![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > ch0 | Structured version Visualization version GIF version |
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 29007 | . 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | |
2 | sh0 28999 | . 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 2111 0ℎc0v 28707 Sℋ csh 28711 Cℋ cch 28712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-hilex 28782 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fv 6332 df-ov 7138 df-sh 28990 df-ch 29004 |
This theorem is referenced by: omlsii 29186 nonbooli 29434 strlem1 30033 |
Copyright terms: Public domain | W3C validator |