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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 30984 | . 2 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻)) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3943 “ cima 5672 (class class class)co 7405 ↑m cmap 8822 ℕcn 12216 ⇝𝑣 chli 30689 Sℋ csh 30690 Cℋ cch 30691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-xp 5675 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fv 6545 df-ov 7408 df-ch 30983 |
This theorem is referenced by: chsssh 30987 chshii 30989 ch0 30990 chss 30991 choccl 31068 chjval 31114 chjcl 31119 pjhth 31155 pjhtheu 31156 pjpreeq 31160 pjpjpre 31181 ch0le 31203 chle0 31205 chslej 31260 chjcom 31268 chub1 31269 chlub 31271 chlej1 31272 chlej2 31273 spansnsh 31323 fh1 31380 fh2 31381 chscllem1 31399 chscllem2 31400 chscllem3 31401 chscllem4 31402 chscl 31403 pjorthi 31431 pjoi0 31479 hstoc 31984 hstnmoc 31985 ch1dle 32114 atomli 32144 chirredlem3 32154 sumdmdii 32177 |
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