Theorem ishil 21761

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 6920	. . . . 5 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = (proj‘𝐻))
2		ishil.k	. . . . 5 ⊢ 𝐾 = (proj‘𝐻)
3	1, 2	eqtr4di 2798	. . . 4 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = 𝐾)
4	3	dmeqd 5930	. . 3 ⊢ (ℎ = 𝐻 → dom (proj‘ℎ) = dom 𝐾)
5		fveq2 6920	. . . 4 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = (ClSubSp‘𝐻))
6		ishil.c	. . . 4 ⊢ 𝐶 = (ClSubSp‘𝐻)
7	5, 6	eqtr4di 2798	. . 3 ⊢ (ℎ = 𝐻 → (ClSubSp‘ℎ) = 𝐶)
8	4, 7	eqeq12d 2756	. 2 ⊢ (ℎ = 𝐻 → (dom (proj‘ℎ) = (ClSubSp‘ℎ) ↔ dom 𝐾 = 𝐶))
9		df-hil 21747	. 2 ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}
10	8, 9	elrab2 3711	1 ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  dom cdm 5700  ‘cfv 6573  PreHilcphl 21665  ClSubSpccss 21702  projcpj 21743  Hilchil 21744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-dm 5710  df-iota 6525  df-fv 6581  df-hil 21747

This theorem is referenced by:  ishil2  21762  hlhil  25496