Theorem ishil 20062

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 = (CSubSp‘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 6191	. . . . 5 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = (proj‘𝐻))
2		ishil.k	. . . . 5 ⊢ 𝐾 = (proj‘𝐻)
3	1, 2	syl6eqr 2674	. . . 4 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = 𝐾)
4	3	dmeqd 5326	. . 3 ⊢ (ℎ = 𝐻 → dom (proj‘ℎ) = dom 𝐾)
5		fveq2 6191	. . . 4 ⊢ (ℎ = 𝐻 → (CSubSp‘ℎ) = (CSubSp‘𝐻))
6		ishil.c	. . . 4 ⊢ 𝐶 = (CSubSp‘𝐻)
7	5, 6	syl6eqr 2674	. . 3 ⊢ (ℎ = 𝐻 → (CSubSp‘ℎ) = 𝐶)
8	4, 7	eqeq12d 2637	. 2 ⊢ (ℎ = 𝐻 → (dom (proj‘ℎ) = (CSubSp‘ℎ) ↔ dom 𝐾 = 𝐶))
9		df-hil 20048	. 2 ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (CSubSp‘ℎ)}
10	8, 9	elrab2 3366	1 ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  dom cdm 5114  ‘cfv 5888  PreHilcphl 19969  CSubSpccss 20005  projcpj 20044  Hilchs 20045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-hil 20048

This theorem is referenced by:  ishil2  20063  hlhil  23214