Theorems · Theorem · global analysis

contMDiff_equivTangentBundleProd_symm

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {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] {E' : Type u_5} [inst_6 : NormedAddCommGroup E']
  [inst_7 : NormedSpace 𝕜 E'] {H' : Type u_6} [inst_8 : TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'}
  {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] [inst_11 : IsManifold I 1 M]
  [inst_12 : IsManifold I' 1 M'],
  ContMDiff (I.tangent.prod I'.tangent) (I.prod I').tangent n ⇑(equivTangentBundleProd I M I' M').symm

The canonical equivalence between the product of tangent bundles and the tangent bundle of a product is smooth.

Defined in

Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Cited by

2 results in Mathlib

Foundations

Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifold