ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContDiff.comp_contMDiffWithinAt
theorem ContDiff.comp_contMDiffWithinAt {g : F → F'} {f : M → F} {s : Set M} {x : M}
    (hg : ContDiff 𝕜 n g) (hf : ContMDiffWithinAt I 𝓘(𝕜, F) n f s x) :
    ContMDiffWithinAt I 𝓘(𝕜, F') n (g ∘ f) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {F' : Type u_11} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {n : ℕ∞} {g : F → F'} {f : M → F} {s : Set M} {x : M} (hg : ContDiff 𝕜 n g) (hf : ContMDiffWithinAt I 𝓘(𝕜, F) n f s x)
Goal: ContMDiffWithinAt I 𝓘(𝕜, F') n (g ∘ f) s x
-- ContDiff.comp_contMDiffAt
theorem ContDiff.comp_contMDiffAt {g : F → F'} {f : M → F} {x : M} (hg : ContDiff 𝕜 n g)
    (hf : ContMDiffAt I 𝓘(𝕜, F) n f x) : ContMDiffAt I 𝓘(𝕜, F') n (g ∘ f) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {F' : Type u_11} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {n : ℕ∞} {g : F → F'} {f : M → F} {x : M} (hg : ContDiff 𝕜 n g) (hf : ContMDiffAt I 𝓘(𝕜, F) n f x)
Goal: ContMDiffAt I 𝓘(𝕜, F') n (g ∘ f) x
-- ContDiff.comp_contMDiff
theorem ContDiff.comp_contMDiff {g : F → F'} {f : M → F} (hg : ContDiff 𝕜 n g)
    (hf : ContMDiff I 𝓘(𝕜, F) n f) : ContMDiff I 𝓘(𝕜, F') n (g ∘ f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {F' : Type u_11} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {n : ℕ∞} {g : F → F'} {f : M → F} (hg : ContDiff 𝕜 n g) (hf : ContMDiff I 𝓘(𝕜, F) n f)
Goal: ContMDiff I 𝓘(𝕜, F') n (g ∘ f)
-- ContinuousLinearMap.contMDiff
theorem ContinuousLinearMap.contMDiff (L : E →L[𝕜] F) : ContMDiff 𝓘(𝕜, E) 𝓘(𝕜, F) n L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (L : E →L[𝕜] F)
Goal: ContMDiff 𝓘(𝕜, E) 𝓘(𝕜, F) n ⇑L
-- ContinuousLinearMap.contMDiffAt
theorem ContinuousLinearMap.contMDiffAt (L : E →L[𝕜] F) {x} : ContMDiffAt 𝓘(𝕜, E) 𝓘(𝕜, F) n L x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (L : E →L[𝕜] F) {x : E}
Goal: ContMDiffAt 𝓘(𝕜, E) 𝓘(𝕜, F) n (⇑L) x
-- ContinuousLinearMap.contMDiffWithinAt
theorem ContinuousLinearMap.contMDiffWithinAt (L : E →L[𝕜] F) {s x} :
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, F) n L s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (L : E →L[𝕜] F) {s : Set E} {x : E}
Goal: ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, F) n (⇑L) s x
-- ContinuousLinearMap.contMDiffOn
theorem ContinuousLinearMap.contMDiffOn (L : E →L[𝕜] F) {s} : ContMDiffOn 𝓘(𝕜, E) 𝓘(𝕜, F) n L s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (L : E →L[𝕜] F) {s : Set E}
Goal: ContMDiffOn 𝓘(𝕜, E) 𝓘(𝕜, F) n (⇑L) s
-- ContinuousLinearMap.smooth
theorem ContinuousLinearMap.smooth (L : E →L[𝕜] F) : Smooth 𝓘(𝕜, E) 𝓘(𝕜, F) L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (L : E →L[𝕜] F)
Goal: Smooth 𝓘(𝕜, E) 𝓘(𝕜, F) ⇑L
-- ContMDiffWithinAt.clm_comp
theorem ContMDiffWithinAt.clm_comp {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₁} {s : Set M} {x : M}
    (hg : ContMDiffWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₃) n g s x)
    (hf : ContMDiffWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₁) n f s x) :
    ContMDiffWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₃) n (fun x => (g x).comp (f x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₁} {s : Set M} {x : M} (hg : ContMDiffWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₃) n g s x) (hf : ContMDiffWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₁) n f s x)
Goal: ContMDiffWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₃) n (fun x => (g x).comp (f x)) s x
-- ContMDiffAt.clm_comp
theorem ContMDiffAt.clm_comp {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₁} {x : M}
    (hg : ContMDiffAt I 𝓘(𝕜, F₁ →L[𝕜] F₃) n g x) (hf : ContMDiffAt I 𝓘(𝕜, F₂ →L[𝕜] F₁) n f x) :
    ContMDiffAt I 𝓘(𝕜, F₂ →L[𝕜] F₃) n (fun x => (g x).comp (f x)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₁} {x : M} (hg : ContMDiffAt I 𝓘(𝕜, F₁ →L[𝕜] F₃) n g x) (hf : ContMDiffAt I 𝓘(𝕜, F₂ →L[𝕜] F₁) n f x)
Goal: ContMDiffAt I 𝓘(𝕜, F₂ →L[𝕜] F₃) n (fun x => (g x).comp (f x)) x
-- ContMDiffOn.clm_comp
theorem ContMDiffOn.clm_comp {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₁} {s : Set M}
    (hg : ContMDiffOn I 𝓘(𝕜, F₁ →L[𝕜] F₃) n g s) (hf : ContMDiffOn I 𝓘(𝕜, F₂ →L[𝕜] F₁) n f s) :
    ContMDiffOn I 𝓘(𝕜, F₂ →L[𝕜] F₃) n (fun x => (g x).comp (f x)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₁} {s : Set M} (hg : ContMDiffOn I 𝓘(𝕜, F₁ →L[𝕜] F₃) n g s) (hf : ContMDiffOn I 𝓘(𝕜, F₂ →L[𝕜] F₁) n f s)
Goal: ContMDiffOn I 𝓘(𝕜, F₂ →L[𝕜] F₃) n (fun x => (g x).comp (f x)) s
-- ContMDiff.clm_comp
theorem ContMDiff.clm_comp {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₁}
    (hg : ContMDiff I 𝓘(𝕜, F₁ →L[𝕜] F₃) n g) (hf : ContMDiff I 𝓘(𝕜, F₂ →L[𝕜] F₁) n f) :
    ContMDiff I 𝓘(𝕜, F₂ →L[𝕜] F₃) n fun x => (g x).comp (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₁} (hg : ContMDiff I 𝓘(𝕜, F₁ →L[𝕜] F₃) n g) (hf : ContMDiff I 𝓘(𝕜, F₂ →L[𝕜] F₁) n f)
Goal: ContMDiff I 𝓘(𝕜, F₂ →L[𝕜] F₃) n fun x => (g x).comp (f x)
-- ContMDiffWithinAt.clm_apply
theorem ContMDiffWithinAt.clm_apply {g : M → F₁ →L[𝕜] F₂} {f : M → F₁} {s : Set M} {x : M}
    (hg : ContMDiffWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n g s x)
    (hf : ContMDiffWithinAt I 𝓘(𝕜, F₁) n f s x) :
    ContMDiffWithinAt I 𝓘(𝕜, F₂) n (fun x => g x (f x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {n : ℕ∞} {g : M → F₁ →L[𝕜] F₂} {f : M → F₁} {s : Set M} {x : M} (hg : ContMDiffWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n g s x) (hf : ContMDiffWithinAt I 𝓘(𝕜, F₁) n f s x)
Goal: ContMDiffWithinAt I 𝓘(𝕜, F₂) n (fun x => (g x) (f x)) s x
-- ContMDiffAt.clm_apply
nonrec theorem ContMDiffAt.clm_apply {g : M → F₁ →L[𝕜] F₂} {f : M → F₁} {x : M}
    (hg : ContMDiffAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n g x) (hf : ContMDiffAt I 𝓘(𝕜, F₁) n f x) :
    ContMDiffAt I 𝓘(𝕜, F₂) n (fun x => g x (f x)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {n : ℕ∞} {g : M → F₁ →L[𝕜] F₂} {f : M → F₁} {x : M} (hg : ContMDiffAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n g x) (hf : ContMDiffAt I 𝓘(𝕜, F₁) n f x)
Goal: ContMDiffAt I 𝓘(𝕜, F₂) n (fun x => (g x) (f x)) x
-- ContMDiffOn.clm_apply
theorem ContMDiffOn.clm_apply {g : M → F₁ →L[𝕜] F₂} {f : M → F₁} {s : Set M}
    (hg : ContMDiffOn I 𝓘(𝕜, F₁ →L[𝕜] F₂) n g s) (hf : ContMDiffOn I 𝓘(𝕜, F₁) n f s) :
    ContMDiffOn I 𝓘(𝕜, F₂) n (fun x => g x (f x)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {n : ℕ∞} {g : M → F₁ →L[𝕜] F₂} {f : M → F₁} {s : Set M} (hg : ContMDiffOn I 𝓘(𝕜, F₁ →L[𝕜] F₂) n g s) (hf : ContMDiffOn I 𝓘(𝕜, F₁) n f s)
Goal: ContMDiffOn I 𝓘(𝕜, F₂) n (fun x => (g x) (f x)) s
-- ContMDiff.clm_apply
theorem ContMDiff.clm_apply {g : M → F₁ →L[𝕜] F₂} {f : M → F₁}
    (hg : ContMDiff I 𝓘(𝕜, F₁ →L[𝕜] F₂) n g) (hf : ContMDiff I 𝓘(𝕜, F₁) n f) :
    ContMDiff I 𝓘(𝕜, F₂) n fun x => g x (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {n : ℕ∞} {g : M → F₁ →L[𝕜] F₂} {f : M → F₁} (hg : ContMDiff I 𝓘(𝕜, F₁ →L[𝕜] F₂) n g) (hf : ContMDiff I 𝓘(𝕜, F₁) n f)
Goal: ContMDiff I 𝓘(𝕜, F₂) n fun x => (g x) (f x)
-- ContMDiffWithinAt.clm_precomp
theorem ContMDiffWithinAt.clm_precomp {f : M → F₁ →L[𝕜] F₂} {s : Set M} {x : M}
    (hf : ContMDiffWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n f s x) :
    ContMDiffWithinAt I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) n
      (fun y ↦ (f y).precomp F₃ : M → (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {f : M → F₁ →L[𝕜] F₂} {s : Set M} {x : M} (hf : ContMDiffWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n f s x)
Goal: ContMDiffWithinAt I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] F₁ →L[𝕜] F₃) n (fun y => ContinuousLinearMap.precomp F₃ (f y)) s x
-- ContMDiffAt.clm_precomp
nonrec theorem ContMDiffAt.clm_precomp {f : M → F₁ →L[𝕜] F₂} {x : M}
    (hf : ContMDiffAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n f x) :
