ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PartitionOfUnity.sum_finsupport_smul_eq_finsum
theorem sum_finsupport_smul_eq_finsum {M : Type*} [AddCommGroup M] [Module ℝ M] (φ : ι → X → M) :
    ∑ i ∈ ρ.finsupport x₀, ρ i x₀ • φ i x₀ = ∑ᶠ i, ρ i x₀ • φ i x₀ 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) {x₀ : X} {M : Type u_2} [AddCommGroup M] [Module ℝ M] (φ : ι → X → M)
Goal: ∑ i ∈ ρ.finsupport x₀, (ρ i) x₀ • φ i x₀ = ∑ᶠ (i : ι), (ρ i) x₀ • φ i x₀
-- PartitionOfUnity.finite_tsupport
theorem finite_tsupport : {i | x₀ ∈ tsupport (ρ i)}.Finite 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X)
Goal: {i | x₀ ∈ tsupport ⇑(ρ i)}.Finite
-- PartitionOfUnity.mem_fintsupport_iff
theorem mem_fintsupport_iff (i : ι) : i ∈ ρ.fintsupport x₀ ↔ x₀ ∈ tsupport (ρ i) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) (i : ι)
Goal: i ∈ ρ.fintsupport x₀ ↔ x₀ ∈ tsupport ⇑(ρ i)
-- PartitionOfUnity.eventually_fintsupport_subset
theorem eventually_fintsupport_subset :
    ∀ᶠ y in 𝓝 x₀, ρ.fintsupport y ⊆ ρ.fintsupport x₀ 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X)
Goal: ∀ᶠ (y : X) in 𝓝 x₀, ρ.fintsupport y ⊆ ρ.fintsupport x₀
-- PartitionOfUnity.finsupport_subset_fintsupport
theorem finsupport_subset_fintsupport : ρ.finsupport x₀ ⊆ ρ.fintsupport x₀ 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X)
Goal: ρ.finsupport x₀ ⊆ ρ.fintsupport x₀
-- PartitionOfUnity.eventually_finsupport_subset
theorem eventually_finsupport_subset : ∀ᶠ y in 𝓝 x₀, ρ.finsupport y ⊆ ρ.fintsupport x₀ 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X)
Goal: ∀ᶠ (y : X) in 𝓝 x₀, ρ.finsupport y ⊆ ρ.fintsupport x₀
-- PartitionOfUnity.continuous_smul
theorem continuous_smul {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport (f i), ContinuousAt g x) :
    Continuous fun x => f i x • g x 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {E : Type u_1} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {s : Set X} (f : PartitionOfUnity ι X s) {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport ⇑(f i), ContinuousAt g x)
Goal: Continuous fun x => (f i) x • g x
-- PartitionOfUnity.continuous_finsum_smul
theorem continuous_finsum_smul [ContinuousAdd E] {g : ι → X → E}
    (hg : ∀ (i), ∀ x ∈ tsupport (f i), ContinuousAt (g i) x) :
    Continuous fun x => ∑ᶠ i, f i x • g i x 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {E : Type u_1} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {s : Set X} (f : PartitionOfUnity ι X s) [ContinuousAdd E] {g : ι → X → E} (hg : ∀ (i : ι), ∀ x ∈ tsupport ⇑(f i), ContinuousAt (g i) x)
Goal: Continuous fun x => ∑ᶠ (i : ι), (f i) x • g i x
-- PartitionOfUnity.exists_finset_nhd'
theorem exists_finset_nhd' {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) :
    ∃ I : Finset ι, (∀ᶠ x in 𝓝[s] x₀, ∑ i ∈ I, ρ i x = 1) ∧
      ∀ᶠ x in 𝓝 x₀, support (ρ · x) ⊆ I 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X)
Goal: ∃ I, (∀ᶠ (x : X) in 𝓝[s] x₀, ∑ i ∈ I, (ρ i) x = 1) ∧ ∀ᶠ (x : X) in 𝓝 x₀, (Function.support fun x_1 => (ρ x_1) x) ⊆ ↑I
-- PartitionOfUnity.exists_finset_nhd
theorem exists_finset_nhd (ρ : PartitionOfUnity ι X univ) (x₀ : X) :
    ∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, ∑ i ∈ I, ρ i x = 1 ∧ support (ρ · x) ⊆ I 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] (ρ : PartitionOfUnity ι X) (x₀ : X)
Goal: ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, ∑ i ∈ I, (ρ i) x = 1 ∧ (Function.support fun x_1 => (ρ x_1) x) ⊆ ↑I
-- PartitionOfUnity.exists_finset_nhd_support_subset
theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
    (ho : ∀ i, IsOpen (U i)) (x : X) :
    ∃ is : Finset ι, ∃ n ∈ 𝓝 x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (support (f · z)) ⊆ is 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} {f : PartitionOfUnity ι X s} {U : ι → Set X} (hso : f.IsSubordinate U) (ho : ∀ (i : ι), IsOpen (U i)) (x : X)
Goal: ∃ is, ∃ n ∈ 𝓝 x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (Function.support fun x => (f x) z) ⊆ ↑is
-- PartitionOfUnity.IsSubordinate.continuous_finsum_smul
theorem IsSubordinate.continuous_finsum_smul [ContinuousAdd E] {U : ι → Set X}
    (ho : ∀ i, IsOpen (U i)) (hf : f.IsSubordinate U) {g : ι → X → E}
    (hg : ∀ i, ContinuousOn (g i) (U i)) : Continuous fun x => ∑ᶠ i, f i x • g i x 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {E : Type u_1} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {s : Set X} {f : PartitionOfUnity ι X s} [ContinuousAdd E] {U : ι → Set X} (ho : ∀ (i : ι), IsOpen (U i)) (hf : f.IsSubordinate U) {g : ι → X → E} (hg : ∀ (i : ι), ContinuousOn (g i) (U i))
Goal: Continuous fun x => ∑ᶠ (i : ι), (f i) x • g i x
-- BumpCovering.locallyFinite
protected theorem locallyFinite : LocallyFinite fun i => support (f i) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s)
Goal: LocallyFinite fun i => Function.support ⇑(f i)
-- BumpCovering.locallyFinite_tsupport
theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s)
Goal: LocallyFinite fun i => tsupport ⇑(f i)
-- BumpCovering.point_finite
protected theorem point_finite (x : X) : { i | f i x ≠ 0 }.Finite 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X)
Goal: {i | (f i) x ≠ 0}.Finite
-- BumpCovering.nonneg
theorem nonneg (i : ι) (x : X) : 0 ≤ f i x 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X)
Goal: 0 ≤ (f i) x
-- BumpCovering.le_one
theorem le_one (i : ι) (x : X) : f i x ≤ 1 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X)
Goal: (f i) x ≤ 1
-- BumpCovering.coe_single
@[simp]
theorem coe_single (i : ι) (s : Set X) : ⇑(BumpCovering.single i s) = Pi.single i 1 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] (i : ι) (s : Set X)
Goal: ⇑(BumpCovering.single i s) = Pi.single i 1
-- BumpCovering.IsSubordinate.mono
theorem IsSubordinate.mono {f : BumpCovering ι X s} {U V : ι → Set X} (hU : f.IsSubordinate U)
    (hV : ∀ i, U i ⊆ V i) : f.IsSubordinate V 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} {f : BumpCovering ι X s} {U V : ι → Set X} (hU : f.IsSubordinate U) (hV : ∀ (i : ι), U i ⊆ V i)
Goal: f.IsSubordinate V
-- BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop
theorem exists_isSubordinate_of_locallyFinite_of_prop [NormalSpace X] (p : (X → ℝ) → Prop)
    (h01 : ∀ s t, IsClosed s → IsClosed t → Disjoint s t →
      ∃ f : C(X, ℝ), p f ∧ EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1)
    (hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U)
    (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, (∀ i, p (f i)) ∧ f.IsSubordinate U 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] (p : (X → ℝ) → Prop) (h01 :
  ∀ (s t : Set X),
    IsClosed s →
      IsClosed t → Disjoint s t → ∃ f, p ⇑f ∧ Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1) (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i)
Goal: ∃ f, (∀ (i : ι), p ⇑(f i)) ∧ f.IsSubordinate U
-- BumpCovering.exists_isSubordinate_of_locallyFinite
theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s) (U : ι → Set X)
    (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) :
