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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 28993 | . 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐻 ⊆ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ⊆ wss 3881 ℋchba 28702 Sℋ csh 28711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-hilex 28782 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-sh 28990 |
This theorem is referenced by: sheli 28997 shelii 28998 chssii 29014 hhssabloilem 29044 hhssabloi 29045 hhssnv 29047 hhssba 29054 shunssji 29152 shsval3i 29171 shjshsi 29275 span0 29325 spanuni 29327 imaelshi 29841 nlelchi 29844 hmopidmchi 29934 pjimai 29959 shatomistici 30144 |
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