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Mirrors > Home > HSE Home > Th. List > chshii | Structured version Visualization version GIF version |
Description: A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chshi.1 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
chshii | ⊢ 𝐻 ∈ Sℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chshi.1 | . 2 ⊢ 𝐻 ∈ Cℋ | |
2 | chsh 31106 | . 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐻 ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Sℋ csh 30810 Cℋ cch 30811 |
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 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fv 6557 df-ov 7422 df-ch 31103 |
This theorem is referenced by: chssii 31113 helsh 31127 h0elsh 31138 hhsscms 31160 hhssbnOLD 31161 chocunii 31183 shsleji 31252 shjshcli 31258 pjhthlem1 31273 pjhthlem2 31274 omlsii 31285 ococi 31287 pjoc1i 31313 chne0i 31335 chocini 31336 chjcli 31339 chsleji 31340 chseli 31341 chunssji 31349 chjcomi 31350 chub1i 31351 chlubi 31353 chlej1i 31355 chlej2i 31356 h1de2bi 31436 h1de2ctlem 31437 spansnpji 31460 spanunsni 31461 h1datomi 31463 pjoml2i 31467 qlaxr3i 31518 osumi 31524 osumcor2i 31526 spansnji 31528 spansnm0i 31532 nonbooli 31533 spansncvi 31534 5oai 31543 3oalem2 31545 3oalem5 31548 3oalem6 31549 pjaddii 31557 pjmulii 31559 pjss2i 31562 pjssmii 31563 pj0i 31575 pjocini 31580 pjjsi 31582 pjpythi 31604 mayete3i 31610 pjnmopi 32030 pjimai 32058 pjclem4 32081 pj3si 32089 sto1i 32118 stlei 32122 strlem1 32132 hatomici 32241 hatomistici 32244 atomli 32264 chirredlem3 32274 sumdmdii 32297 sumdmdlem 32300 sumdmdlem2 32301 |
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