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Mirrors > Home > HSE Home > Th. List > chsh | Structured version Visualization version GIF version |
Description: A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chsh | ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isch 28651 | . 2 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)) ⊆ 𝐻)) | |
2 | 1 | simplbi 493 | 1 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3791 “ cima 5358 (class class class)co 6922 ↑𝑚 cmap 8140 ℕcn 11374 ⇝𝑣 chli 28356 Sℋ csh 28357 Cℋ cch 28358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-xp 5361 df-cnv 5363 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fv 6143 df-ov 6925 df-ch 28650 |
This theorem is referenced by: chsssh 28654 chshii 28656 ch0 28657 chss 28658 choccl 28737 chjval 28783 chjcl 28788 pjhth 28824 pjhtheu 28825 pjpreeq 28829 pjpjpre 28850 ch0le 28872 chle0 28874 chslej 28929 chjcom 28937 chub1 28938 chlub 28940 chlej1 28941 chlej2 28942 spansnsh 28992 fh1 29049 fh2 29050 chscllem1 29068 chscllem2 29069 chscllem3 29070 chscllem4 29071 chscl 29072 pjorthi 29100 pjoi0 29148 hstoc 29653 hstnmoc 29654 ch1dle 29783 atomli 29813 chirredlem3 29823 sumdmdii 29846 |
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