ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousMultilinearMap.toMultilinearMap_injective
theorem toMultilinearMap_injective :
    Function.Injective
      (ContinuousMultilinearMap.toMultilinearMap :
        ContinuousMultilinearMap R M₁ M₂ → MultilinearMap R M₁ M₂)
  
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂]
Goal: Function.Injective ContinuousMultilinearMap.toMultilinearMap
-- ContinuousMultilinearMap.coe_continuous
@[continuity]
theorem coe_continuous : Continuous (f : (∀ i, M₁ i) → M₂) 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂)
Goal: Continuous ⇑f
-- ContinuousMultilinearMap.coe_coe
@[simp]
theorem coe_coe : (f.toMultilinearMap : (∀ i, M₁ i) → M₂) = f 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂)
Goal: ⇑f.toMultilinearMap = ⇑f
-- ContinuousMultilinearMap.ext
@[ext]
theorem ext {f f' : ContinuousMultilinearMap R M₁ M₂} (H : ∀ x, f x = f' x) : f = f' 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {f f' : ContinuousMultilinearMap R M₁ M₂} (H : ∀ (x : (i : ι) → M₁ i), f x = f' x)
Goal: f = f'
-- ContinuousMultilinearMap.ext_iff
theorem ext_iff {f f' : ContinuousMultilinearMap R M₁ M₂} : f = f' ↔ ∀ x, f x = f' x 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {f f' : ContinuousMultilinearMap R M₁ M₂}
Goal: f = f' ↔ ∀ (x : (i : ι) → M₁ i), f x = f' x
-- ContinuousMultilinearMap.map_add
@[simp]
theorem map_add [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (x y : M₁ i) :
    f (update m i (x + y)) = f (update m i x) + f (update m i y) 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i)
Goal: f (Function.update m i (x + y)) = f (Function.update m i x) + f (Function.update m i y)
-- ContinuousMultilinearMap.map_smul
@[simp]
theorem map_smul [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (c : R) (x : M₁ i) :
    f (update m i (c • x)) = c • f (update m i x) 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i)
Goal: f (Function.update m i (c • x)) = c • f (Function.update m i x)
-- ContinuousMultilinearMap.map_coord_zero
theorem map_coord_zero {m : ∀ i, M₁ i} (i : ι) (h : m i = 0) : f m = 0 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) {m : (i : ι) → M₁ i} (i : ι) (h : m i = 0)
Goal: f m = 0
-- ContinuousMultilinearMap.map_zero
@[simp]
theorem map_zero [Nonempty ι] : f 0 = 0 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [Nonempty ι]
Goal: f 0 = 0
-- ContinuousMultilinearMap.zero_apply
@[simp]
theorem zero_apply (m : ∀ i, M₁ i) : (0 : ContinuousMultilinearMap R M₁ M₂) m = 0 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (m : (i : ι) → M₁ i)
Goal: 0 m = 0
-- ContinuousMultilinearMap.toMultilinearMap_zero
@[simp]
theorem toMultilinearMap_zero : (0 : ContinuousMultilinearMap R M₁ M₂).toMultilinearMap = 0 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂]
Goal: ContinuousMultilinearMap.toMultilinearMap 0 = 0
-- ContinuousMultilinearMap.smul_apply
@[simp]
theorem smul_apply (f : ContinuousMultilinearMap A M₁ M₂) (c : R') (m : ∀ i, M₁ i) :
    (c • f) m = c • f m 
Arguments: {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {R' : Type u_1} {A : Type u_3} [Monoid R'] [Semiring A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [DistribMulAction R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] (f : ContinuousMultilinearMap A M₁ M₂) (c : R') (m : (i : ι) → M₁ i)
Goal: (c • f) m = c • f m
-- ContinuousMultilinearMap.toMultilinearMap_smul
@[simp]
theorem toMultilinearMap_smul (c : R') (f : ContinuousMultilinearMap A M₁ M₂) :