    ContMDiffAt I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) n
      (fun y ↦ (f y).precomp F₃ : M → (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {f : M → F₁ →L[𝕜] F₂} {x : M} (hf : ContMDiffAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) n f x)
Goal: ContMDiffAt I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] F₁ →L[𝕜] F₃) n (fun y => ContinuousLinearMap.precomp F₃ (f y)) x
-- ContMDiffOn.clm_precomp
theorem ContMDiffOn.clm_precomp {f : M → F₁ →L[𝕜] F₂} {s : Set M}
    (hf : ContMDiffOn I 𝓘(𝕜, F₁ →L[𝕜] F₂) n f s) :
    ContMDiffOn I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) n
      (fun y ↦ (f y).precomp F₃ : M → (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {f : M → F₁ →L[𝕜] F₂} {s : Set M} (hf : ContMDiffOn I 𝓘(𝕜, F₁ →L[𝕜] F₂) n f s)
Goal: ContMDiffOn I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] F₁ →L[𝕜] F₃) n (fun y => ContinuousLinearMap.precomp F₃ (f y)) s
-- ContMDiff.clm_precomp
theorem ContMDiff.clm_precomp {f : M → F₁ →L[𝕜] F₂} (hf : ContMDiff I 𝓘(𝕜, F₁ →L[𝕜] F₂) n f) :
    ContMDiff I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) n
      (fun y ↦ (f y).precomp F₃ : M → (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {f : M → F₁ →L[𝕜] F₂} (hf : ContMDiff I 𝓘(𝕜, F₁ →L[𝕜] F₂) n f)
Goal: ContMDiff I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] F₁ →L[𝕜] F₃) n fun y => ContinuousLinearMap.precomp F₃ (f y)
-- ContMDiffWithinAt.clm_postcomp
theorem ContMDiffWithinAt.clm_postcomp {f : M → F₂ →L[𝕜] F₃} {s : Set M} {x : M}
    (hf : ContMDiffWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₃) n f s x) :
    ContMDiffWithinAt I 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃)) n
      (fun y ↦ (f y).postcomp F₁ : M → (F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {f : M → F₂ →L[𝕜] F₃} {s : Set M} {x : M} (hf : ContMDiffWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₃) n f s x)
Goal: ContMDiffWithinAt I 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₃) n (fun y => ContinuousLinearMap.postcomp F₁ (f y)) s x
-- ContMDiffAt.clm_postcomp
nonrec theorem ContMDiffAt.clm_postcomp {f : M → F₂ →L[𝕜] F₃} {x : M}
    (hf : ContMDiffAt I 𝓘(𝕜, F₂ →L[𝕜] F₃) n f x) :
    ContMDiffAt I 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃)) n
      (fun y ↦ (f y).postcomp F₁ : M → (F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {f : M → F₂ →L[𝕜] F₃} {x : M} (hf : ContMDiffAt I 𝓘(𝕜, F₂ →L[𝕜] F₃) n f x)
Goal: ContMDiffAt I 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₃) n (fun y => ContinuousLinearMap.postcomp F₁ (f y)) x
-- ContMDiffOn.clm_postcomp
nonrec theorem ContMDiffOn.clm_postcomp {f : M → F₂ →L[𝕜] F₃} {s : Set M}
    (hf : ContMDiffOn I 𝓘(𝕜, F₂ →L[𝕜] F₃) n f s) :
    ContMDiffOn I 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃)) n
      (fun y ↦ (f y).postcomp F₁ : M → (F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {f : M → F₂ →L[𝕜] F₃} {s : Set M} (hf : ContMDiffOn I 𝓘(𝕜, F₂ →L[𝕜] F₃) n f s)
Goal: ContMDiffOn I 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₃) n (fun y => ContinuousLinearMap.postcomp F₁ (f y)) s
-- ContMDiff.clm_postcomp
theorem ContMDiff.clm_postcomp {f : M → F₂ →L[𝕜] F₃} (hf : ContMDiff I 𝓘(𝕜, F₂ →L[𝕜] F₃) n f) :
    ContMDiff I 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃)) n
      (fun y ↦ (f y).postcomp F₁ : M → (F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {n : ℕ∞} {f : M → F₂ →L[𝕜] F₃} (hf : ContMDiff I 𝓘(𝕜, F₂ →L[𝕜] F₃) n f)
Goal: ContMDiff I 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₃) n fun y => ContinuousLinearMap.postcomp F₁ (f y)
-- ContMDiffWithinAt.cle_arrowCongr
theorem ContMDiffWithinAt.cle_arrowCongr {f : M → F₁ ≃L[𝕜] F₂} {g : M → F₃ ≃L[𝕜] F₄}
    {s : Set M} {x : M}
    (hf : ContMDiffWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₁) n (fun x ↦ ((f x).symm : F₂ →L[𝕜] F₁)) s x)
    (hg : ContMDiffWithinAt I 𝓘(𝕜, F₃ →L[𝕜] F₄) n (fun x ↦ (g x : F₃ →L[𝕜] F₄)) s x) :
    ContMDiffWithinAt I 𝓘(𝕜, (F₁ →L[𝕜] F₃) →L[𝕜] (F₂ →L[𝕜] F₄)) n
      (fun y ↦ (f y).arrowCongr (g y) : M → (F₁ →L[𝕜] F₃) →L[𝕜] (F₂ →L[𝕜] F₄)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {F₄ : Type u_17} [NormedAddCommGroup F₄] [NormedSpace 𝕜 F₄] {n : ℕ∞} {f : M → F₁ ≃L[𝕜] F₂} {g : M → F₃ ≃L[𝕜] F₄} {s : Set M} {x : M} (hf : ContMDiffWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₁) n (fun x => ↑(f x).symm) s x) (hg : ContMDiffWithinAt I 𝓘(𝕜, F₃ →L[𝕜] F₄) n (fun x => ↑(g x)) s x)
Goal: ContMDiffWithinAt I 𝓘(𝕜, (F₁ →L[𝕜] F₃) →L[𝕜] F₂ →L[𝕜] F₄) n (fun y => ↑((f y).arrowCongr (g y))) s x
-- ContMDiffAt.cle_arrowCongr
nonrec theorem ContMDiffAt.cle_arrowCongr {f : M → F₁ ≃L[𝕜] F₂} {g : M → F₃ ≃L[𝕜] F₄} {x : M}
    (hf : ContMDiffAt I 𝓘(𝕜, F₂ →L[𝕜] F₁) n (fun x ↦ ((f x).symm : F₂ →L[𝕜] F₁)) x)
    (hg : ContMDiffAt I 𝓘(𝕜, F₃ →L[𝕜] F₄) n (fun x ↦ (g x : F₃ →L[𝕜] F₄)) x) :
    ContMDiffAt I 𝓘(𝕜, (F₁ →L[𝕜] F₃) →L[𝕜] (F₂ →L[𝕜] F₄)) n
      (fun y ↦ (f y).arrowCongr (g y) : M → (F₁ →L[𝕜] F₃) →L[𝕜] (F₂ →L[𝕜] F₄)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {F₄ : Type u_17} [NormedAddCommGroup F₄] [NormedSpace 𝕜 F₄] {n : ℕ∞} {f : M → F₁ ≃L[𝕜] F₂} {g : M → F₃ ≃L[𝕜] F₄} {x : M} (hf : ContMDiffAt I 𝓘(𝕜, F₂ →L[𝕜] F₁) n (fun x => ↑(f x).symm) x) (hg : ContMDiffAt I 𝓘(𝕜, F₃ →L[𝕜] F₄) n (fun x => ↑(g x)) x)
Goal: ContMDiffAt I 𝓘(𝕜, (F₁ →L[𝕜] F₃) →L[𝕜] F₂ →L[𝕜] F₄) n (fun y => ↑((f y).arrowCongr (g y))) x
-- ContMDiffOn.cle_arrowCongr
theorem ContMDiffOn.cle_arrowCongr {f : M → F₁ ≃L[𝕜] F₂} {g : M → F₃ ≃L[𝕜] F₄} {s : Set M}
    (hf : ContMDiffOn I 𝓘(𝕜, F₂ →L[𝕜] F₁) n (fun x ↦ ((f x).symm : F₂ →L[𝕜] F₁)) s)
    (hg : ContMDiffOn I 𝓘(𝕜, F₃ →L[𝕜] F₄) n (fun x ↦ (g x : F₃ →L[𝕜] F₄)) s) :
    ContMDiffOn I 𝓘(𝕜, (F₁ →L[𝕜] F₃) →L[𝕜] (F₂ →L[𝕜] F₄)) n
      (fun y ↦ (f y).arrowCongr (g y) : M → (F₁ →L[𝕜] F₃) →L[𝕜] (F₂ →L[𝕜] F₄)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {F₄ : Type u_17} [NormedAddCommGroup F₄] [NormedSpace 𝕜 F₄] {n : ℕ∞} {f : M → F₁ ≃L[𝕜] F₂} {g : M → F₃ ≃L[𝕜] F₄} {s : Set M} (hf : ContMDiffOn I 𝓘(𝕜, F₂ →L[𝕜] F₁) n (fun x => ↑(f x).symm) s) (hg : ContMDiffOn I 𝓘(𝕜, F₃ →L[𝕜] F₄) n (fun x => ↑(g x)) s)
Goal: ContMDiffOn I 𝓘(𝕜, (F₁ →L[𝕜] F₃) →L[𝕜] F₂ →L[𝕜] F₄) n (fun y => ↑((f y).arrowCongr (g y))) s
-- ContMDiff.cle_arrowCongr
theorem ContMDiff.cle_arrowCongr {f : M → F₁ ≃L[𝕜] F₂} {g : M → F₃ ≃L[𝕜] F₄}
    (hf : ContMDiff I 𝓘(𝕜, F₂ →L[𝕜] F₁) n (fun x ↦ ((f x).symm : F₂ →L[𝕜] F₁)))
    (hg : ContMDiff I 𝓘(𝕜, F₃ →L[𝕜] F₄) n (fun x ↦ (g x : F₃ →L[𝕜] F₄))) :
    ContMDiff I 𝓘(𝕜, (F₁ →L[𝕜] F₃) →L[𝕜] (F₂ →L[𝕜] F₄)) n
      (fun y ↦ (f y).arrowCongr (g y) : M → (F₁ →L[𝕜] F₃) →L[𝕜] (F₂ →L[𝕜] F₄)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {F₄ : Type u_17} [NormedAddCommGroup F₄] [NormedSpace 𝕜 F₄] {n : ℕ∞} {f : M → F₁ ≃L[𝕜] F₂} {g : M → F₃ ≃L[𝕜] F₄} (hf : ContMDiff I 𝓘(𝕜, F₂ →L[𝕜] F₁) n fun x => ↑(f x).symm) (hg : ContMDiff I 𝓘(𝕜, F₃ →L[𝕜] F₄) n fun x => ↑(g x))
Goal: ContMDiff I 𝓘(𝕜, (F₁ →L[𝕜] F₃) →L[𝕜] F₂ →L[𝕜] F₄) n fun y => ↑((f y).arrowCongr (g y))
-- ContMDiffWithinAt.clm_prodMap
theorem ContMDiffWithinAt.clm_prodMap {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₄} {s : Set M}
    {x : M} (hg : ContMDiffWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₃) n g s x)
    (hf : ContMDiffWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₄) n f s x) :
    ContMDiffWithinAt I 𝓘(𝕜, F₁ × F₂ →L[𝕜] F₃ × F₄) n (fun x => (g x).prodMap (f x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {F₄ : Type u_17} [NormedAddCommGroup F₄] [NormedSpace 𝕜 F₄] {n : ℕ∞} {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₄} {s : Set M} {x : M} (hg : ContMDiffWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₃) n g s x) (hf : ContMDiffWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₄) n f s x)
Goal: ContMDiffWithinAt I 𝓘(𝕜, F₁ × F₂ →L[𝕜] F₃ × F₄) n (fun x => (g x).prodMap (f x)) s x