Theorem issh 31822

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 31613	. . . 4 ⊢ ℋ ∈ V
2	1	elpw2 5299	. . 3 ⊢ (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3		3anass 1111	. . 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 640	. 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 2852	. . . 4 ⊢ (ℎ = 𝐻 → (0ℎ ∈ ℎ ↔ 0ℎ ∈ 𝐻))
6		id 23	. . . . . . 7 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
7	6	sqxpeqd 5686	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ × ℎ) = (𝐻 × 𝐻))
8	7	imaeq2d 6057	. . . . 5 ⊢ (ℎ = 𝐻 → ( +ℎ “ (ℎ × ℎ)) = ( +ℎ “ (𝐻 × 𝐻)))
9	8, 6	sseq12d 3945	. . . 4 ⊢ (ℎ = 𝐻 → (( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ↔ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻))
10		xpeq2 5675	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℂ × ℎ) = (ℂ × 𝐻))
11	10	imaeq2d 6057	. . . . 5 ⊢ (ℎ = 𝐻 → ( ·ℎ “ (ℂ × ℎ)) = ( ·ℎ “ (ℂ × 𝐻)))
12	11, 6	sseq12d 3945	. . . 4 ⊢ (ℎ = 𝐻 → (( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ ↔ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))
13	5, 9, 12	3anbi123d 1464	. . 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 31821	. . 3 ⊢ Sℋ = {ℎ ∈ 𝒫 ℋ ∣ (0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ)}
15	13, 14	elrab2 3651	. 2 ⊢ (𝐻 ∈ Sℋ ↔ (𝐻 ∈ 𝒫 ℋ ∧ (0ℎ ∈ 𝐻 ∧ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
16		anass 474	. 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)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146   ⊆ wss 3899  𝒫 cpw 4559   × cxp 5652   “ cima 5657  ℂcc 11208   ℋchba 31533   +_ℎ cva 31534   ·_ℎ csm 31535  0_ℎc0v 31538   S_ℋ csh 31542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-hilex 31613

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-sh 31821

This theorem is used by:  issh2  31823  shss  31824  sh0  31830