Theorem issh 31501

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 31292	. . . 4 ⊢ ℋ ∈ V
2	1	elpw2 5305	. . 3 ⊢ (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3		3anass 1109	. . 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 639	. 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 2858	. . . 4 ⊢ (ℎ = 𝐻 → (0ℎ ∈ ℎ ↔ 0ℎ ∈ 𝐻))
6		id 23	. . . . . . 7 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
7	6	sqxpeqd 5694	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ × ℎ) = (𝐻 × 𝐻))
8	7	imaeq2d 6063	. . . . 5 ⊢ (ℎ = 𝐻 → ( +ℎ “ (ℎ × ℎ)) = ( +ℎ “ (𝐻 × 𝐻)))
9	8, 6	sseq12d 3978	. . . 4 ⊢ (ℎ = 𝐻 → (( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ↔ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻))
10		xpeq2 5683	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℂ × ℎ) = (ℂ × 𝐻))
11	10	imaeq2d 6063	. . . . 5 ⊢ (ℎ = 𝐻 → ( ·ℎ “ (ℂ × ℎ)) = ( ·ℎ “ (ℂ × 𝐻)))
12	11, 6	sseq12d 3978	. . . 4 ⊢ (ℎ = 𝐻 → (( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ ↔ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))
13	5, 9, 12	3anbi123d 1462	. . 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 31500	. . 3 ⊢ Sℋ = {ℎ ∈ 𝒫 ℋ ∣ (0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ)}
15	13, 14	elrab2 3663	. 2 ⊢ (𝐻 ∈ Sℋ ↔ (𝐻 ∈ 𝒫 ℋ ∧ (0ℎ ∈ 𝐻 ∧ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
16		anass 473	. 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 306	1 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913  𝒫 cpw 4567   × cxp 5660   “ cima 5665  ℂcc 11098   ℋchba 31212   +_ℎ cva 31213   ·_ℎ csm 31214  0_ℎc0v 31217   S_ℋ csh 31221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-hilex 31292

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-sh 31500

This theorem is referenced by:  issh2  31502  shss  31503  sh0  31509