c, x = ∑ i : ι, c i • ↑(b i)
-- Basis.coord_equivFun_symm
theorem Basis.coord_equivFun_symm [Finite ι] (b : Basis ι R M) (i : ι) (f : ι → R) :
    b.coord i (b.equivFun.symm f) = f i 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Finite ι] (b : Basis ι R M) (i : ι) (f : ι → R)
Goal: (b.coord i) (b.equivFun.symm f) = f i
-- Basis.equiv'_apply
@[simp]
theorem equiv'_apply (f : M → M') (g : M' → M) (hf hg hgf hfg) (i : ι) :
    b.equiv' b' f g hf hg hgf hfg (b i) = f (b i) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (b' : Basis ι' R M') (f : M → M') (g : M' → M) (hf : ∀ (i : ι), f (b i) ∈ Set.range ⇑b') (hg : ∀ (i : ι'), g (b' i) ∈ Set.range ⇑b) (hgf : ∀ (i : ι), g (f (b i)) = b i) (hfg : ∀ (i : ι'), f (g (b' i)) = b' i) (i : ι)
Goal: (b.equiv' b' f g hf hg hgf hfg) (b i) = f (b i)
-- Basis.equiv'_symm_apply
@[simp]
theorem equiv'_symm_apply (f : M → M') (g : M' → M) (hf hg hgf hfg) (i : ι') :
    (b.equiv' b' f g hf hg hgf hfg).symm (b' i) = g (b' i) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (b' : Basis ι' R M') (f : M → M') (g : M' → M) (hf : ∀ (i : ι), f (b i) ∈ Set.range ⇑b') (hg : ∀ (i : ι'), g (b' i) ∈ Set.range ⇑b) (hgf : ∀ (i : ι), g (f (b i)) = b i) (hfg : ∀ (i : ι'), f (g (b' i)) = b' i) (i : ι')
Goal: (b.equiv' b' f g hf hg hgf hfg).symm (b' i) = g (b' i)
-- Basis.sum_repr_mul_repr
theorem sum_repr_mul_repr {ι'} [Fintype ι'] (b' : Basis ι' R M) (x : M) (i : ι) :
    (∑ j : ι', b.repr (b' j) i * b'.repr x j) = b.repr x i 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {ι' : Type u_10} [Fintype ι'] (b' : Basis ι' R M) (x : M) (i : ι)
Goal: ∑ j : ι', (b.repr (b' j)) i * (b'.repr x) j = (b.repr x) i
-- Basis.maximal
theorem maximal [Nontrivial R] (b : Basis ι R M) : b.linearIndependent.Maximal 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Ring R] [AddCommGroup M] [Module R M] [Nontrivial R] (b : Basis ι R M)
Goal: ⋯.Maximal
-- Basis.mk_repr
@[simp]
theorem mk_repr : (Basis.mk hli hsp).repr x = hli.repr ⟨x, hsp Submodule.mem_top⟩ 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] {x : M} (hli : LinearIndependent R v) (hsp : ⊤ ≤ Submodule.span R (Set.range v))
Goal: (Basis.mk hli hsp).repr x = hli.repr ⟨x, ⋯⟩
-- Basis.mk_apply
theorem mk_apply (i : ι) : Basis.mk hli hsp i = v i 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hli : LinearIndependent R v) (hsp : ⊤ ≤ Submodule.span R (Set.range v)) (i : ι)
Goal: (Basis.mk hli hsp) i = v i
-- Basis.coe_mk
@[simp]
theorem coe_mk : ⇑(Basis.mk hli hsp) = v 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hli : LinearIndependent R v) (hsp : ⊤ ≤ Submodule.span R (Set.range v))
Goal: ⇑(Basis.mk hli hsp) = v
-- Basis.mk_coord_apply_eq
theorem mk_coord_apply_eq (i : ι) : (Basis.mk hli hsp).coord i (v i) = 1 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] {hli : LinearIndependent R v} {hsp : ⊤ ≤ Submodule.span R (Set.range v)} (i : ι)
Goal: ((Basis.mk hli hsp).coord i) (v i) = 1
-- Basis.mk_coord_apply_ne
theorem mk_coord_apply_ne {i j : ι} (h : j ≠ i) : (Basis.mk hli hsp).coord i (v j) = 0 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] {hli : LinearIndependent R v} {hsp : ⊤ ≤ Submodule.span R (Set.range v)} {i j : ι} (h : j ≠ i)
Goal: ((Basis.mk hli hsp).coord i) (v j) = 0
-- Basis.mk_coord_apply
theorem mk_coord_apply [DecidableEq ι] {i j : ι} :
    (Basis.mk hli hsp).coord i (v j) = if j = i then 1 else 0 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] {hli : LinearIndependent R v} {hsp : ⊤ ≤ Submodule.span R (Set.range v)} [DecidableEq ι] {i j : ι}
Goal: ((Basis.mk hli hsp).coord i) (v j) = if j = i then 1 else 0
-- Basis.span_apply
protected theorem span_apply (i : ι) : (Basis.span hli i : M) = v i 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hli : LinearIndependent R v) (i : ι)
Goal: ↑((Basis.span hli) i) = v i
-- Basis.groupSMul_span_eq_top
theorem groupSMul_span_eq_top {G : Type*} [Group G] [DistribMulAction G R] [DistribMulAction G M]
    [IsScalarTower G R M] {v : ι → M} (hv : Submodule.span R (Set.range v) = ⊤) {w : ι → G} :
    Submodule.span R (Set.range (w • v)) = ⊤ 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Ring R] [AddCommGroup M] [Module R M] {G : Type u_10} [Group G] [DistribMulAction G R] [DistribMulAction G M] [IsScalarTower G R M] {v : ι → M} (hv : Submodule.span R (Set.range v) = ⊤) {w : ι → G}
Goal: Submodule.span R (Set.range (w • v)) = ⊤
-- Basis.groupSMul_apply
theorem groupSMul_apply {G : Type*} [Group G] [DistribMulAction G R] [DistribMulAction G M]
    [IsScalarTower G R M] [SMulCommClass G R M] {v : Basis ι R M} {w : ι → G} (i : ι) :
    v.groupSMul w i = (w • (v : ι → M)) i 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Ring R] [AddCommGroup M] [Module R M] {G : Type u_10} [Group G] [DistribMulAction G R] [DistribMulAction G M] [IsScalarTower G R M] [SMulCommClass G R M] {v : Basis ι R M} {w : ι → G} (i : ι)
Goal: (v.groupSMul w) i = (w • ⇑v) i
-- Basis.units_smul_span_eq_top
theorem units_smul_span_eq_top {v : ι → M} (hv : Submodule.span R (Set.range v) = ⊤) {w : ι → Rˣ} :
    Submodule.span R (Set.range (w • v)) = ⊤ 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} (hv : Submodule.span R (Set.range v) = ⊤) {w : ι → Rˣ}
Goal: Submodule.span R (Set.range (w • v)) = ⊤
-- Basis.unitsSMul_apply
theorem unitsSMul_apply {v : Basis ι R M} {w : ι → Rˣ} (i : ι) : unitsSMul v w i = w i • v i 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Ring R] [AddCommGroup M] [Module R M] {v : Basis ι R M} {w : ι → Rˣ} (i : ι)
Goal: (v.unitsSMul w) i = w i • v i
-- Basis.coord_unitsSMul
@[simp]
theorem coord_unitsSMul (e : Basis ι R₂ M) (w : ι → R₂ˣ) (i : ι) :
    (unitsSMul e w).coord i = (w i)⁻¹ • e.coord i 
Arguments: {ι : Type u_1} {R₂ : Type u_4} {M : Type u_6} [CommRing R₂] [AddCommGroup M] [Module R₂ M] (e : Basis ι R₂ M) (w : ι → R₂ˣ) (i : ι)
Goal: (e.unitsSMul w).coord i = (w i)⁻¹ • e.coord i
-- Basis.repr_unitsSMul
@[simp]
theorem repr_unitsSMul (e : Basis ι R₂ M) (w : ι → R₂ˣ) (v : M) (i : ι) :
    (e.unitsSMul w).repr v i = (w i)⁻¹ • e.repr v i 
Arguments: {ι : Type u_1} {R₂ : Type u_4} {M : Type u_6} [CommRing R₂] [AddCommGroup M] [Module R₂ M] (e : Basis ι R₂ M) (w : ι → R₂ˣ) (v : M) (i : ι)
Goal: ((e.unitsSMul w).repr v) i = (w i)⁻¹ • (e.repr v) i
-- Basis.isUnitSMul_apply
theorem isUnitSMul_apply {v : Basis ι R M} {w : ι → R} (hw : ∀ i, IsUnit (w i)) (i : ι) :
    v.isUnitSMul hw i = w i • v i 
Arguments: {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Ring R] [AddCommGroup M] [Module R M] {v : Basis ι R M} {w : ι → R} (hw : ∀ (i : ι), IsUnit (w i)) (i : ι)