    (c • f).toMultilinearMap = c • f.toMultilinearMap 
Arguments: {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {R' : Type u_1} {A : Type u_3} [Monoid R'] [Semiring A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [DistribMulAction R' M₂] [ContinuousConstSMul R' M₂] [SMulCommClass A R' M₂] (c : R') (f : ContinuousMultilinearMap A M₁ M₂)
Goal: (c • f).toMultilinearMap = c • f.toMultilinearMap
-- ContinuousMultilinearMap.add_apply
@[simp]
theorem add_apply (m : ∀ i, M₁ i) : (f + f') m = f m + f' m 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f f' : ContinuousMultilinearMap R M₁ M₂) [ContinuousAdd M₂] (m : (i : ι) → M₁ i)
Goal: (f + f') m = f m + f' m
-- ContinuousMultilinearMap.toMultilinearMap_add
@[simp]
theorem toMultilinearMap_add (f g : ContinuousMultilinearMap R M₁ M₂) :
    (f + g).toMultilinearMap = f.toMultilinearMap + g.toMultilinearMap 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousAdd M₂] (f g : ContinuousMultilinearMap R M₁ M₂)
Goal: (f + g).toMultilinearMap = f.toMultilinearMap + g.toMultilinearMap
-- ContinuousMultilinearMap.sum_apply
@[simp]
theorem sum_apply {α : Type*} (f : α → ContinuousMultilinearMap R M₁ M₂) (m : ∀ i, M₁ i)
    {s : Finset α} : (∑ a ∈ s, f a) m = ∑ a ∈ s, f a m 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [ContinuousAdd M₂] {α : Type u_1} (f : α → ContinuousMultilinearMap R M₁ M₂) (m : (i : ι) → M₁ i) {s : Finset α}
Goal: (∑ a ∈ s, f a) m = ∑ a ∈ s, (f a) m
-- ContinuousMultilinearMap.prod_apply
@[simp]
theorem prod_apply (f : ContinuousMultilinearMap R M₁ M₂) (g : ContinuousMultilinearMap R M₁ M₃)
    (m : ∀ i, M₁ i) : (f.prod g) m = (f m, g m) 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [AddCommMonoid M₃] [(i : ι) → Module R (M₁ i)] [Module R M₂] [Module R M₃] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [TopologicalSpace M₃] (f : ContinuousMultilinearMap R M₁ M₂) (g : ContinuousMultilinearMap R M₁ M₃) (m : (i : ι) → M₁ i)
Goal: (f.prod g) m = (f m, g m)
-- ContinuousMultilinearMap.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, ContinuousMultilinearMap R M₁ (M' i)) : ⇑(pi f) = fun m j => f j m 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : (i : ι') → ContinuousMultilinearMap R M₁ (M' i))
Goal: ⇑(ContinuousMultilinearMap.pi f) = fun m j => (f j) m
-- ContinuousMultilinearMap.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, ContinuousMultilinearMap R M₁ (M' i)) (m : ∀ i, M₁ i) (j : ι') : pi f m j = f j m 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : (i : ι') → ContinuousMultilinearMap R M₁ (M' i)) (m : (i : ι) → M₁ i) (j : ι')
Goal: (ContinuousMultilinearMap.pi f) m j = (f j) m
-- ContinuousMultilinearMap.codRestrict_toMultilinearMap
@[simps! toMultilinearMap apply_coe]
def codRestrict (f : ContinuousMultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ v, f v ∈ p) :
    ContinuousMultilinearMap R M₁ p 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ (v : (i : ι) → M₁ i), f v ∈ p)
Goal: (f.codRestrict p h).toMultilinearMap = f.codRestrict p h
-- ContinuousMultilinearMap.codRestrict_apply_coe
@[simps! toMultilinearMap apply_coe]
def codRestrict (f : ContinuousMultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ v, f v ∈ p) :
    ContinuousMultilinearMap R M₁ p 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ (v : (i : ι) → M₁ i), f v ∈ p) (v : (i : ι) → M₁ i)
Goal: ↑((f.codRestrict p h) v) = f v
-- ContinuousMultilinearMap.ofSubsingleton_symm_apply_apply
@[simps! apply_toMultilinearMap apply_apply symm_apply_apply]
def ofSubsingleton [Subsingleton ι] (i : ι) :
    (M₂ →L[R] M₃) ≃ ContinuousMultilinearMap R (fun _ : ι => M₂) M₃ 
Arguments: (R : Type u) {ι : Type v} (M₂ : Type w₂) (M₃ : Type w₃) [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] [Subsingleton ι] (i : ι) (f : ContinuousMultilinearMap R (fun x => M₂) M₃) (x : M₂)
Goal: ((ContinuousMultilinearMap.ofSubsingleton R M₂ M₃ i).symm f) x = f fun x_1 => x
-- ContinuousMultilinearMap.ofSubsingleton_apply_apply
@[simps! apply_toMultilinearMap apply_apply symm_apply_apply]
def ofSubsingleton [Subsingleton ι] (i : ι) :
    (M₂ →L[R] M₃) ≃ ContinuousMultilinearMap R (fun _ : ι => M₂) M₃ 
Arguments: (R : Type u) {ι : Type v} (M₂ : Type w₂) (M₃ : Type w₃) [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] [Subsingleton ι] (i : ι) (f : M₂ →L[R] M₃) (x : ι → M₂)
Goal: ((ContinuousMultilinearMap.ofSubsingleton R M₂ M₃ i) f) x = f (x i)
-- ContinuousMultilinearMap.ofSubsingleton_apply_toMultilinearMap
@[simps! apply_toMultilinearMap apply_apply symm_apply_apply]
def ofSubsingleton [Subsingleton ι] (i : ι) :
    (M₂ →L[R] M₃) ≃ ContinuousMultilinearMap R (fun _ : ι => M₂) M₃ 
Arguments: (R : Type u) {ι : Type v} (M₂ : Type w₂) (M₃ : Type w₃) [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] [Subsingleton ι] (i : ι) (f : M₂ →L[R] M₃)
Goal: ((ContinuousMultilinearMap.ofSubsingleton R M₂ M₃ i) f).toMultilinearMap = (MultilinearMap.ofSubsingleton R M₂ M₃ i) ↑f
-- ContinuousMultilinearMap.constOfIsEmpty_apply
@[simps! toMultilinearMap apply]
def constOfIsEmpty [IsEmpty ι] (m : M₂) : ContinuousMultilinearMap R M₁ M₂ 
Arguments: (R : Type u) {ι : Type v} (M₁ : ι → Type w₁) {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [IsEmpty ι] (m : M₂)
Goal: ∀ (a : (i : ι) → M₁ i), (ContinuousMultilinearMap.constOfIsEmpty R M₁ m) a = m
-- ContinuousMultilinearMap.constOfIsEmpty_toMultilinearMap
@[simps! toMultilinearMap apply]
def constOfIsEmpty [IsEmpty ι] (m : M₂) : ContinuousMultilinearMap R M₁ M₂ 
Arguments: (R : Type u) {ι : Type v} (M₁ : ι → Type w₁) {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [IsEmpty ι] (m : M₂)
Goal: (ContinuousMultilinearMap.constOfIsEmpty R M₁ m).toMultilinearMap = MultilinearMap.constOfIsEmpty R M₁ m
-- ContinuousMultilinearMap.compContinuousLinearMap_apply
@[simp]
theorem compContinuousLinearMap_apply (g : ContinuousMultilinearMap R M₁' M₄)
    (f : ∀ i : ι, M₁ i →L[R] M₁' i) (m : ∀ i, M₁ i) :
    g.compContinuousLinearMap f m = g fun i => f i <| m i 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₁' : ι → Type w₁'} {M₄ : Type w₄} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → AddCommMonoid (M₁' i)] [AddCommMonoid M₄] [(i : ι) → Module R (M₁ i)] [(i : ι) → Module R (M₁' i)] [Module R M₄] [(i : ι) → TopologicalSpace (M₁ i)] [(i : ι) → TopologicalSpace (M₁' i)] [TopologicalSpace M₄] (g : ContinuousMultilinearMap R M₁' M₄) (f : (i : ι) → M₁ i →L[R] M₁' i) (m : (i : ι) → M₁ i)
Goal: (g.compContinuousLinearMap f) m = g fun i => (f i) (m i)
-- ContinuousLinearMap.compContinuousMultilinearMap_coe
@[simp]
theorem _root_.ContinuousLinearMap.compContinuousMultilinearMap_coe (g : M₂ →L[R] M₃)
    (f : ContinuousMultilinearMap R M₁ M₂) :
    (g.compContinuousMultilinearMap f : (∀ i, M₁ i) → M₃) =
      (g : M₂ → M₃) ∘ (f : (∀ i, M₁ i) → M₂) 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [AddCommMonoid M₃] [(i : ι) → Module R (M₁ i)] [Module R M₂] [Module R M₃] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] [TopologicalSpace M₃] (g : M₂ →L[R] M₃) (f : ContinuousMultilinearMap R M₁ M₂)
Goal: ⇑(g.compContinuousMultilinearMap f) = ⇑g ∘ ⇑f
-- ContinuousMultilinearMap.piEquiv_symm_apply
@[simps]
def piEquiv {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
    [∀ i, TopologicalSpace (M' i)] [∀ i, Module R (M' i)] :
    (∀ i, ContinuousMultilinearMap R M₁ (M' i)) ≃ ContinuousMultilinearMap R M₁ (∀ i, M' i) 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : ContinuousMultilinearMap R M₁ ((i : ι') → M' i)) (i : ι')
Goal: ContinuousMultilinearMap.piEquiv.symm f i = (ContinuousLinearMap.proj i).compContinuousMultilinearMap f
-- ContinuousMultilinearMap.piEquiv_apply
@[simps]
def piEquiv {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
    [∀ i, TopologicalSpace (M' i)] [∀ i, Module R (M' i)] :
    (∀ i, ContinuousMultilinearMap R M₁ (M' i)) ≃ ContinuousMultilinearMap R M₁ (∀ i, M' i) 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [(i : ι) → TopologicalSpace (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [(i : ι') → Module R (M' i)] (f : (i : ι') → ContinuousMultilinearMap R M₁ (M' i))
Goal: ContinuousMultilinearMap.piEquiv f = ContinuousMultilinearMap.pi f
-- ContinuousMultilinearMap.domDomCongr_toMultilinearMap
@[simps! toMultilinearMap apply]
nonrec def domDomCongr {ι' : Type*} (e : ι ≃ ι')
    (f : ContinuousMultilinearMap R (fun _ : ι => M₂) M₃) :
    ContinuousMultilinearMap R (fun _ : ι' => M₂) M₃ 
Arguments: {R : Type u} {ι : Type v} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] {ι' : Type u_1} (e : ι ≃ ι') (f : ContinuousMultilinearMap R (fun x => M₂) M₃)
Goal: (ContinuousMultilinearMap.domDomCongr e f).toMultilinearMap = MultilinearMap.domDomCongr e f.toMultilinearMap
-- ContinuousMultilinearMap.domDomCongr_apply
@[simps! toMultilinearMap apply]
nonrec def domDomCongr {ι' : Type*} (e : ι ≃ ι')
    (f : ContinuousMultilinearMap R (fun _ : ι => M₂) M₃) :
    ContinuousMultilinearMap R (fun _ : ι' => M₂) M₃ 
Arguments: {R : Type u} {ι : Type v} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] {ι' : Type u_1} (e : ι ≃ ι') (f : ContinuousMultilinearMap R (fun x => M₂) M₃) (v : ι' → M₂)
Goal: (ContinuousMultilinearMap.domDomCongr e f) v = f fun i => v (e i)
-- ContinuousMultilinearMap.domDomCongrEquiv_apply
@[simps]
def domDomCongrEquiv {ι' : Type*} (e : ι ≃ ι') :
    ContinuousMultilinearMap R (fun _ : ι => M₂) M₃ ≃
      ContinuousMultilinearMap R (fun _ : ι' => M₂) M₃ 
Arguments: {R : Type u} {ι : Type v} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] {ι' : Type u_1} (e : ι ≃ ι') (f : ContinuousMultilinearMap R (fun x => M₂) M₃)
Goal: (ContinuousMultilinearMap.domDomCongrEquiv e) f = ContinuousMultilinearMap.domDomCongr e f
-- ContinuousMultilinearMap.domDomCongrEquiv_symm_apply
@[simps]
def domDomCongrEquiv {ι' : Type*} (e : ι ≃ ι') :
    ContinuousMultilinearMap R (fun _ : ι => M₂) M₃ ≃
      ContinuousMultilinearMap R (fun _ : ι' => M₂) M₃ 
Arguments: {R : Type u} {ι : Type v} {M₂ : Type w₂} {M₃ : Type w₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [TopologicalSpace M₂] [TopologicalSpace M₃] {ι' : Type u_1} (e : ι ≃ ι') (f : ContinuousMultilinearMap R (fun x => M₂) M₃)
Goal: (ContinuousMultilinearMap.domDomCongrEquiv e).symm f = ContinuousMultilinearMap.domDomCongr e.symm f
-- ContinuousMultilinearMap.linearDeriv_apply
@[simp]
lemma linearDeriv_apply : f.linearDeriv x y = ∑ i, f (Function.update x i (y i)) 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [ContinuousAdd M₂] [DecidableEq ι] [Fintype ι] (x y : (i : ι) → M₁ i)
Goal: (f.linearDeriv x) y = ∑ i : ι, f (Function.update x i (y i))
-- ContinuousMultilinearMap.cons_add
theorem cons_add (f : ContinuousMultilinearMap R M M₂) (m : ∀ i : Fin n, M i.succ) (x y : M 0) :
    f (cons (x + y) m) = f (cons x m) + f (cons y m) 
Arguments: {R : Type u} {n : ℕ} {M : Fin n.succ → Type w} {M₂ : Type w₂} [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [AddCommMonoid M₂] [(i : Fin n.succ) → Module R (M i)] [Module R M₂] [(i : Fin n.succ) → TopologicalSpace (M i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M M₂) (m : (i : Fin n) → M i.succ) (x y : M 0)
Goal: f (Fin.cons (x + y) m) = f (Fin.cons x m) + f (Fin.cons y m)
-- ContinuousMultilinearMap.cons_smul
theorem cons_smul (f : ContinuousMultilinearMap R M M₂) (m : ∀ i : Fin n, M i.succ) (c : R)
    (x : M 0) : f (cons (c • x) m) = c • f (cons x m) 
Arguments: {R : Type u} {n : ℕ} {M : Fin n.succ → Type w} {M₂ : Type w₂} [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [AddCommMonoid M₂] [(i : Fin n.succ) → Module R (M i)] [Module R M₂] [(i : Fin n.succ) → TopologicalSpace (M i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M M₂) (m : (i : Fin n) → M i.succ) (c : R) (x : M 0)
Goal: f (Fin.cons (c • x) m) = c • f (Fin.cons x m)
-- ContinuousMultilinearMap.map_piecewise_add
theorem map_piecewise_add [DecidableEq ι] (m m' : ∀ i, M₁ i) (t : Finset ι) :
    f (t.piecewise (m + m') m') = ∑ s ∈ t.powerset, f (s.piecewise m m') 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] (m m' : (i : ι) → M₁ i) (t : Finset ι)
Goal: f (t.piecewise (m + m') m') = ∑ s ∈ t.powerset, f (s.piecewise m m')
-- ContinuousMultilinearMap.map_add_univ
theorem map_add_univ [DecidableEq ι] [Fintype ι] (m m' : ∀ i, M₁ i) :
    f (m + m') = ∑ s : Finset ι, f (s.piecewise m m') 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) [DecidableEq ι] [Fintype ι] (m m' : (i : ι) → M₁ i)
Goal: f (m + m') = ∑ s : Finset ι, f (s.piecewise m m')
-- ContinuousMultilinearMap.map_sum_finset
theorem map_sum_finset [DecidableEq ι] :
    (f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i => g i (r i) 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) {α : ι → Type u_1} [Fintype ι] (g : (i : ι) → α i → M₁ i) (A : (i : ι) → Finset (α i)) [DecidableEq ι]
Goal: (f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ Fintype.piFinset A, f fun i => g i (r i)
-- ContinuousMultilinearMap.map_sum
theorem map_sum [DecidableEq ι] [∀ i, Fintype (α i)] :
    (f fun i => ∑ j, g i j) = ∑ r : ∀ i, α i, f fun i => g i (r i) 
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (f : ContinuousMultilinearMap R M₁ M₂) {α : ι → Type u_1} [Fintype ι] (g : (i : ι) → α i → M₁ i) [DecidableEq ι] [(i : ι) → Fintype (α i)]
Goal: (f fun i => ∑ j : α i, g i j) = ∑ r : (i : ι) → α i, f fun i => g i (r i)
-- ContinuousMultilinearMap.coe_restrictScalars
@[simp]
theorem coe_restrictScalars (f : ContinuousMultilinearMap A M₁ M₂) : ⇑(f.restrictScalars R) = f 
Arguments: (R : Type u) {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] {A : Type u_1} [Semiring A] [SMul R A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [∀ (i : ι), IsScalarTower R A (M₁ i)] [IsScalarTower R A M₂] (f : ContinuousMultilinearMap A M₁ M₂)
Goal: ⇑(ContinuousMultilinearMap.restrictScalars R f) = ⇑f