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| 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 31246 | . 2 ⊢ 𝐻 ∈ Sℋ | 
| 3 | 2 | shssii 31232 | 1 ⊢ 𝐻 ⊆ ℋ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 ⊆ wss 3951 ℋchba 30938 Cℋ cch 30948 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-hilex 31018 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 df-ov 7434 df-sh 31226 df-ch 31240 | 
| This theorem is referenced by: cheli 31251 chelii 31252 hhsscms 31297 chocvali 31318 chm1i 31475 chsscon3i 31480 chsscon2i 31482 chjoi 31507 chj1i 31508 shjshsi 31511 sshhococi 31565 h1dei 31569 spansnpji 31597 spanunsni 31598 h1datomi 31600 spansnji 31665 pjfi 31723 riesz3i 32081 hmopidmpji 32171 pjoccoi 32197 pjinvari 32210 stcltr2i 32294 mdsymi 32430 mdcompli 32448 dmdcompli 32449 | 
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