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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 28079 | . 2 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)) ⊆ 𝐻)) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ⊆ wss 3574 “ cima 5117 (class class class)co 6650 ↑𝑚 cmap 7857 ℕcn 11020 ⇝𝑣 chli 27784 Sℋ csh 27785 Cℋ cch 27786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fv 5896 df-ov 6653 df-ch 28078 |
This theorem is referenced by: chsssh 28082 chshii 28084 ch0 28085 chss 28086 choccl 28165 chjval 28211 chjcl 28216 pjhth 28252 pjhtheu 28253 pjpreeq 28257 pjpjpre 28278 ch0le 28300 chle0 28302 chslej 28357 chjcom 28365 chub1 28366 chlub 28368 chlej1 28369 chlej2 28370 spansnsh 28420 fh1 28477 fh2 28478 chscllem1 28496 chscllem2 28497 chscllem3 28498 chscllem4 28499 chscl 28500 pjorthi 28528 pjoi0 28576 hstoc 29081 hstnmoc 29082 ch1dle 29211 atomli 29241 chirredlem3 29251 sumdmdii 29274 |
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