Goal: (v.isUnitSMul hw) i = w i • v i
-- Basis.coe_mkFinCons
@[simp]
theorem coe_mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R N)
    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :
    (mkFinCons y b hli hsp : Fin (n + 1) → M) = Fin.cons y ((↑) ∘ b) 
Arguments: {R : Type u_3} {M : Type u_6} [Ring R] [AddCommGroup M] [Module R M] {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R ↥N) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ (z : M), ∃ c, z + c • y ∈ N)
Goal: ⇑(Basis.mkFinCons y b hli hsp) = Fin.cons y (Subtype.val ∘ ⇑b)
-- Basis.coe_mkFinConsOfLE
@[simp]
theorem coe_mkFinConsOfLE {n : ℕ} {N O : Submodule R M} (y : M) (yO : y ∈ O) (b : Basis (Fin n) R N)
    (hNO : N ≤ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)
    (hsp : ∀ z ∈ O, ∃ c : R, z + c • y ∈ N) :
    (mkFinConsOfLE y yO b hNO hli hsp : Fin (n + 1) → O) =
      Fin.cons ⟨y, yO⟩ (Submodule.inclusion hNO ∘ b) 
Arguments: {R : Type u_3} {M : Type u_6} [Ring R] [AddCommGroup M] [Module R M] {n : ℕ} {N O : Submodule R M} (y : M) (yO : y ∈ O) (b : Basis (Fin n) R ↥N) (hNO : N ≤ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z ∈ O, ∃ c, z + c • y ∈ N)
Goal: ⇑(Basis.mkFinConsOfLE y yO b hNO hli hsp) = Fin.cons ⟨y, yO⟩ (⇑(Submodule.inclusion hNO) ∘ ⇑b)
-- Basis.finTwoProd_zero
@[simp]
theorem finTwoProd_zero (R : Type*) [Semiring R] : Basis.finTwoProd R 0 = (1, 0) 
Arguments: (R : Type u_10) [Semiring R]
Goal: (Basis.finTwoProd R) 0 = (1, 0)
-- Basis.finTwoProd_one
@[simp]
theorem finTwoProd_one (R : Type*) [Semiring R] : Basis.finTwoProd R 1 = (0, 1) 
Arguments: (R : Type u_10) [Semiring R]
Goal: (Basis.finTwoProd R) 1 = (0, 1)
-- Basis.coe_finTwoProd_repr
@[simp]
theorem coe_finTwoProd_repr {R : Type*} [Semiring R] (x : R × R) :
    ⇑((Basis.finTwoProd R).repr x) = ![x.fst, x.snd] 
Arguments: {R : Type u_10} [Semiring R] (x : R × R)
Goal: ⇑((Basis.finTwoProd R).repr x) = ![x.1, x.2]
-- Basis.mem_center_iff
lemma Basis.mem_center_iff {A}
    [Semiring R] [NonUnitalNonAssocSemiring A]
    [Module R A] [SMulCommClass R A A] [SMulCommClass R R A] [IsScalarTower R A A]
    (b : Basis ι R A) {z : A} :
    z ∈ Set.center A ↔
      (∀ i, Commute (b i) z) ∧ ∀ i j,
        z * (b i * b j) = (z * b i) * b j
          ∧ (b i * z) * b j = b i * (z * b j)
          ∧ (b i * b j) * z = b i * (b j * z) 
Arguments: {ι : Type u_1} {R : Type u_3} {A : Type u_10} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [SMulCommClass R R A] [IsScalarTower R A A] (b : Basis ι R A) {z : A}
Goal: z ∈ Set.center A ↔
  (∀ (i : ι), Commute (b i) z) ∧
    ∀ (i j : ι), z * (b i * b j) = z * b i * b j ∧ b i * z * b j = b i * (z * b j) ∧ b i * b j * z = b i * (b j * z)
-- Basis.restrictScalars_apply
@[simp]
theorem Basis.restrictScalars_apply (i : ι) : (b.restrictScalars R i : M) = b i 
Arguments: {ι : Type u_1} (R : Type u_3) {M : Type u_6} {S : Type u_10} [CommRing R] [Ring S] [Nontrivial S] [AddCommGroup M] [Algebra R S] [Module S M] [Module R M] [IsScalarTower R S M] [NoZeroSMulDivisors R S] (b : Basis ι S M) (i : ι)
Goal: ↑((Basis.restrictScalars R b) i) = b i
-- Basis.restrictScalars_repr_apply
@[simp]
theorem Basis.restrictScalars_repr_apply (m : span R (Set.range b)) (i : ι) :
    algebraMap R S ((b.restrictScalars R).repr m i) = b.repr m i 
