Theorem ishil 21005

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 6811	. . . . 5 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = (proj‘𝐻))
2		ishil.k	. . . . 5 ⊢ 𝐾 = (proj‘𝐻)
3	1, 2	eqtr4di 2794	. . . 4 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = 𝐾)
4	3	dmeqd 5834	. . 3 ⊢ (ℎ = 𝐻 → dom (proj‘ℎ) = dom 𝐾)
5		fveq2 6811	. . . 4 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = (ClSubSp‘𝐻))
6		ishil.c	. . . 4 ⊢ 𝐶 = (ClSubSp‘𝐻)
7	5, 6	eqtr4di 2794	. . 3 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = 𝐶)
8	4, 7	eqeq12d 2752	. 2 ⊢ (ℎ = 𝐻 → (dom (proj‘ℎ) = (ClSubSp‘ℎ) ↔ dom 𝐾 = 𝐶))
9		df-hil 20991	. 2 ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}
10	8, 9	elrab2 3636	1 ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  dom cdm 5607  ‘cfv 6465  PreHilcphl 20909  ClSubSpccss 20946  projcpj 20987  Hilchil 20988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-br 5087  df-dm 5617  df-iota 6417  df-fv 6473  df-hil 20991

This theorem is referenced by:  ishil2  21006  hlhil  24687