Theorems · Theorem · global analysis

mfderivWithin_prodMap

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {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'] {F : Type u_11}
  [inst_11 : NormedAddCommGroup F] [inst_12 : NormedSpace 𝕜 F] {G : Type u_12} [inst_13 : TopologicalSpace G]
  {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [inst_14 : TopologicalSpace N] [inst_15 : ChartedSpace G N]
  {F' : Type u_14} [inst_16 : NormedAddCommGroup F'] [inst_17 : NormedSpace 𝕜 F'] {G' : Type u_15}
  [inst_18 : TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [inst_19 : TopologicalSpace N']
  [inst_20 : ChartedSpace G' N'] {s : Set M} {p : M × M'} {t : Set M'} {f : M → N} {g : M' → N'},
  MDiffAt[s] f p.1 →
    MDiffAt[t] g p.2 →
      UniqueMDiffAt[s] p.1 →
        UniqueMDiffAt[t] p.2 → mfderiv[s ×ˢ t] (Prod.map f g) p = (mfderiv[s] f p.1).prodMap (mfderiv[t] g p.2)

Defined in

Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Cited by

1 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpace