Theorems · Theorem · global analysis

Manifold.IsSmoothEmbedding.prodMap

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E₁ : Type u_2} {E₂ : Type u_3} {E₃ : Type u_4} {E₄ : Type u_5}
  [inst_1 : NormedAddCommGroup E₁] [inst_2 : NormedSpace 𝕜 E₁] [inst_3 : NormedAddCommGroup E₂]
  [inst_4 : NormedSpace 𝕜 E₂] [inst_5 : NormedAddCommGroup E₃] [inst_6 : NormedSpace 𝕜 E₃]
  [inst_7 : NormedAddCommGroup E₄] [inst_8 : NormedSpace 𝕜 E₄] {H : Type u_6} {H' : Type u_7} {G : Type u_8}
  {G' : Type u_9} [inst_9 : TopologicalSpace H] [inst_10 : TopologicalSpace H'] [inst_11 : TopologicalSpace G]
  [inst_12 : TopologicalSpace G'] {I : ModelWithCorners 𝕜 E₁ H} {I' : ModelWithCorners 𝕜 E₂ H'}
  {J : ModelWithCorners 𝕜 E₃ G} {J' : ModelWithCorners 𝕜 E₄ G'} {M : Type u_10} {M' : Type u_11} {N : Type u_12}
  {N' : Type u_13} [inst_13 : TopologicalSpace M] [inst_14 : ChartedSpace H M] [inst_15 : TopologicalSpace M']
  [inst_16 : ChartedSpace H' M'] [inst_17 : TopologicalSpace N] [inst_18 : ChartedSpace G N]
  [inst_19 : TopologicalSpace N'] [inst_20 : ChartedSpace G' N'] {n : WithTop ℕ∞} {f : M → N} {g : M' → N'}
  [IsManifold I n M] [IsManifold I' n M'] [IsManifold J n N] [IsManifold J' n N'],
  Manifold.IsSmoothEmbedding I J n f →
    Manifold.IsSmoothEmbedding I' J' n g → Manifold.IsSmoothEmbedding (I.prod I') (J.prod J') n (Prod.map f g)

If f: M → N and g: M' × N' are smooth embeddings, respectively, then so is f × g: M × M' → N × N'.

Defined in

Mathlib.Geometry.Manifold.SmoothEmbedding

Cited by

0 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifoldIsManifoldIsManifold