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Mirrors > Home > HSE Home > Th. List > chss | Structured version Visualization version GIF version |
Description: A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chss | ⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chsh 28995 | . 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | |
2 | shss 28981 | . 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3935 ℋchba 28690 Sℋ csh 28699 Cℋ cch 28700 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-hilex 28770 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fv 6357 df-ov 7153 df-sh 28978 df-ch 28992 |
This theorem is referenced by: chel 29001 pjhcl 29172 dfch2 29178 shlub 29185 chsscon2 29273 chscllem2 29409 pjvec 29467 pjocvec 29468 pjhf 29479 elpjrn 29961 |
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