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| Description: The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| ch0 | ⊢ (𝐻 ∈ Cℋ → 0ℎ ∈ 𝐻) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | chsh 31244 | . 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | |
| 2 | sh0 31236 | . 2 ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝐻 ∈ Cℋ → 0ℎ ∈ 𝐻) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 0ℎc0v 30944 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 ax-sep 5295 ax-hilex 31019 | 
| 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-pw 4601 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-sh 31227 df-ch 31241 | 
| This theorem is referenced by: omlsii 31423 nonbooli 31671 strlem1 32270 | 
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