Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > chsh | Structured version Visualization version GIF version |
Description: A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chsh | ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isch 28926 | . 2 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻)) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3933 “ cima 5551 (class class class)co 7145 ↑m cmap 8395 ℕcn 11626 ⇝𝑣 chli 28631 Sℋ csh 28632 Cℋ cch 28633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 df-ov 7148 df-ch 28925 |
This theorem is referenced by: chsssh 28929 chshii 28931 ch0 28932 chss 28933 choccl 29010 chjval 29056 chjcl 29061 pjhth 29097 pjhtheu 29098 pjpreeq 29102 pjpjpre 29123 ch0le 29145 chle0 29147 chslej 29202 chjcom 29210 chub1 29211 chlub 29213 chlej1 29214 chlej2 29215 spansnsh 29265 fh1 29322 fh2 29323 chscllem1 29341 chscllem2 29342 chscllem3 29343 chscllem4 29344 chscl 29345 pjorthi 29373 pjoi0 29421 hstoc 29926 hstnmoc 29927 ch1dle 30056 atomli 30086 chirredlem3 30096 sumdmdii 30119 |
Copyright terms: Public domain | W3C validator |