    ∃ f : BumpCovering ι X s, f.IsSubordinate U 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i)
Goal: ∃ f, f.IsSubordinate U
-- BumpCovering.exists_isSubordinate_of_prop
theorem exists_isSubordinate_of_prop [NormalSpace X] [ParacompactSpace X] (p : (X → ℝ) → Prop)
    (h01 : ∀ s t, IsClosed s → IsClosed t → Disjoint s t →
      ∃ f : C(X, ℝ), p f ∧ EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1)
    (hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hU : s ⊆ ⋃ i, U i) :
    ∃ f : BumpCovering ι X s, (∀ i, p (f i)) ∧ f.IsSubordinate U 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] [ParacompactSpace X] (p : (X → ℝ) → Prop) (h01 :
  ∀ (s t : Set X),
    IsClosed s →
      IsClosed t → Disjoint s t → ∃ f, p ⇑f ∧ Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1) (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hU : s ⊆ ⋃ i, U i)
Goal: ∃ f, (∀ (i : ι), p ⇑(f i)) ∧ f.IsSubordinate U
-- BumpCovering.exists_isSubordinate
theorem exists_isSubordinate [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X)
    (ho : ∀ i, IsOpen (U i)) (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, f.IsSubordinate U 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hU : s ⊆ ⋃ i, U i)
Goal: ∃ f, f.IsSubordinate U
-- BumpCovering.eventuallyEq_one
theorem eventuallyEq_one (x : X) (hx : x ∈ s) : f (f.ind x hx) =ᶠ[𝓝 x] 1 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X) (hx : x ∈ s)
Goal: ⇑(f (f.ind x hx)) =ᶠ[𝓝 x] 1
-- BumpCovering.ind_apply
theorem ind_apply (x : X) (hx : x ∈ s) : f (f.ind x hx) x = 1 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X) (hx : x ∈ s)
Goal: (f (f.ind x hx)) x = 1
-- BumpCovering.toPOUFun_zero_of_zero
theorem toPOUFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPOUFun i x = 0 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) {i : ι} {x : X} (h : (f i) x = 0)
Goal: f.toPOUFun i x = 0
-- BumpCovering.support_toPOUFun_subset
theorem support_toPOUFun_subset (i : ι) : support (f.toPOUFun i) ⊆ support (f i) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι)
Goal: Function.support (f.toPOUFun i) ⊆ Function.support ⇑(f i)
-- BumpCovering.toPOUFun_eq_mul_prod
theorem toPOUFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
    (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
    f.toPOUFun i x = f i x * ∏ j ∈ t.filter fun j => WellOrderingRel j i, (1 - f j x) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X) (t : Finset ι) (ht : ∀ (j : ι), WellOrderingRel j i → (f j) x ≠ 0 → j ∈ t)
Goal: f.toPOUFun i x = (f i) x * ∏ j ∈ Finset.filter (fun j => WellOrderingRel j i) t, (1 - (f j) x)
-- BumpCovering.sum_toPOUFun_eq
theorem sum_toPOUFun_eq (x : X) : ∑ᶠ i, f.toPOUFun i x = 1 - ∏ᶠ i, (1 - f i x) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X)
Goal: ∑ᶠ (i : ι), f.toPOUFun i x = 1 - ∏ᶠ (i : ι), (1 - (f i) x)
-- BumpCovering.exists_finset_toPOUFun_eventuallyEq
theorem exists_finset_toPOUFun_eventuallyEq (i : ι) (x : X) : ∃ t : Finset ι,
    f.toPOUFun i =ᶠ[𝓝 x] f i * ∏ j ∈ t.filter fun j => WellOrderingRel j i, (1 - f j) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X)
Goal: ∃ t, f.toPOUFun i =ᶠ[𝓝 x] ⇑(f i) * ⇑(∏ j ∈ Finset.filter (fun j => WellOrderingRel j i) t, (1 - f j))
-- BumpCovering.continuous_toPOUFun
theorem continuous_toPOUFun (i : ι) : Continuous (f.toPOUFun i) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι)
Goal: Continuous (f.toPOUFun i)
-- BumpCovering.toPartitionOfUnity_apply
theorem toPartitionOfUnity_apply (i : ι) (x : X) :
    f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (_ : WellOrderingRel j i), (1 - f j x) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X)
Goal: (f.toPartitionOfUnity i) x = (f i) x * ∏ᶠ (j : ι) (_ : WellOrderingRel j i), (1 - (f j) x)
-- BumpCovering.toPartitionOfUnity_eq_mul_prod
theorem toPartitionOfUnity_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
    (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
    f.toPartitionOfUnity i x = f i x * ∏ j ∈ t.filter fun j => WellOrderingRel j i, (1 - f j x) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X) (t : Finset ι) (ht : ∀ (j : ι), WellOrderingRel j i → (f j) x ≠ 0 → j ∈ t)
Goal: (f.toPartitionOfUnity i) x = (f i) x * ∏ j ∈ Finset.filter (fun j => WellOrderingRel j i) t, (1 - (f j) x)
-- BumpCovering.exists_finset_toPartitionOfUnity_eventuallyEq
theorem exists_finset_toPartitionOfUnity_eventuallyEq (i : ι) (x : X) : ∃ t : Finset ι,
    f.toPartitionOfUnity i =ᶠ[𝓝 x] f i * ∏ j ∈ t.filter fun j => WellOrderingRel j i, (1 - f j) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι) (x : X)
Goal: ∃ t, ⇑(f.toPartitionOfUnity i) =ᶠ[𝓝 x] ⇑(f i) * ⇑(∏ j ∈ Finset.filter (fun j => WellOrderingRel j i) t, (1 - f j))
-- BumpCovering.toPartitionOfUnity_zero_of_zero
theorem toPartitionOfUnity_zero_of_zero {i : ι} {x : X} (h : f i x = 0) :
    f.toPartitionOfUnity i x = 0 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) {i : ι} {x : X} (h : (f i) x = 0)
Goal: (f.toPartitionOfUnity i) x = 0
-- BumpCovering.support_toPartitionOfUnity_subset
theorem support_toPartitionOfUnity_subset (i : ι) :
    support (f.toPartitionOfUnity i) ⊆ support (f i) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (i : ι)
Goal: Function.support ⇑(f.toPartitionOfUnity i) ⊆ Function.support ⇑(f i)
-- BumpCovering.sum_toPartitionOfUnity_eq
theorem sum_toPartitionOfUnity_eq (x : X) :
    ∑ᶠ i, f.toPartitionOfUnity i x = 1 - ∏ᶠ i, (1 - f i x) 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) (x : X)
Goal: ∑ᶠ (i : ι), (f.toPartitionOfUnity i) x = 1 - ∏ᶠ (i : ι), (1 - (f i) x)
-- BumpCovering.IsSubordinate.toPartitionOfUnity
theorem IsSubordinate.toPartitionOfUnity {f : BumpCovering ι X s} {U : ι → Set X}
    (h : f.IsSubordinate U) : f.toPartitionOfUnity.IsSubordinate U 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} {f : BumpCovering ι X s} {U : ι → Set X} (h : f.IsSubordinate U)
Goal: f.toPartitionOfUnity.IsSubordinate U
-- PartitionOfUnity.exists_isSubordinate_of_locallyFinite
theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s) (U : ι → Set X)
    (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) :
    ∃ f : PartitionOfUnity ι X s, f.IsSubordinate U 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i)
Goal: ∃ f, f.IsSubordinate U
-- PartitionOfUnity.exists_isSubordinate
theorem exists_isSubordinate [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X)
    (ho : ∀ i, IsOpen (U i)) (hU : s ⊆ ⋃ i, U i) :
    ∃ f : PartitionOfUnity ι X s, f.IsSubordinate U 
Arguments: {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hU : s ⊆ ⋃ i, U i)
Goal: ∃ f, f.IsSubordinate U
-- Pretrivialization.continuousOn_continuousLinearMapCoordChange
theorem continuousOn_continuousLinearMapCoordChange [VectorBundle 𝕜₁ F₁ E₁] [VectorBundle 𝕜₂ F₂ E₂]
    [MemTrivializationAtlas e₁] [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂]
    [MemTrivializationAtlas e₂'] :
    ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂')
      (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) 
Arguments: {𝕜₁ : Type u_1} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₂] {σ : 𝕜₁ →+* 𝕜₂} [iσ : RingHomIsometric σ] {B : Type u_3} {F₁ : Type u_4} [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] {E₁ : B → Type u_5} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜₁ (E₁ x)] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [NormedAddCommGroup F₂] [NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜₂ (E₂ x)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [TopologicalSpace B] {e₁ e₁' : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ e₂' : Trivialization F₂ Bundle.TotalSpace.proj} [(x : B) → TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [VectorBundle 𝕜₁ F₁ E₁] [VectorBundle 𝕜₂ F₂ E₂] [MemTrivializationAtlas e₁] [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂] [MemTrivializationAtlas e₂']
Goal: ContinuousOn (Pretrivialization.continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂')
  (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))
-- Pretrivialization.continuousLinearMap_apply
theorem continuousLinearMap_apply (p : TotalSpace (F₁ →SL[σ] F₂) fun x => E₁ x →SL[σ] E₂ x) :
    (continuousLinearMap σ e₁ e₂) p =
      ⟨p.1, .comp (e₂.continuousLinearMapAt 𝕜₂ p.1) (p.2.comp (e₁.symmL 𝕜₁ p.1))⟩ 
Arguments: {𝕜₁ : Type u_1} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {B : Type u_3} {F₁ : Type u_4} [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] {E₁ : B → Type u_5} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜₁ (E₁ x)] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [NormedAddCommGroup F₂] [NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜₂ (E₂ x)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [TopologicalSpace B] (e₁ : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ : Trivialization F₂ Bundle.TotalSpace.proj) [(x : B) → TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [Trivialization.IsLinear 𝕜₁ e₁] [Trivialization.IsLinear 𝕜₂ e₂] (p : Bundle.TotalSpace (F₁ →SL[σ] F₂) fun x => E₁ x →SL[σ] E₂ x)
Goal: ↑(Pretrivialization.continuousLinearMap σ e₁ e₂) p =
  (p.proj, (Trivialization.continuousLinearMapAt 𝕜₂ e₂ p.proj).comp (p.snd.comp (Trivialization.symmL 𝕜₁ e₁ p.proj)))
-- Pretrivialization.continuousLinearMap_symm_apply
theorem continuousLinearMap_symm_apply (p : B × (F₁ →SL[σ] F₂)) :
    (continuousLinearMap σ e₁ e₂).toPartialEquiv.symm p =
      ⟨p.1, .comp (e₂.symmL 𝕜₂ p.1) (p.2.comp (e₁.continuousLinearMapAt 𝕜₁ p.1))⟩ 
Arguments: {𝕜₁ : Type u_1} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {B : Type u_3} {F₁ : Type u_4} [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] {E₁ : B → Type u_5} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜₁ (E₁ x)] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [NormedAddCommGroup F₂] [NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜₂ (E₂ x)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [TopologicalSpace B] (e₁ : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ : Trivialization F₂ Bundle.TotalSpace.proj) [(x : B) → TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [Trivialization.IsLinear 𝕜₁ e₁] [Trivialization.IsLinear 𝕜₂ e₂] (p : B × (F₁ →SL[σ] F₂))
Goal: ↑(Pretrivialization.continuousLinearMap σ e₁ e₂).symm p =
  { proj := p.1,
    snd := (Trivialization.symmL 𝕜₂ e₂ p.1).comp (p.2.comp (Trivialization.continuousLinearMapAt 𝕜₁ e₁ p.1)) }
-- Pretrivialization.continuousLinearMap_symm_apply'
theorem continuousLinearMap_symm_apply' {b : B} (hb : b ∈ e₁.baseSet ∩ e₂.baseSet)
    (L : F₁ →SL[σ] F₂) :
    (continuousLinearMap σ e₁ e₂).symm b L =
      (e₂.symmL 𝕜₂ b).comp (L.comp <| e₁.continuousLinearMapAt 𝕜₁ b) 
Arguments: {𝕜₁ : Type u_1} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {B : Type u_3} {F₁ : Type u_4} [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] {E₁ : B → Type u_5} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜₁ (E₁ x)] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [NormedAddCommGroup F₂] [NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜₂ (E₂ x)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [TopologicalSpace B] (e₁ : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ : Trivialization F₂ Bundle.TotalSpace.proj) [(x : B) → TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [Trivialization.IsLinear 𝕜₁ e₁] [Trivialization.IsLinear 𝕜₂ e₂] {b : B} (hb : b ∈ e₁.baseSet ∩ e₂.baseSet) (L : F₁ →SL[σ] F₂)
Goal: (Pretrivialization.continuousLinearMap σ e₁ e₂).symm b L =
  (Trivialization.symmL 𝕜₂ e₂ b).comp (L.comp (Trivialization.continuousLinearMapAt 𝕜₁ e₁ b))
-- Pretrivialization.continuousLinearMapCoordChange_apply
theorem continuousLinearMapCoordChange_apply (b : B)
    (hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) (L : F₁ →SL[σ] F₂) :
    continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b L =
      (continuousLinearMap σ e₁' e₂' ⟨b, (continuousLinearMap σ e₁ e₂).symm b L⟩).2 
Arguments: {𝕜₁ : Type u_1} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {B : Type u_3} {F₁ : Type u_4} [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] {E₁ : B → Type u_5} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜₁ (E₁ x)] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [NormedAddCommGroup F₂] [NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜₂ (E₂ x)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [TopologicalSpace B] (e₁ e₁' : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ e₂' : Trivialization F₂ Bundle.TotalSpace.proj) [(x : B) → TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [Trivialization.IsLinear 𝕜₁ e₁] [Trivialization.IsLinear 𝕜₁ e₁'] [Trivialization.IsLinear 𝕜₂ e₂] [Trivialization.IsLinear 𝕜₂ e₂'] (b : B) (hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)) (L : F₁ →SL[σ] F₂)
Goal: (Pretrivialization.continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂' b) L =
  (↑(Pretrivialization.continuousLinearMap σ e₁' e₂')
      { proj := b, snd := (Pretrivialization.continuousLinearMap σ e₁ e₂).symm b L }).2
-- Trivialization.baseSet_continuousLinearMap
@[simp]
theorem Trivialization.baseSet_continuousLinearMap :
    (e₁.continuousLinearMap σ e₂).baseSet = e₁.baseSet ∩ e₂.baseSet 
Arguments: {𝕜₁ : Type u_1} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) [iσ : RingHomIsometric σ] {B : Type u_3} {F₁ : Type u_4} [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] {E₁ : B → Type u_5} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜₁ (E₁ x)] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [NormedAddCommGroup F₂] [NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜₂ (E₂ x)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [TopologicalSpace B] {e₁ : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ : Trivialization F₂ Bundle.TotalSpace.proj} [(x : B) → TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [VectorBundle 𝕜₁ F₁ E₁] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [VectorBundle 𝕜₂ F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)] [he₁ : MemTrivializationAtlas e₁] [he₂ : MemTrivializationAtlas e₂]
Goal: (Trivialization.continuousLinearMap σ e₁ e₂).baseSet = e₁.baseSet ∩ e₂.baseSet