|   | Hilbert Space Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > HSE Home > Th. List > isch | Structured version Visualization version GIF version | ||
| Description: Closed subspace 𝐻 of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| isch | ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | oveq1 7439 | . . . 4 ⊢ (ℎ = 𝐻 → (ℎ ↑m ℕ) = (𝐻 ↑m ℕ)) | |
| 2 | 1 | imaeq2d 6077 | . . 3 ⊢ (ℎ = 𝐻 → ( ⇝𝑣 “ (ℎ ↑m ℕ)) = ( ⇝𝑣 “ (𝐻 ↑m ℕ))) | 
| 3 | id 22 | . . 3 ⊢ (ℎ = 𝐻 → ℎ = 𝐻) | |
| 4 | 2, 3 | sseq12d 4016 | . 2 ⊢ (ℎ = 𝐻 → (( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ ↔ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻)) | 
| 5 | df-ch 31241 | . 2 ⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ} | |
| 6 | 4, 5 | elrab2 3694 | 1 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 “ cima 5687 (class class class)co 7432 ↑m cmap 8867 ℕcn 12267 ⇝𝑣 chli 30947 Sℋ csh 30948 Cℋ cch 30949 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fv 6568 df-ov 7435 df-ch 31241 | 
| This theorem is referenced by: isch2 31243 chsh 31244 | 
| Copyright terms: Public domain | W3C validator |