Theorems · Theorem · global analysis

ContMDiffAt.cle_arrowCongr

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {F₁ : Type u_8} [inst_6 : NormedAddCommGroup F₁]
  [inst_7 : NormedSpace 𝕜 F₁] {F₂ : Type u_9} [inst_8 : NormedAddCommGroup F₂] [inst_9 : NormedSpace 𝕜 F₂]
  {F₃ : Type u_10} [inst_10 : NormedAddCommGroup F₃] [inst_11 : NormedSpace 𝕜 F₃] {F₄ : Type u_11}
  [inst_12 : NormedAddCommGroup F₄] [inst_13 : NormedSpace 𝕜 F₄] {n : WithTop ℕ∞} {f : M → F₁ ≃L[𝕜] F₂}
  {g : M → F₃ ≃L[𝕜] F₄} {x : M},
  ContMDiffAt I (modelWithCornersSelf 𝕜 (F₂ →L[𝕜] F₁)) n (fun x => ↑(f x).symm) x →
    ContMDiffAt I (modelWithCornersSelf 𝕜 (F₃ →L[𝕜] F₄)) n (fun x => ↑(g x)) x →
      ContMDiffAt I (modelWithCornersSelf 𝕜 ((F₁ →L[𝕜] F₃) →L[𝕜] F₂ →L[𝕜] F₄)) n (fun y => ↑((f y).arrowCongr (g y))) x

Defined in

Mathlib.Geometry.Manifold.ContMDiff.NormedSpace

Cited by

1 results in Mathlib

Foundations

Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace