ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- locallyConnectedSpace_iff_open_connected_subsets
theorem locallyConnectedSpace_iff_open_connected_subsets :
    LocallyConnectedSpace α ↔
      ∀ x, ∀ U ∈ 𝓝 x, ∃ V : Set α, V ⊆ U ∧ IsOpen V ∧ x ∈ V ∧ IsConnected V 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen V ∧ x ∈ V ∧ IsConnected V
-- connectedComponentIn_mem_nhds
theorem connectedComponentIn_mem_nhds [LocallyConnectedSpace α] {F : Set α} {x : α} (h : F ∈ 𝓝 x) :
    connectedComponentIn F x ∈ 𝓝 x 
Arguments: {α : Type u} [TopologicalSpace α] [LocallyConnectedSpace α] {F : Set α} {x : α} (h : F ∈ 𝓝 x)
Goal: connectedComponentIn F x ∈ 𝓝 x
-- IsOpen.connectedComponentIn
protected theorem IsOpen.connectedComponentIn [LocallyConnectedSpace α] {F : Set α} {x : α}
    (hF : IsOpen F) : IsOpen (connectedComponentIn F x) 
Arguments: {α : Type u} [TopologicalSpace α] [LocallyConnectedSpace α] {F : Set α} {x : α} (hF : IsOpen F)
Goal: IsOpen (connectedComponentIn F x)
-- isOpen_connectedComponent
theorem isOpen_connectedComponent [LocallyConnectedSpace α] {x : α} :
    IsOpen (connectedComponent x) 
Arguments: {α : Type u} [TopologicalSpace α] [LocallyConnectedSpace α] {x : α}
Goal: IsOpen (connectedComponent x)
-- isClopen_connectedComponent
theorem isClopen_connectedComponent [LocallyConnectedSpace α] {x : α} :
    IsClopen (connectedComponent x) 
Arguments: {α : Type u} [TopologicalSpace α] [LocallyConnectedSpace α] {x : α}
Goal: IsClopen (connectedComponent x)
-- locallyConnectedSpace_iff_connectedComponentIn_open
theorem locallyConnectedSpace_iff_connectedComponentIn_open :
    LocallyConnectedSpace α ↔
      ∀ F : Set α, IsOpen F → ∀ x ∈ F, IsOpen (connectedComponentIn F x) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: LocallyConnectedSpace α ↔ ∀ (F : Set α), IsOpen F → ∀ x ∈ F, IsOpen (connectedComponentIn F x)
-- locallyConnectedSpace_iff_connected_subsets
theorem locallyConnectedSpace_iff_connected_subsets :
    LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U
-- locallyConnectedSpace_iff_connected_basis
theorem locallyConnectedSpace_iff_connected_basis :
    LocallyConnectedSpace α ↔
      ∀ x, (𝓝 x).HasBasis (fun s : Set α => s ∈ 𝓝 x ∧ IsPreconnected s) id 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: LocallyConnectedSpace α ↔ ∀ (x : α), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsPreconnected s) id
-- locallyConnectedSpace_of_connected_bases
theorem locallyConnectedSpace_of_connected_bases {ι : Type*} (b : α → ι → Set α) (p : α → ι → Prop)
    (hbasis : ∀ x, (𝓝 x).HasBasis (p x) (b x))
    (hconnected : ∀ x i, p x i → IsPreconnected (b x i)) : LocallyConnectedSpace α 
Arguments: {α : Type u} [TopologicalSpace α] {ι : Type u_3} (b : α → ι → Set α) (p : α → ι → Prop) (hbasis : ∀ (x : α), (𝓝 x).HasBasis (p x) (b x)) (hconnected : ∀ (x : α) (i : ι), p x i → IsPreconnected (b x i))
Goal: LocallyConnectedSpace α
-- OpenEmbedding.locallyConnectedSpace
theorem OpenEmbedding.locallyConnectedSpace [LocallyConnectedSpace α] [TopologicalSpace β]
    {f : β → α} (h : OpenEmbedding f) : LocallyConnectedSpace β 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [LocallyConnectedSpace α] [TopologicalSpace β] {f : β → α} (h : OpenEmbedding f)
Goal: LocallyConnectedSpace β
-- IsOpen.locallyConnectedSpace
theorem IsOpen.locallyConnectedSpace [LocallyConnectedSpace α] {U : Set α} (hU : IsOpen U) :
    LocallyConnectedSpace U 
Arguments: {α : Type u} [TopologicalSpace α] [LocallyConnectedSpace α] {U : Set α} (hU : IsOpen U)
Goal: LocallyConnectedSpace ↑U
-- ContinuousAlternatingMap.toContinuousMultilinearMap_injective
theorem toContinuousMultilinearMap_injective :
    Injective (ContinuousAlternatingMap.toContinuousMultilinearMap :
      M [⋀^ι]→L[R] N → ContinuousMultilinearMap R (fun _ : ι => M) N)
  
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N]
Goal: Function.Injective ContinuousAlternatingMap.toContinuousMultilinearMap
-- ContinuousAlternatingMap.range_toContinuousMultilinearMap
theorem range_toContinuousMultilinearMap :
    Set.range
        (toContinuousMultilinearMap :
          M [⋀^ι]→L[R] N → ContinuousMultilinearMap R (fun _ : ι => M) N) =
      {f | ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → f v = 0} 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N]
Goal: Set.range ContinuousAlternatingMap.toContinuousMultilinearMap =
  {f | ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → f v = 0}
-- ContinuousAlternatingMap.coe_continuous
@[continuity]
theorem coe_continuous : Continuous f 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N)
Goal: Continuous ⇑f
-- ContinuousAlternatingMap.coe_toContinuousMultilinearMap
@[simp]
theorem coe_toContinuousMultilinearMap : ⇑f.toContinuousMultilinearMap = f 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N)
Goal: ⇑f.toContinuousMultilinearMap = ⇑f
-- ContinuousAlternatingMap.coe_mk
@[simp]
theorem coe_mk (f : ContinuousMultilinearMap R (fun _ : ι => M) N) (h) : ⇑(mk f h) = f 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : ContinuousMultilinearMap R (fun x => M) N) (h : ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → f.toFun v = 0)
Goal: ⇑{ toContinuousMultilinearMap := f, map_eq_zero_of_eq' := h } = ⇑f
-- ContinuousAlternatingMap.coe_toAlternatingMap
theorem coe_toAlternatingMap : ⇑f.toAlternatingMap = f 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N)
Goal: ⇑f.toAlternatingMap = ⇑f
-- ContinuousAlternatingMap.ext
@[ext]
theorem ext {f g : M [⋀^ι]→L[R] N} (H : ∀ x, f x = g x) : f = g 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] {f g : M [⋀^ι]→L[R] N} (H : ∀ (x : ι → M), f x = g x)
Goal: f = g
-- ContinuousAlternatingMap.ext_iff
theorem ext_iff {f g : M [⋀^ι]→L[R] N} : f = g ↔ ∀ x, f x = g x 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] {f g : M [⋀^ι]→L[R] N}
Goal: f = g ↔ ∀ (x : ι → M), f x = g x
-- ContinuousAlternatingMap.toAlternatingMap_injective
theorem toAlternatingMap_injective :
    Injective (toAlternatingMap : (M [⋀^ι]→L[R] N) → (M [⋀^ι]→ₗ[R] N)) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N]
Goal: Function.Injective ContinuousAlternatingMap.toAlternatingMap
-- ContinuousAlternatingMap.range_toAlternatingMap
@[simp]
theorem range_toAlternatingMap :
    Set.range (toAlternatingMap : M [⋀^ι]→L[R] N → (M [⋀^ι]→ₗ[R] N)) =
      {f : M [⋀^ι]→ₗ[R] N | Continuous f} 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N]
Goal: Set.range ContinuousAlternatingMap.toAlternatingMap = {f | Continuous ⇑f}
-- ContinuousAlternatingMap.map_add
@[simp]
theorem map_add [DecidableEq ι] (m : ι → M) (i : ι) (x y : M) :
    f (update m i (x + y)) = f (update m i x) + f (update m i y) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [DecidableEq ι] (m : ι → M) (i : ι) (x y : M)
Goal: f (Function.update m i (x + y)) = f (Function.update m i x) + f (Function.update m i y)
-- ContinuousAlternatingMap.map_smul
@[simp]
theorem map_smul [DecidableEq ι] (m : ι → M) (i : ι) (c : R) (x : M) :
    f (update m i (c • x)) = c • f (update m i x) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [DecidableEq ι] (m : ι → M) (i : ι) (c : R) (x : M)
Goal: f (Function.update m i (c • x)) = c • f (Function.update m i x)
-- ContinuousAlternatingMap.map_coord_zero
theorem map_coord_zero {m : ι → M} (i : ι) (h : m i = 0) : f m = 0 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) {m : ι → M} (i : ι) (h : m i = 0)
Goal: f m = 0
-- ContinuousAlternatingMap.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} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [DecidableEq ι] (m : ι → M) (i : ι)
Goal: f (Function.update m i 0) = 0
-- ContinuousAlternatingMap.map_zero
@[simp]
theorem map_zero [Nonempty ι] : f 0 = 0 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [Nonempty ι]
Goal: f 0 = 0
-- ContinuousAlternatingMap.map_eq_zero_of_eq
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} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) (v : ι → M) {i j : ι} (h : v i = v j) (hij : i ≠ j)
Goal: f v = 0
-- ContinuousAlternatingMap.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} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) (v : ι → M) (hv : ¬Function.Injective v)
Goal: f v = 0
-- ContinuousAlternatingMap.codRestrict_apply_coe
@[simps!]
def codRestrict (f : M [⋀^ι]→L[R] N) (p : Submodule R N) (h : ∀ v, f v ∈ p) : M [⋀^ι]→L[R] p 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) (p : Submodule R N) (h : ∀ (v : ι → M), f v ∈ p) (v : (i : ι) → (fun x => M) i)
Goal: ↑((f.codRestrict p h) v) = f v
-- ContinuousAlternatingMap.coe_zero
@[simp]
theorem coe_zero : ⇑(0 : M [⋀^ι]→L[R] N) = 0 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N]
Goal: ⇑0 = 0
-- ContinuousAlternatingMap.toContinuousMultilinearMap_zero
@[simp]
theorem toContinuousMultilinearMap_zero : (0 : M [⋀^ι]→L[R] N).toContinuousMultilinearMap = 0 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N]
Goal: ContinuousAlternatingMap.toContinuousMultilinearMap 0 = 0
-- ContinuousAlternatingMap.toAlternatingMap_zero
@[simp]
theorem toAlternatingMap_zero : (0 : M [⋀^ι]→L[R] N).toAlternatingMap = 0 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N]
Goal: ContinuousAlternatingMap.toAlternatingMap 0 = 0
-- ContinuousAlternatingMap.coe_smul
@[simp]
theorem coe_smul (f : M [⋀^ι]→L[A] N) (c : R') : ⇑(c • f) = c • ⇑f 
Arguments: {M : Type u_2} {N : Type u_4} {ι : Type u_6} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {R' : Type u_7} {A : Type u_9} [Monoid R'] [Semiring A] [Module A M] [Module A N] [DistribMulAction R' N] [ContinuousConstSMul R' N] [SMulCommClass A R' N] (f : M [⋀^ι]→L[A] N) (c : R')
Goal: ⇑(c • f) = c • ⇑f
-- ContinuousAlternatingMap.smul_apply
theorem smul_apply (f : M [⋀^ι]→L[A] N) (c : R') (v : ι → M) : (c • f) v = c • f v 
Arguments: {M : Type u_2} {N : Type u_4} {ι : Type u_6} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {R' : Type u_7} {A : Type u_9} [Monoid R'] [Semiring A] [Module A M] [Module A N] [DistribMulAction R' N] [ContinuousConstSMul R' N] [SMulCommClass A R' N] (f : M [⋀^ι]→L[A] N) (c : R') (v : ι → M)
Goal: (c • f) v = c • f v
-- ContinuousAlternatingMap.toContinuousMultilinearMap_smul
@[simp]
theorem toContinuousMultilinearMap_smul (c : R') (f : M [⋀^ι]→L[A] N) :
    (c • f).toContinuousMultilinearMap = c • f.toContinuousMultilinearMap 
Arguments: {M : Type u_2} {N : Type u_4} {ι : Type u_6} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {R' : Type u_7} {A : Type u_9} [Monoid R'] [Semiring A] [Module A M] [Module A N] [DistribMulAction R' N] [ContinuousConstSMul R' N] [SMulCommClass A R' N] (c : R') (f : M [⋀^ι]→L[A] N)
Goal: (c • f).toContinuousMultilinearMap = c • f.toContinuousMultilinearMap
-- ContinuousAlternatingMap.toAlternatingMap_smul
@[simp]
theorem toAlternatingMap_smul (c : R') (f : M [⋀^ι]→L[A] N) :
    (c • f).toAlternatingMap = c • f.toAlternatingMap 
Arguments: {M : Type u_2} {N : Type u_4} {ι : Type u_6} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {R' : Type u_7} {A : Type u_9} [Monoid R'] [Semiring A] [Module A M] [Module A N] [DistribMulAction R' N] [ContinuousConstSMul R' N] [SMulCommClass A R' N] (c : R') (f : M [⋀^ι]→L[A] N)
Goal: (c • f).toAlternatingMap = c • f.toAlternatingMap
-- ContinuousAlternatingMap.coe_add
@[simp]
theorem coe_add : ⇑(f + g) = ⇑f + ⇑g 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f g : M [⋀^ι]→L[R] N) [ContinuousAdd N]
Goal: ⇑(f + g) = ⇑f + ⇑g
-- ContinuousAlternatingMap.add_apply
@[simp]
theorem add_apply (v : ι → M) : (f + g) v = f v + g v 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f g : M [⋀^ι]→L[R] N) [ContinuousAdd N] (v : ι → M)
Goal: (f + g) v = f v + g v
-- ContinuousAlternatingMap.toContinuousMultilinearMap_add
@[simp]
theorem toContinuousMultilinearMap_add (f g : M [⋀^ι]→L[R] N) : (f + g).1 = f.1 + g.1 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [ContinuousAdd N] (f g : M [⋀^ι]→L[R] N)
Goal: (f + g).toContinuousMultilinearMap = f.toContinuousMultilinearMap + g.toContinuousMultilinearMap
-- ContinuousAlternatingMap.toAlternatingMap_add
@[simp]
theorem toAlternatingMap_add (f g : M [⋀^ι]→L[R] N) :
    (f + g).toAlternatingMap = f.toAlternatingMap + g.toAlternatingMap 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [ContinuousAdd N] (f g : M [⋀^ι]→L[R] N)
Goal: (f + g).toAlternatingMap = f.toAlternatingMap + g.toAlternatingMap
-- ContinuousAlternatingMap.sum_apply
@[simp]
theorem sum_apply {α : Type*} (f : α → M [⋀^ι]→L[R] N) (m : ι → M) {s : Finset α} :
    (∑ a ∈ s, f a) m = ∑ a ∈ s, f a m 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [ContinuousAdd N] {α : Type u_7} (f : α → M [⋀^ι]→L[R] N) (m : ι → M) {s : Finset α}
Goal: (∑ a ∈ s, f a) m = ∑ a ∈ s, (f a) m
-- ContinuousAlternatingMap.toMultilinearAddHom_apply
@[simps]
def toMultilinearAddHom : M [⋀^ι]→L[R] N →+ ContinuousMultilinearMap R (fun _ : ι => M) N 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [ContinuousAdd N] (f : M [⋀^ι]→L[R] N)
Goal: ContinuousAlternatingMap.toMultilinearAddHom f = f.toContinuousMultilinearMap
-- ContinuousAlternatingMap.toContinuousLinearMap_apply
@[simps! apply]
def toContinuousLinearMap [DecidableEq ι] (m : ι → M) (i : ι) : M →L[R] N 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [DecidableEq ι] (m : ι → M) (i : ι) (x : M)
Goal: (f.toContinuousLinearMap m i) x = f (Function.update m i x)
-- ContinuousAlternatingMap.prod_apply
@[simps!]
