Theorem ishil 22079

Description: The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)

Hypotheses

Ref	Expression
ishil.k	⊢ 𝐾 = (proj‘𝐻)
ishil.c	⊢ 𝐶 = (ClSubSp‘𝐻)

Assertion

Ref	Expression
ishil	⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))

Proof of Theorem ishil

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6916	. . . . 5 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = (proj‘𝐻))
2		ishil.k	. . . . 5 ⊢ 𝐾 = (proj‘𝐻)
3	1, 2	eqtr4di 2816	. . . 4 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = 𝐾)
4	3	dmeqd 5899	. . 3 ⊢ (ℎ = 𝐻 → dom (proj‘ℎ) = dom 𝐾)
5		fveq2 6916	. . . 4 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = (ClSubSp‘𝐻))
6		ishil.c	. . . 4 ⊢ 𝐶 = (ClSubSp‘𝐻)
7	5, 6	eqtr4di 2816	. . 3 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = 𝐶)
8	4, 7	eqeq12d 2779	. 2 ⊢ (ℎ = 𝐻 → (dom (proj‘ℎ) = (ClSubSp‘ℎ) ↔ dom 𝐾 = 𝐶))
9		df-hil 22065	. 2 ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}
10	8, 9	elrab2 3652	1 ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  dom cdm 5658  ‘cfv 6566  PreHilcphl 21985  ClSubSpccss 22022  projcpj 22061  Hilchil 22062

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-dm 5668  df-iota 6520  df-fv 6574  df-hil 22065

This theorem is used by:  ishil2  22080  hlhil  25819