ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Diffeomorph.symm_apply_apply
@[simp]
theorem symm_apply_apply (h : M ≃ₘ^n⟮I, J⟯ N) (x : M) : h.symm (h x) = x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) (x : M)
Goal: h.symm (h x) = x
-- Diffeomorph.symm_refl
@[simp]
theorem symm_refl : (Diffeomorph.refl I M n).symm = Diffeomorph.refl I M n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {n : ℕ∞}
Goal: (Diffeomorph.refl I M n).symm = Diffeomorph.refl I M n
-- Diffeomorph.self_trans_symm
@[simp]
theorem self_trans_symm (h : M ≃ₘ^n⟮I, J⟯ N) : h.trans h.symm = Diffeomorph.refl I M n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N)
Goal: h.trans h.symm = Diffeomorph.refl I M n
-- Diffeomorph.symm_trans_self
@[simp]
theorem symm_trans_self (h : M ≃ₘ^n⟮I, J⟯ N) : h.symm.trans h = Diffeomorph.refl J N n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N)
Goal: h.symm.trans h = Diffeomorph.refl J N n
-- Diffeomorph.symm_trans'
@[simp]
theorem symm_trans' (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : M' ≃ₘ^n⟮I', J⟯ N) :
    (h₁.trans h₂).symm = h₂.symm.trans h₁.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : M' ≃ₘ^n⟮I', J⟯ N)
Goal: (h₁.trans h₂).symm = h₂.symm.trans h₁.symm
-- Diffeomorph.symm_toEquiv
@[simp]
theorem symm_toEquiv (h : M ≃ₘ^n⟮I, J⟯ N) : h.symm.toEquiv = h.toEquiv.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N)
Goal: h.symm.toEquiv = h.symm
-- Diffeomorph.toEquiv_coe_symm
@[simp, mfld_simps]
theorem toEquiv_coe_symm (h : M ≃ₘ^n⟮I, J⟯ N) : ⇑h.toEquiv.symm = h.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N)
Goal: ⇑h.symm = ⇑h.symm
-- Diffeomorph.image_eq_preimage
theorem image_eq_preimage (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set M) : h '' s = h.symm ⁻¹' s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set M)
Goal: ⇑h '' s = ⇑h.symm ⁻¹' s
-- Diffeomorph.symm_image_eq_preimage
theorem symm_image_eq_preimage (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set N) : h.symm '' s = h ⁻¹' s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set N)
Goal: ⇑h.symm '' s = ⇑h ⁻¹' s
-- Diffeomorph.range_comp
@[simp, mfld_simps]
nonrec theorem range_comp {α} (h : M ≃ₘ^n⟮I, J⟯ N) (f : α → M) :
    range (h ∘ f) = h.symm ⁻¹' range f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {α : Sort u_13} (h : M ≃ₘ^n⟮I, J⟯ N) (f : α → M)
Goal: Set.range (⇑h ∘ f) = ⇑h.symm ⁻¹' Set.range f
-- Diffeomorph.image_symm_image
@[simp]
theorem image_symm_image (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set N) : h '' (h.symm '' s) = s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set N)
Goal: ⇑h '' (⇑h.symm '' s) = s
-- Diffeomorph.symm_image_image
@[simp]
theorem symm_image_image (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set M) : h.symm '' (h '' s) = s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set M)
Goal: ⇑h.symm '' (⇑h '' s) = s
-- Diffeomorph.toHomeomorph_toEquiv
@[simp]
theorem toHomeomorph_toEquiv (h : M ≃ₘ^n⟮I, J⟯ N) : h.toHomeomorph.toEquiv = h.toEquiv 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N)
Goal: h.toHomeomorph.toEquiv = h.toEquiv
-- Diffeomorph.symm_toHomeomorph
@[simp]
theorem symm_toHomeomorph (h : M ≃ₘ^n⟮I, J⟯ N) : h.symm.toHomeomorph = h.toHomeomorph.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N)
Goal: h.symm.toHomeomorph = h.toHomeomorph.symm
-- Diffeomorph.coe_toHomeomorph
@[simp]
theorem coe_toHomeomorph (h : M ≃ₘ^n⟮I, J⟯ N) : ⇑h.toHomeomorph = h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N)
Goal: ⇑h.toHomeomorph = ⇑h
-- Diffeomorph.coe_toHomeomorph_symm
@[simp]
theorem coe_toHomeomorph_symm (h : M ≃ₘ^n⟮I, J⟯ N) : ⇑h.toHomeomorph.symm = h.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N)
Goal: ⇑h.toHomeomorph.symm = ⇑h.symm
-- Diffeomorph.contMDiffWithinAt_comp_diffeomorph_iff
@[simp]
theorem contMDiffWithinAt_comp_diffeomorph_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} {s x}
    (hm : m ≤ n) :
    ContMDiffWithinAt I I' m (f ∘ h) s x ↔ ContMDiffWithinAt J I' m f (h.symm ⁻¹' s) (h x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n m : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} {s : Set M} {x : M} (hm : m ≤ n)
Goal: ContMDiffWithinAt I I' m (f ∘ ⇑h) s x ↔ ContMDiffWithinAt J I' m f (⇑h.symm ⁻¹' s) (h x)
-- Diffeomorph.contMDiffOn_comp_diffeomorph_iff
@[simp]
theorem contMDiffOn_comp_diffeomorph_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} {s} (hm : m ≤ n) :
    ContMDiffOn I I' m (f ∘ h) s ↔ ContMDiffOn J I' m f (h.symm ⁻¹' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n m : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} {s : Set M} (hm : m ≤ n)
Goal: ContMDiffOn I I' m (f ∘ ⇑h) s ↔ ContMDiffOn J I' m f (⇑h.symm ⁻¹' s)
-- Diffeomorph.contMDiffAt_comp_diffeomorph_iff
@[simp]
theorem contMDiffAt_comp_diffeomorph_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} {x} (hm : m ≤ n) :
    ContMDiffAt I I' m (f ∘ h) x ↔ ContMDiffAt J I' m f (h x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n m : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} {x : M} (hm : m ≤ n)
Goal: ContMDiffAt I I' m (f ∘ ⇑h) x ↔ ContMDiffAt J I' m f (h x)
-- Diffeomorph.contMDiff_comp_diffeomorph_iff
@[simp]
theorem contMDiff_comp_diffeomorph_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} (hm : m ≤ n) :
    ContMDiff I I' m (f ∘ h) ↔ ContMDiff J I' m f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n m : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} (hm : m ≤ n)
Goal: ContMDiff I I' m (f ∘ ⇑h) ↔ ContMDiff J I' m f
-- Diffeomorph.contMDiffWithinAt_diffeomorph_comp_iff
@[simp]
theorem contMDiffWithinAt_diffeomorph_comp_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n)
    {s x} : ContMDiffWithinAt I' J m (h ∘ f) s x ↔ ContMDiffWithinAt I' I m f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n m : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n) {s : Set M'} {x : M'}
Goal: ContMDiffWithinAt I' J m (⇑h ∘ f) s x ↔ ContMDiffWithinAt I' I m f s x
-- Diffeomorph.contMDiffAt_diffeomorph_comp_iff
@[simp]
theorem contMDiffAt_diffeomorph_comp_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n) {x} :
    ContMDiffAt I' J m (h ∘ f) x ↔ ContMDiffAt I' I m f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n m : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n) {x : M'}
Goal: ContMDiffAt I' J m (⇑h ∘ f) x ↔ ContMDiffAt I' I m f x
-- Diffeomorph.contMDiffOn_diffeomorph_comp_iff
@[simp]
theorem contMDiffOn_diffeomorph_comp_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n) {s} :
    ContMDiffOn I' J m (h ∘ f) s ↔ ContMDiffOn I' I m f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n m : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n) {s : Set M'}
Goal: ContMDiffOn I' J m (⇑h ∘ f) s ↔ ContMDiffOn I' I m f s
-- Diffeomorph.contMDiff_diffeomorph_comp_iff
@[simp]
theorem contMDiff_diffeomorph_comp_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n) :
    ContMDiff I' J m (h ∘ f) ↔ ContMDiff I' I m f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n m : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n)
Goal: ContMDiff I' J m (⇑h ∘ f) ↔ ContMDiff I' I m f
-- Diffeomorph.toPartialHomeomorph_mdifferentiable
theorem toPartialHomeomorph_mdifferentiable (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) :
    h.toHomeomorph.toPartialHomeomorph.MDifferentiable I J 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n)
Goal: PartialHomeomorph.MDifferentiable I J h.toHomeomorph.toPartialHomeomorph
-- Diffeomorph.prodCongr_symm
@[simp]
theorem prodCongr_symm (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : N ≃ₘ^n⟮J, J'⟯ N') :
    (h₁.prodCongr h₂).symm = h₁.symm.prodCongr h₂.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {G' : Type u_8} [TopologicalSpace G'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {J' : ModelWithCorners 𝕜 F G'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {N' : Type u_12} [TopologicalSpace N'] [ChartedSpace G' N'] {n : ℕ∞} (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : N ≃ₘ^n⟮J, J'⟯ N')
Goal: (h₁.prodCongr h₂).symm = h₁.symm.prodCongr h₂.symm
-- Diffeomorph.coe_prodCongr
@[simp]
theorem coe_prodCongr (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : N ≃ₘ^n⟮J, J'⟯ N') :
    ⇑(h₁.prodCongr h₂) = Prod.map h₁ h₂ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {G' : Type u_8} [TopologicalSpace G'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {J' : ModelWithCorners 𝕜 F G'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {N' : Type u_12} [TopologicalSpace N'] [ChartedSpace G' N'] {n : ℕ∞} (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : N ≃ₘ^n⟮J, J'⟯ N')
Goal: ⇑(h₁.prodCongr h₂) = Prod.map ⇑h₁ ⇑h₂
-- Diffeomorph.prodComm_symm
@[simp]
theorem prodComm_symm : (prodComm I J M N n).symm = prodComm J I N M n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] (I : ModelWithCorners 𝕜 E H) (J : ModelWithCorners 𝕜 F G) (M : Type u_9) [TopologicalSpace M] [ChartedSpace H M] (N : Type u_11) [TopologicalSpace N] [ChartedSpace G N] (n : ℕ∞)
Goal: (Diffeomorph.prodComm I J M N n).symm = Diffeomorph.prodComm J I N M n
-- Diffeomorph.coe_prodComm
@[simp]
theorem coe_prodComm : ⇑(prodComm I J M N n) = Prod.swap 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] (I : ModelWithCorners 𝕜 E H) (J : ModelWithCorners 𝕜 F G) (M : Type u_9) [TopologicalSpace M] [ChartedSpace H M] (N : Type u_11) [TopologicalSpace N] [ChartedSpace G N] (n : ℕ∞)
Goal: ⇑(Diffeomorph.prodComm I J M N n) = Prod.swap
-- Diffeomorph.uniqueMDiffOn_image_aux
theorem uniqueMDiffOn_image_aux (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set M}
    (hs : UniqueMDiffOn I s) : UniqueMDiffOn J (h '' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners J N] (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set M} (hs : UniqueMDiffOn I s)
Goal: UniqueMDiffOn J (⇑h '' s)
-- Diffeomorph.uniqueMDiffOn_image
@[simp]
theorem uniqueMDiffOn_image (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set M} :
    UniqueMDiffOn J (h '' s) ↔ UniqueMDiffOn I s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners J N] (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set M}
Goal: UniqueMDiffOn J (⇑h '' s) ↔ UniqueMDiffOn I s
-- Diffeomorph.uniqueMDiffOn_preimage
@[simp]
theorem uniqueMDiffOn_preimage (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set N} :
    UniqueMDiffOn I (h ⁻¹' s) ↔ UniqueMDiffOn J s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners J N] (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set N}
Goal: UniqueMDiffOn I (⇑h ⁻¹' s) ↔ UniqueMDiffOn J s
-- Diffeomorph.uniqueDiffOn_image
@[simp]
theorem uniqueDiffOn_image (h : E ≃ₘ^n⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) (hn : 1 ≤ n) {s : Set E} :
    UniqueDiffOn 𝕜 (h '' s) ↔ UniqueDiffOn 𝕜 s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (h : E ≃ₘ^n⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) (hn : 1 ≤ n) {s : Set E}
Goal: UniqueDiffOn 𝕜 (⇑h '' s) ↔ UniqueDiffOn 𝕜 s
-- Diffeomorph.uniqueDiffOn_preimage
@[simp]
-- Porting note (#11215): TODO: should use `E ≃ₘ^n[𝕜] F` notation
theorem uniqueDiffOn_preimage (h : E ≃ₘ^n⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) (hn : 1 ≤ n) {s : Set F} :
    UniqueDiffOn 𝕜 (h ⁻¹' s) ↔ UniqueDiffOn 𝕜 s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (h : E ≃ₘ^n⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) (hn : 1 ≤ n) {s : Set F}
Goal: UniqueDiffOn 𝕜 (⇑h ⁻¹' s) ↔ UniqueDiffOn 𝕜 s