Theorems · Theorem · global analysis

ContMDiffAt.iff_comp_isImmersionAt

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} {E''' : Type u_4} {E'' : Type u}
  [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'''] {H : Type u_7}
  [inst_7 : TopologicalSpace H] {G : Type u_9} [inst_8 : TopologicalSpace G] {G' : Type u_10}
  [inst_9 : TopologicalSpace G'] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 E'' G}
  {J' : ModelWithCorners 𝕜 E''' G'} {M : Type u_11} [inst_10 : TopologicalSpace M] [inst_11 : ChartedSpace H M]
  {N : Type u_13} [inst_12 : TopologicalSpace N] [inst_13 : ChartedSpace G N] {N' : Type u_14}
  [inst_14 : TopologicalSpace N'] [inst_15 : ChartedSpace G' N'] {n : WithTop ℕ∞} {x : M} {f : M → N} {φ : N → N'},
  Manifold.IsImmersionAt 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

0 results in Mathlib

Foundations

Depth 211 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpace