Theorem issh 31144

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 30935	. . . 4 ⊢ ℋ ∈ V
2	1	elpw2 5292	. . 3 ⊢ (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3		3anass 1094	. . 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 2818	. . . 4 ⊢ (ℎ = 𝐻 → (0ℎ ∈ ℎ ↔ 0ℎ ∈ 𝐻))
6		id 22	. . . . . . 7 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
7	6	sqxpeqd 5673	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ × ℎ) = (𝐻 × 𝐻))
8	7	imaeq2d 6034	. . . . 5 ⊢ (ℎ = 𝐻 → ( +ℎ “ (ℎ × ℎ)) = ( +ℎ “ (𝐻 × 𝐻)))
9	8, 6	sseq12d 3983	. . . 4 ⊢ (ℎ = 𝐻 → (( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ↔ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻))
10		xpeq2 5662	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℂ × ℎ) = (ℂ × 𝐻))
11	10	imaeq2d 6034	. . . . 5 ⊢ (ℎ = 𝐻 → ( ·ℎ “ (ℂ × ℎ)) = ( ·ℎ “ (ℂ × 𝐻)))
12	11, 6	sseq12d 3983	. . . 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 31143	. . 3 ⊢ Sℋ = {ℎ ∈ 𝒫 ℋ ∣ (0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ)}
15	13, 14	elrab2 3665	. 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 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917  𝒫 cpw 4566   × cxp 5639   “ cima 5644  ℂcc 11073   ℋchba 30855   +_ℎ cva 30856   ·_ℎ csm 30857  0_ℎc0v 30860   S_ℋ csh 30864

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-hilex 30935

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-sh 31143

This theorem is referenced by:  issh2  31145  shss  31146  sh0  31152