Theorems · Theorem · global analysis

Manifold.IsImmersionAt.prodMap

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} {E' : Type u_3} {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'']
  [inst_7 : NormedAddCommGroup E'''] [inst_8 : NormedSpace 𝕜 E'''] {H : Type u_7} [inst_9 : TopologicalSpace H]
  {H' : Type u_8} [inst_10 : TopologicalSpace H'] {G : Type u_9} [inst_11 : TopologicalSpace G] {G' : Type u_10}
  [inst_12 : 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} [inst_13 : TopologicalSpace M]
  [inst_14 : ChartedSpace H M] {M' : Type u_12} [inst_15 : TopologicalSpace M'] [inst_16 : ChartedSpace H' M']
  {N : Type u_13} [inst_17 : TopologicalSpace N] [inst_18 : ChartedSpace G N] {N' : Type u_14}
  [inst_19 : TopologicalSpace N'] [inst_20 : ChartedSpace G' N'] {n : WithTop ℕ∞} {x : M} {f : M → N} {g : M' → N'}
  {x' : M'} [IsManifold I n M] [IsManifold I' n M'] [IsManifold J n N] [IsManifold J' n N'],
  Manifold.IsImmersionAt I J n f x →
    Manifold.IsImmersionAt I' J' n g x' → Manifold.IsImmersionAt (I.prod I') (J.prod J') n (Prod.map f g) (x, x')

If f: M → N and g: M' × N' are immersions at x and x', respectively, then f × g: M × N → M' × N' is an immersion at (x, x').

Defined in

Mathlib.Geometry.Manifold.Immersion

Cited by

0 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifoldIsManifoldIsManifold