Theorems · Theorem · functional analysis

DirectSum.IsInternal.collectedOrthonormalBasis.congr_simp

∀ {ι : Type u_1} {𝕜 : Type u_3} [inst : RCLike 𝕜] {E : Type u_4} [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] [inst_3 : Fintype ι] {A : ι → Submodule 𝕜 E}
  (hV : OrthogonalFamily 𝕜 (fun i => ↥(A i)) fun i => (A i).subtypeₗᵢ) [inst_4 : DecidableEq ι]
  (hV_sum : DirectSum.IsInternal fun i => A i) {α : ι → Type u_7} [inst_5 : (i : ι) → Fintype (α i)]
  (v_family v_family_1 : (i : ι) → OrthonormalBasis (α i) 𝕜 ↥(A i)),
  v_family = v_family_1 →
    DirectSum.IsInternal.collectedOrthonormalBasis hV hV_sum v_family =
      DirectSum.IsInternal.collectedOrthonormalBasis hV hV_sum v_family_1

Defined in

Mathlib.Analysis.InnerProductSpace.PiL2

Cited by

0 results in Mathlib

Foundations

Depth 233 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFintypeDecidableEqFintype