Theorem issh 31227

Description: Subspace 𝐻 of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
issh	⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))

Proof of Theorem issh

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 31018	. . . 4 ⊢ ℋ ∈ V
2	1	elpw2 5334	. . 3 ⊢ (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3		3anass 1095	. . 3 ⊢ ((0ℎ ∈ 𝐻 ∧ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (0ℎ ∈ 𝐻 ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
4	2, 3	anbi12i 628	. 2 ⊢ ((𝐻 ∈ 𝒫 ℋ ∧ (0ℎ ∈ 𝐻 ∧ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0ℎ ∈ 𝐻 ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))))
5		eleq2 2830	. . . 4 ⊢ (ℎ = 𝐻 → (0ℎ ∈ ℎ ↔ 0ℎ ∈ 𝐻))
6		id 22	. . . . . . 7 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
7	6	sqxpeqd 5717	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ × ℎ) = (𝐻 × 𝐻))
8	7	imaeq2d 6078	. . . . 5 ⊢ (ℎ = 𝐻 → ( +ℎ “ (ℎ × ℎ)) = ( +ℎ “ (𝐻 × 𝐻)))
9	8, 6	sseq12d 4017	. . . 4 ⊢ (ℎ = 𝐻 → (( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ↔ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻))
10		xpeq2 5706	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℂ × ℎ) = (ℂ × 𝐻))
11	10	imaeq2d 6078	. . . . 5 ⊢ (ℎ = 𝐻 → ( ·ℎ “ (ℂ × ℎ)) = ( ·ℎ “ (ℂ × 𝐻)))
12	11, 6	sseq12d 4017	. . . 4 ⊢ (ℎ = 𝐻 → (( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ ↔ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))
13	5, 9, 12	3anbi123d 1438	. . 3 ⊢ (ℎ = 𝐻 → ((0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ) ↔ (0ℎ ∈ 𝐻 ∧ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
14		df-sh 31226	. . 3 ⊢ Sℋ = {ℎ ∈ 𝒫 ℋ ∣ (0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ)}
15	13, 14	elrab2 3695	. 2 ⊢ (𝐻 ∈ Sℋ ↔ (𝐻 ∈ 𝒫 ℋ ∧ (0ℎ ∈ 𝐻 ∧ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
16		anass 468	. 2 ⊢ (((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0ℎ ∈ 𝐻 ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))))
17	4, 15, 16	3bitr4i 303	1 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  𝒫 cpw 4600   × cxp 5683   “ cima 5688  ℂcc 11153   ℋchba 30938   +_ℎ cva 30939   ·_ℎ csm 30940  0_ℎc0v 30943   S_ℋ csh 30947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-hilex 31018

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-sh 31226

This theorem is referenced by:  issh2  31228  shss  31229  sh0  31235