![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > chssii | Structured version Visualization version GIF version |
Description: A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chssi.1 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
chssii | ⊢ 𝐻 ⊆ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chssi.1 | . . 3 ⊢ 𝐻 ∈ Cℋ | |
2 | 1 | chshii 31256 | . 2 ⊢ 𝐻 ∈ Sℋ |
3 | 2 | shssii 31242 | 1 ⊢ 𝐻 ⊆ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ⊆ wss 3963 ℋchba 30948 Cℋ cch 30958 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-hilex 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fv 6571 df-ov 7434 df-sh 31236 df-ch 31250 |
This theorem is referenced by: cheli 31261 chelii 31262 hhsscms 31307 chocvali 31328 chm1i 31485 chsscon3i 31490 chsscon2i 31492 chjoi 31517 chj1i 31518 shjshsi 31521 sshhococi 31575 h1dei 31579 spansnpji 31607 spanunsni 31608 h1datomi 31610 spansnji 31675 pjfi 31733 riesz3i 32091 hmopidmpji 32181 pjoccoi 32207 pjinvari 32220 stcltr2i 32304 mdsymi 32440 mdcompli 32458 dmdcompli 32459 |
Copyright terms: Public domain | W3C validator |