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Mirrors > Home > HSE Home > Th. List > shssii | 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 |
---|---|
shssi.1 | ⊢ 𝐻 ∈ Sℋ |
Ref | Expression |
---|---|
shssii | ⊢ 𝐻 ⊆ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shssi.1 | . 2 ⊢ 𝐻 ∈ Sℋ | |
2 | shss 31112 | . 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐻 ⊆ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ⊆ wss 3944 ℋchba 30821 Sℋ csh 30830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-hilex 30901 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-sh 31109 |
This theorem is referenced by: sheli 31116 shelii 31117 chssii 31133 hhssabloilem 31163 hhssabloi 31164 hhssnv 31166 hhssba 31173 shunssji 31271 shsval3i 31290 shjshsi 31394 span0 31444 spanuni 31446 imaelshi 31960 nlelchi 31963 hmopidmchi 32053 pjimai 32078 shatomistici 32263 |
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