Arguments: {ι : Type u_1} (R : Type u_3) {M : Type u_6} {S : Type u_10} [CommRing R] [Ring S] [Nontrivial S] [AddCommGroup M] [Algebra R S] [Module S M] [Module R M] [IsScalarTower R S M] [NoZeroSMulDivisors R S] (b : Basis ι S M) (m : ↥(Submodule.span R (Set.range ⇑b))) (i : ι)
Goal: (algebraMap R S) (((Basis.restrictScalars R b).repr m) i) = (b.repr ↑m) i
-- Basis.mem_span_iff_repr_mem
theorem Basis.mem_span_iff_repr_mem (m : M) :
    m ∈ span R (Set.range b) ↔ ∀ i, b.repr m i ∈ Set.range (algebraMap R S) 
Arguments: {ι : Type u_1} (R : Type u_3) {M : Type u_6} {S : Type u_10} [CommRing R] [Ring S] [Nontrivial S] [AddCommGroup M] [Algebra R S] [Module S M] [Module R M] [IsScalarTower R S M] [NoZeroSMulDivisors R S] (b : Basis ι S M) (m : M)
Goal: m ∈ Submodule.span R (Set.range ⇑b) ↔ ∀ (i : ι), (b.repr m) i ∈ Set.range ⇑(algebraMap R S)
-- basis_finite_of_finite_spans
lemma basis_finite_of_finite_spans (w : Set M) (hw : w.Finite) (s : span R w = ⊤) {ι : Type w}
    (b : Basis ι R M) : Finite ι 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Nontrivial R] [Module R M] (w : Set M) (hw : w.Finite) (s : Submodule.span R w = ⊤) {ι : Type w} (b : Basis ι R M)
Goal: Finite ι
-- union_support_maximal_linearIndependent_eq_range_basis
theorem union_support_maximal_linearIndependent_eq_range_basis {ι : Type w} (b : Basis ι R M)
    {κ : Type w'} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal) :
    ⋃ k, ((b.repr (v k)).support : Set ι) = Set.univ 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Nontrivial R] [Module R M] {ι : Type w} (b : Basis ι R M) {κ : Type w'} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal)
Goal: ⋃ k, ↑(b.repr (v k)).support = Set.univ
-- infinite_basis_le_maximal_linearIndependent'
theorem infinite_basis_le_maximal_linearIndependent' {ι : Type w} (b : Basis ι R M) [Infinite ι]
    {κ : Type w'} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal) :
    Cardinal.lift.{w'} #ι ≤ Cardinal.lift.{w} #κ 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Nontrivial R] [Module R M] {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w'} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal)
Goal: Cardinal.lift.{w', w} (Cardinal.mk ι) ≤ Cardinal.lift.{w, w'} (Cardinal.mk κ)
-- infinite_basis_le_maximal_linearIndependent
theorem infinite_basis_le_maximal_linearIndependent {ι : Type w} (b : Basis ι R M) [Infinite ι]
    {κ : Type w} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal) : #ι ≤ #κ 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Nontrivial R] [Module R M] {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal)
Goal: Cardinal.mk ι ≤ Cardinal.mk κ
-- AlternatingMap.map_eq_zero_of_eq'
structure AlternatingMap extends MultilinearMap R (fun _ : ι => M) N where
    map_eq_zero_of_eq' : ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → toFun v = 0
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (self : M [⋀^ι]→ₗ[R] N) (v : ι → M) (i j : ι)
Goal: v i = v j → i ≠ j → (↑self).toFun v = 0
-- AlternatingMap.toFun_eq_coe
@[simp]
theorem toFun_eq_coe : f.toFun = f 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N)
Goal: (↑f).toFun = ⇑f
-- AlternatingMap.coe_mk
@[simp]
theorem coe_mk (f : MultilinearMap R (fun _ : ι => M) N) (h) :
    ⇑(⟨f, h⟩ : M [⋀^ι]→ₗ[R] N) = f 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : MultilinearMap R (fun x => M) N) (h : ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → f.toFun v = 0)
Goal: ⇑{ toMultilinearMap := f, map_eq_zero_of_eq' := h } = ⇑f
-- AlternatingMap.congr_fun
protected theorem congr_fun {f g : M [⋀^ι]→ₗ[R] N} (h : f = g) (x : ι → M) : f x = g x 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {f g : M [⋀^ι]→ₗ[R] N} (h : f = g) (x : ι → M)
Goal: f x = g x
-- AlternatingMap.congr_arg
protected theorem congr_arg (f : M [⋀^ι]→ₗ[R] N) {x y : ι → M} (h : x = y) : f x = f y 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) {x y : ι → M} (h : x = y)
Goal: f x = f y
-- AlternatingMap.coe_injective
theorem coe_injective : Injective ((↑) : M [⋀^ι]→ₗ[R] N → (ι → M) → N) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7}
Goal: Function.Injective DFunLike.coe
-- AlternatingMap.coe_inj
@[norm_cast] -- @[simp] -- Porting note (#10618): simp can prove this
theorem coe_inj {f g : M [⋀^ι]→ₗ[R] N} : (f : (ι → M) → N) = g ↔ f = g 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {f g : M [⋀^ι]→ₗ[R] N}
Goal: ⇑f = ⇑g ↔ f = g
-- AlternatingMap.ext
@[ext]
theorem ext {f f' : M [⋀^ι]→ₗ[R] N} (H : ∀ x, f x = f' x) : f = f' 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {f f' : M [⋀^ι]→ₗ[R] N} (H : ∀ (x : ι → M), f x = f' x)
Goal: f = f'
-- AlternatingMap.ext_iff
protected theorem ext_iff {f g : M [⋀^ι]→ₗ[R] N} : f = g ↔ ∀ x, f x = g x 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {f g : M [⋀^ι]→ₗ[R] N}
Goal: f = g ↔ ∀ (x : ι → M), f x = g x
-- AlternatingMap.coe_multilinearMap
@[simp, norm_cast]
theorem coe_multilinearMap : ⇑(f : MultilinearMap R (fun _ : ι => M) N) = f 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N)
Goal: ⇑↑f = ⇑f
-- AlternatingMap.coe_multilinearMap_injective
theorem coe_multilinearMap_injective :
    Function.Injective ((↑) : M [⋀^ι]→ₗ[R] N → MultilinearMap R (fun _ : ι => M) N) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7}
Goal: Function.Injective AlternatingMap.toMultilinearMap
-- AlternatingMap.coe_multilinearMap_mk
theorem coe_multilinearMap_mk (f : (ι → M) → N) (h₁ h₂ h₃) :
    ((⟨⟨f, h₁, h₂⟩, h₃⟩ : M [⋀^ι]→ₗ[R] N) : MultilinearMap R (fun _ : ι => M) N) =
      ⟨f, @h₁, @h₂⟩ 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : (ι → M) → N) (h₁ :
  ∀ [inst : DecidableEq ι] (m : ι → M) (i : ι) (x y : M),
    f (Function.update m i (x + y)) = f (Function.update m i x) + f (Function.update m i y)) (h₂ :
  ∀ [inst : DecidableEq ι] (m : ι → M) (i : ι) (c : R) (x : M),
    f (Function.update m i (c • x)) = c • f (Function.update m i x)) (h₃ : ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → { toFun := f, map_add' := ⋯, map_smul' := ⋯ }.toFun v = 0)
Goal: ↑{ toFun := f, map_add' := ⋯, map_smul' := ⋯, map_eq_zero_of_eq' := h₃ } =
  { toFun := f, map_add' := h₁, map_smul' := h₂ }
-- AlternatingMap.map_add
@[simp]
theorem map_add [DecidableEq ι] (i : ι) (x y : M) :
    f (update v i (x + y)) = f (update v i x) + f (update v i y) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] (i : ι) (x y : M)
Goal: f (Function.update v i (x + y)) = f (Function.update v i x) + f (Function.update v i y)
-- AlternatingMap.map_sub
@[simp]
theorem map_sub [DecidableEq ι] (i : ι) (x y : M') :
    g' (update v' i (x - y)) = g' (update v' i x) - g' (update v' i y) 
Arguments: {R : Type u_1} [Semiring R] {M' : Type u_5} [AddCommGroup M'] [Module R M'] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g' : M' [⋀^ι]→ₗ[R] N') (v' : ι → M') [DecidableEq ι] (i : ι) (x y : M')
Goal: g' (Function.update v' i (x - y)) = g' (Function.update v' i x) - g' (Function.update v' i y)
-- AlternatingMap.map_neg
@[simp]
theorem map_neg [DecidableEq ι] (i : ι) (x : M') : g' (update v' i (-x)) = -g' (update v' i x) 
Arguments: {R : Type u_1} [Semiring R] {M' : Type u_5} [AddCommGroup M'] [Module R M'] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g' : M' [⋀^ι]→ₗ[R] N') (v' : ι → M') [DecidableEq ι] (i : ι) (x : M')
Goal: g' (Function.update v' i (-x)) = -g' (Function.update v' i x)
-- AlternatingMap.map_smul
@[simp]
theorem map_smul [DecidableEq ι] (i : ι) (r : R) (x : M) :
    f (update v i (r • x)) = r • f (update v i x) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] (i : ι) (r : R) (x : M)
Goal: f (Function.update v i (r • x)) = r • f (Function.update v i x)
-- AlternatingMap.map_eq_zero_of_eq
@[simp]
theorem map_eq_zero_of_eq (v : ι → M) {i j : ι} (h : v i = v j) (hij : i ≠ j) : f v = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) {i j : ι} (h : v i = v j) (hij : i ≠ j)
Goal: f v = 0
-- AlternatingMap.map_coord_zero
theorem map_coord_zero {m : ι → M} (i : ι) (h : m i = 0) : f m = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) {m : ι → M} (i : ι) (h : m i = 0)
Goal: f m = 0
-- AlternatingMap.map_update_zero
@[simp]
theorem map_update_zero [DecidableEq ι] (m : ι → M) (i : ι) : f (update m i 0) = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) [DecidableEq ι] (m : ι → M) (i : ι)
Goal: f (Function.update m i 0) = 0
-- AlternatingMap.map_zero
@[simp]
theorem map_zero [Nonempty ι] : f 0 = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) [Nonempty ι]
Goal: f 0 = 0
-- AlternatingMap.map_eq_zero_of_not_injective
theorem map_eq_zero_of_not_injective (v : ι → M) (hv : ¬Function.Injective v) : f v = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) (hv : ¬Function.Injective v)
Goal: f v = 0
-- AlternatingMap.smul_apply
@[simp]
theorem smul_apply (c : S) (m : ι → M) : (c • f) m = c • f m 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) {S : Type u_10} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] (c : S) (m : ι → M)
Goal: (c • f) m = c • f m
-- AlternatingMap.coe_smul
@[norm_cast]
theorem coe_smul (c : S) : ↑(c • f) = c • (f : MultilinearMap R (fun _ : ι => M) N) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) {S : Type u_10} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] (c : S)
Goal: ↑(c • f) = c • ↑f
-- AlternatingMap.coeFn_smul
theorem coeFn_smul (c : S) (f : M [⋀^ι]→ₗ[R] N) : ⇑(c • f) = c • ⇑f 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {S : Type u_10} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] (c : S) (f : M [⋀^ι]→ₗ[R] N)
Goal: ⇑(c • f) = c • ⇑f
-- AlternatingMap.prod_apply
@[simps!]
def prod (f : M [⋀^ι]→ₗ[R] N) (g : M [⋀^ι]→ₗ[R] P) : M [⋀^ι]→ₗ[R] (N × P) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {P : Type u_4} [AddCommMonoid P] [Module R P] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (g : M [⋀^ι]→ₗ[R] P) (m : (i : ι) → (fun x => M) i)
Goal: (f.prod g) m = (f m, g m)