def prod (f : M [⋀^ι]→L[R] N) (g : M [⋀^ι]→L[R] N') : M [⋀^ι]→L[R] (N × N') 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {N' : Type u_5} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [AddCommMonoid N'] [Module R N'] [TopologicalSpace N'] (f : M [⋀^ι]→L[R] N) (g : M [⋀^ι]→L[R] N') (m : (i : ι) → (fun x => M) i)
Goal: (f.prod g) m = (f m, g m)
-- ContinuousAlternatingMap.coe_pi
@[simp]
theorem coe_pi {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
    [∀ i, TopologicalSpace (M' i)] [∀ i, Module R (M' i)] (f : ∀ i, M [⋀^ι]→L[R] M' i) :
    ⇑(pi f) = fun m j => f j m 
Arguments: {R : Type u_1} {M : Type u_2} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] {ι' : Type u_7} {M' : ι' → Type u_8} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : (i : ι') → M [⋀^ι]→L[R] M' i)
Goal: ⇑(ContinuousAlternatingMap.pi f) = fun m j => (f j) m
-- ContinuousAlternatingMap.pi_apply
theorem pi_apply {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
    [∀ i, TopologicalSpace (M' i)] [∀ i, Module R (M' i)] (f : ∀ i, M [⋀^ι]→L[R] M' i) (m : ι → M)
    (j : ι') : pi f m j = f j m 
Arguments: {R : Type u_1} {M : Type u_2} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] {ι' : Type u_7} {M' : ι' → Type u_8} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : (i : ι') → M [⋀^ι]→L[R] M' i) (m : ι → M) (j : ι')
Goal: (ContinuousAlternatingMap.pi f) m j = (f j) m
-- ContinuousAlternatingMap.ofSubsingleton_apply_toContinuousMultilinearMap
@[simps! apply_apply symm_apply_apply apply_toContinuousMultilinearMap]
def ofSubsingleton [Subsingleton ι] (i : ι) :
    (M →L[R] N) ≃ M [⋀^ι]→L[R] N 
Arguments: (R : Type u_1) (M : Type u_2) (N : Type u_4) {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [Subsingleton ι] (i : ι) (f : M →L[R] N)
Goal: ((ContinuousAlternatingMap.ofSubsingleton R M N i) f).toContinuousMultilinearMap =
  (ContinuousMultilinearMap.ofSubsingleton R M N i) f
-- ContinuousAlternatingMap.ofSubsingleton_apply_apply
@[simps! apply_apply symm_apply_apply apply_toContinuousMultilinearMap]
def ofSubsingleton [Subsingleton ι] (i : ι) :
    (M →L[R] N) ≃ M [⋀^ι]→L[R] N 
Arguments: (R : Type u_1) (M : Type u_2) (N : Type u_4) {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [Subsingleton ι] (i : ι) (f : M →L[R] N) (x : ι → M)
Goal: ((ContinuousAlternatingMap.ofSubsingleton R M N i) f) x = f (x i)
-- ContinuousAlternatingMap.ofSubsingleton_symm_apply_apply
@[simps! apply_apply symm_apply_apply apply_toContinuousMultilinearMap]
def ofSubsingleton [Subsingleton ι] (i : ι) :
    (M →L[R] N) ≃ M [⋀^ι]→L[R] N 
Arguments: (R : Type u_1) (M : Type u_2) (N : Type u_4) {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [Subsingleton ι] (i : ι) (f : M [⋀^ι]→L[R] N) (x : M)
Goal: ((ContinuousAlternatingMap.ofSubsingleton R M N i).symm f) x = f fun x_1 => x
-- ContinuousAlternatingMap.ofSubsingleton_toAlternatingMap
@[simp]
theorem ofSubsingleton_toAlternatingMap [Subsingleton ι] (i : ι) (f : M →L[R] N) :
    (ofSubsingleton R M N i f).toAlternatingMap = AlternatingMap.ofSubsingleton R M N i f 
Arguments: (R : Type u_1) (M : Type u_2) (N : Type u_4) {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [Subsingleton ι] (i : ι) (f : M →L[R] N)
Goal: ((ContinuousAlternatingMap.ofSubsingleton R M N i) f).toAlternatingMap = (AlternatingMap.ofSubsingleton R M N i) ↑f
-- ContinuousAlternatingMap.constOfIsEmpty_apply
@[simps! toContinuousMultilinearMap apply]
def constOfIsEmpty [IsEmpty ι] (m : N) : M [⋀^ι]→L[R] N 
Arguments: (R : Type u_1) (M : Type u_2) {N : Type u_4} (ι : Type u_6) [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [IsEmpty ι] (m : N)
Goal: ∀ (a : (i : ι) → (fun x => M) i), (ContinuousAlternatingMap.constOfIsEmpty R M ι m) a = m
-- ContinuousAlternatingMap.constOfIsEmpty_toContinuousMultilinearMap
@[simps! toContinuousMultilinearMap apply]
def constOfIsEmpty [IsEmpty ι] (m : N) : M [⋀^ι]→L[R] N 
Arguments: (R : Type u_1) (M : Type u_2) {N : Type u_4} (ι : Type u_6) [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [IsEmpty ι] (m : N)
Goal: (ContinuousAlternatingMap.constOfIsEmpty R M ι m).toContinuousMultilinearMap =
  ContinuousMultilinearMap.constOfIsEmpty R (fun x => M) m
-- ContinuousAlternatingMap.constOfIsEmpty_toAlternatingMap
@[simp]
theorem constOfIsEmpty_toAlternatingMap [IsEmpty ι] (m : N) :
    (constOfIsEmpty R M ι m).toAlternatingMap = AlternatingMap.constOfIsEmpty R M ι m 
Arguments: (R : Type u_1) (M : Type u_2) {N : Type u_4} (ι : Type u_6) [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [IsEmpty ι] (m : N)
Goal: (ContinuousAlternatingMap.constOfIsEmpty R M ι m).toAlternatingMap = AlternatingMap.constOfIsEmpty R M ι m
-- ContinuousAlternatingMap.compContinuousLinearMap_apply
@[simp]
theorem compContinuousLinearMap_apply (g : M [⋀^ι]→L[R] N) (f : M' →L[R] M) (m : ι → M') :
    g.compContinuousLinearMap f m = g (f ∘ m) 
Arguments: {R : Type u_1} {M : Type u_2} {M' : Type u_3} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid M'] [Module R M'] [TopologicalSpace M'] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (g : M [⋀^ι]→L[R] N) (f : M' →L[R] M) (m : ι → M')
Goal: (g.compContinuousLinearMap f) m = g (⇑f ∘ m)
-- ContinuousLinearMap.compContinuousAlternatingMap_coe
@[simp]
theorem _root_.ContinuousLinearMap.compContinuousAlternatingMap_coe (g : N →L[R] N')
    (f : M [⋀^ι]→L[R] N) : ⇑(g.compContinuousAlternatingMap f) = g ∘ f 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {N' : Type u_5} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [AddCommMonoid N'] [Module R N'] [TopologicalSpace N'] (g : N →L[R] N') (f : M [⋀^ι]→L[R] N)
Goal: ⇑(g.compContinuousAlternatingMap f) = ⇑g ∘ ⇑f