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Mirrors > Home > HSE Home > Th. List > issh | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
issh | ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hilex 28770 | . . . 4 ⊢ ℋ ∈ V | |
2 | 1 | elpw2 5241 | . . 3 ⊢ (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ) |
3 | 3anass 1091 | . . 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 2901 | . . . 4 ⊢ (ℎ = 𝐻 → (0ℎ ∈ ℎ ↔ 0ℎ ∈ 𝐻)) | |
6 | id 22 | . . . . . . 7 ⊢ (ℎ = 𝐻 → ℎ = 𝐻) | |
7 | 6 | sqxpeqd 5582 | . . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ × ℎ) = (𝐻 × 𝐻)) |
8 | 7 | imaeq2d 5924 | . . . . 5 ⊢ (ℎ = 𝐻 → ( +ℎ “ (ℎ × ℎ)) = ( +ℎ “ (𝐻 × 𝐻))) |
9 | 8, 6 | sseq12d 4000 | . . . 4 ⊢ (ℎ = 𝐻 → (( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ↔ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻)) |
10 | xpeq2 5571 | . . . . . 6 ⊢ (ℎ = 𝐻 → (ℂ × ℎ) = (ℂ × 𝐻)) | |
11 | 10 | imaeq2d 5924 | . . . . 5 ⊢ (ℎ = 𝐻 → ( ·ℎ “ (ℂ × ℎ)) = ( ·ℎ “ (ℂ × 𝐻))) |
12 | 11, 6 | sseq12d 4000 | . . . 4 ⊢ (ℎ = 𝐻 → (( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ ↔ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)) |
13 | 5, 9, 12 | 3anbi123d 1432 | . . 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 28978 | . . 3 ⊢ Sℋ = {ℎ ∈ 𝒫 ℋ ∣ (0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ)} | |
15 | 13, 14 | elrab2 3683 | . 2 ⊢ (𝐻 ∈ Sℋ ↔ (𝐻 ∈ 𝒫 ℋ ∧ (0ℎ ∈ 𝐻 ∧ ( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) |
16 | anass 471 | . 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 305 | 1 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 𝒫 cpw 4539 × cxp 5548 “ cima 5553 ℂcc 10529 ℋchba 28690 +ℎ cva 28691 ·ℎ csm 28692 0ℎc0v 28695 Sℋ csh 28699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-hilex 28770 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-sh 28978 |
This theorem is referenced by: issh2 28980 shss 28981 sh0 28987 |
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