Theorems · Theorem · global analysis

Manifold.IsSubmersionAtOfComplement.prodMap

∀ {𝕜 : Type u_1} {E' : Type u_2} {E'' : Type u_3} {E''' : Type u_4} {F : Type u_5} {F' : Type u_6} {H : Type u_7}
  {H' : Type u_8} {G : Type u_9} {G' : Type u_10} {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 E''] [inst_6 : NormedSpace 𝕜 E'']
  [inst_7 : NormedAddCommGroup E'''] [inst_8 : NormedSpace 𝕜 E'''] [inst_9 : NormedAddCommGroup F]
  [inst_10 : NormedSpace 𝕜 F] [inst_11 : NormedAddCommGroup F'] [inst_12 : NormedSpace 𝕜 F']
  [inst_13 : TopologicalSpace H] [inst_14 : TopologicalSpace H'] [inst_15 : TopologicalSpace G]
  [inst_16 : TopologicalSpace G'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'}
  {J : ModelWithCorners 𝕜 E'' G} {J' : ModelWithCorners 𝕜 E''' G'} {M : Type u_11} {M' : Type u_12} {N : Type u_13}
  {N' : Type u_14} [inst_17 : TopologicalSpace M] [inst_18 : ChartedSpace H M] [inst_19 : TopologicalSpace M']
  [inst_20 : ChartedSpace H' M'] [inst_21 : TopologicalSpace N] [inst_22 : ChartedSpace G N]
  [inst_23 : TopologicalSpace N'] [inst_24 : ChartedSpace G' N'] {n : WithTop ℕ∞} {x : M} {f : M → N} {g : M' → N'}
  {x' : M'} [IsManifold I n M] [⋯], ⋯

If f: M → N and g: M' → N' are submersions at x and x', respectively, then f × g: M × M' → N × N' is a submersion at (x, x').

Defined in

Mathlib.Geometry.Manifold.Submersion

Cited by

2 results in Mathlib

Foundations

Depth 208 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifoldIsManifoldIsManifold