Theorems · Theorem · global analysis

ContMDiffAt.iff_comp_isImmersionAtOfComplement

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} {E''' : Type u_4} {E'' : Type u} {F : Type u_5}
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup E'']
  [inst_4 : NormedSpace 𝕜 E''] [inst_5 : NormedAddCommGroup E'''] [inst_6 : NormedSpace 𝕜 E''']
  [inst_7 : NormedAddCommGroup F] [inst_8 : NormedSpace 𝕜 F] {H : Type u_7} [inst_9 : TopologicalSpace H] {G : Type u_9}
  [inst_10 : TopologicalSpace G] {G' : Type u_10} [inst_11 : TopologicalSpace G'] {I : ModelWithCorners 𝕜 E H}
  {J : ModelWithCorners 𝕜 E'' G} {J' : ModelWithCorners 𝕜 E''' G'} {M : Type u_11} [inst_12 : TopologicalSpace M]
  [inst_13 : ChartedSpace H M] {N : Type u_13} [inst_14 : TopologicalSpace N] [inst_15 : ChartedSpace G N]
  {N' : Type u_14} [inst_16 : TopologicalSpace N'] [inst_17 : ChartedSpace G' N'] {n : WithTop ℕ∞} {x : M} {f : M → N}
  {φ : N → N'},
  Manifold.IsImmersionAtOfComplement F J J' n φ (f x) →
    (ContMDiffAt I J n f x ↔ ContinuousAt f x ∧ ContMDiffAt I J' n (φ ∘ f) x)

A function f : M → N between C^n manifolds is C^n at x if and only if it is continuous at x and its composition φ ∘ f with a C^n immersion φ : N → N' at f x is C^n at x.

Defined in

Mathlib.Geometry.Manifold.Immersion

Cited by

2 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpace