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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 29302 | . 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | |
2 | shss 29288 | . 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 3863 ℋchba 28997 Sℋ csh 29006 Cℋ cch 29007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5189 ax-hilex 29077 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6335 df-fv 6385 df-ov 7213 df-sh 29285 df-ch 29299 |
This theorem is referenced by: chel 29308 pjhcl 29479 dfch2 29485 shlub 29492 chsscon2 29580 chscllem2 29716 pjvec 29774 pjocvec 29775 pjhf 29786 elpjrn 30268 |
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