Theorems · Theorem · global analysis

ContinuousLinearMap.equivProdOfSurjectiveOfIsCompl.congr_simp

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] [inst_7 : CompleteSpace E] [inst_8 : CompleteSpace (F × G)]
  (f f_1 : E →L[𝕜] F) (e_f : f = f_1) (g g_1 : E →L[𝕜] G) (e_g : g = g_1) (hf : (↑f).range = ⊤) (hg : (↑g).range = ⊤)
  (hfg : IsCompl (↑f).ker (↑g).ker),
  f.equivProdOfSurjectiveOfIsCompl g hf hg hfg = f_1.equivProdOfSurjectiveOfIsCompl g_1 ⋯ ⋯ ⋯

Defined in

Mathlib.Analysis.Calculus.Implicit

Cited by

0 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceCompleteSpace