Theorem ishil 19881

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 6103	. . . . 5 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = (proj‘𝐻))
2		ishil.k	. . . . 5 ⊢ 𝐾 = (proj‘𝐻)
3	1, 2	syl6eqr 2662	. . . 4 ⊢ (ℎ = 𝐻 → (proj‘ℎ) = 𝐾)
4	3	dmeqd 5248	. . 3 ⊢ (ℎ = 𝐻 → dom (proj‘ℎ) = dom 𝐾)
5		fveq2 6103	. . . 4 ⊢ (ℎ = 𝐻 → (CSubSp‘ℎ) = (CSubSp‘𝐻))
6		ishil.c	. . . 4 ⊢ 𝐶 = (CSubSp‘𝐻)
7	5, 6	syl6eqr 2662	. . 3 ⊢ (ℎ = 𝐻 → (CSubSp‘ℎ) = 𝐶)
8	4, 7	eqeq12d 2625	. 2 ⊢ (ℎ = 𝐻 → (dom (proj‘ℎ) = (CSubSp‘ℎ) ↔ dom 𝐾 = 𝐶))
9		df-hil 19867	. 2 ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (CSubSp‘ℎ)}
10	8, 9	elrab2 3333	1 ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  dom cdm 5038  ‘cfv 5804  PreHilcphl 19788  CSubSpccss 19824  projcpj 19863  Hilchs 19864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-hil 19867

This theorem is referenced by:  ishil2  19882  hlhil  23022