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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 31160 | . 2 ⊢ 𝐻 ∈ Sℋ |
3 | 2 | shssii 31146 | 1 ⊢ 𝐻 ⊆ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ⊆ wss 3947 ℋchba 30852 Cℋ cch 30862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-hilex 30932 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fv 6562 df-ov 7427 df-sh 31140 df-ch 31154 |
This theorem is referenced by: cheli 31165 chelii 31166 hhsscms 31211 chocvali 31232 chm1i 31389 chsscon3i 31394 chsscon2i 31396 chjoi 31421 chj1i 31422 shjshsi 31425 sshhococi 31479 h1dei 31483 spansnpji 31511 spanunsni 31512 h1datomi 31514 spansnji 31579 pjfi 31637 riesz3i 31995 hmopidmpji 32085 pjoccoi 32111 pjinvari 32124 stcltr2i 32208 mdsymi 32344 mdcompli 32362 dmdcompli 32363 |
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