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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 28928 | . 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | |
2 | shss 28914 | . 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 2105 ⊆ wss 3933 ℋchba 28623 Sℋ csh 28632 Cℋ cch 28633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-hilex 28703 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 df-ov 7148 df-sh 28911 df-ch 28925 |
This theorem is referenced by: chel 28934 pjhcl 29105 dfch2 29111 shlub 29118 chsscon2 29206 chscllem2 29342 pjvec 29400 pjocvec 29401 pjhf 29412 elpjrn 29894 |
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