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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 28993 | . 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐻 ∈ Sℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 Sℋ csh 28697 Cℋ cch 28698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 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 7151 df-ch 28990 |
This theorem is referenced by: chssii 29000 helsh 29014 h0elsh 29025 hhsscms 29047 hhssbnOLD 29048 chocunii 29070 shsleji 29139 shjshcli 29145 pjhthlem1 29160 pjhthlem2 29161 omlsii 29172 ococi 29174 pjoc1i 29200 chne0i 29222 chocini 29223 chjcli 29226 chsleji 29227 chseli 29228 chunssji 29236 chjcomi 29237 chub1i 29238 chlubi 29240 chlej1i 29242 chlej2i 29243 h1de2bi 29323 h1de2ctlem 29324 spansnpji 29347 spanunsni 29348 h1datomi 29350 pjoml2i 29354 qlaxr3i 29405 osumi 29411 osumcor2i 29413 spansnji 29415 spansnm0i 29419 nonbooli 29420 spansncvi 29421 5oai 29430 3oalem2 29432 3oalem5 29435 3oalem6 29436 pjaddii 29444 pjmulii 29446 pjss2i 29449 pjssmii 29450 pj0i 29462 pjocini 29467 pjjsi 29469 pjpythi 29491 mayete3i 29497 pjnmopi 29917 pjimai 29945 pjclem4 29968 pj3si 29976 sto1i 30005 stlei 30009 strlem1 30019 hatomici 30128 hatomistici 30131 atomli 30151 chirredlem3 30161 sumdmdii 30184 sumdmdlem 30187 sumdmdlem2 30188 |
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