Theorems · Theorem · global analysis

Manifold.IsSubmersionAtOfComplement.target_subset_preimage_target

∀ {𝕜 : Type u_1} {E'' : Type u_3} {F : Type u_5} {H : Type u_7} {G : Type u_9} {E : Type u}
  [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E]
  [inst_3 : NormedAddCommGroup E''] [inst_4 : NormedSpace 𝕜 E''] [inst_5 : NormedAddCommGroup F]
  [inst_6 : NormedSpace 𝕜 F] [inst_7 : TopologicalSpace H] [inst_8 : TopologicalSpace G] {I : ModelWithCorners 𝕜 E H}
  {J : ModelWithCorners 𝕜 E'' G} {M : Type u_11} {N : Type u_13} [inst_9 : TopologicalSpace M]
  [inst_10 : ChartedSpace H M] [inst_11 : TopologicalSpace N] [inst_12 : ChartedSpace G N] {n : WithTop ℕ∞} {f : M → N}
  {x : M} (h : Manifold.IsSubmersionAtOfComplement F I J n f x),
  (h.domChart.extend I).target ⊆ Prod.fst ∘ ⇑h.equiv ⁻¹' (h.codChart.extend J).target

If f is a submersion at x, its domain chart's target (h.domChart.extend I).target is mapped to its codomain chart's target (h.domChart.extend J).target: see image_target_subset_target for a version stated using images.

Defined in

Mathlib.Geometry.Manifold.Submersion

Cited by

0 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpace