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Mirrors > Home > HSE Home > Th. List > choccl | Structured version Visualization version GIF version |
Description: Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.) |
Ref | Expression |
---|---|
choccl | ⊢ (𝐴 ∈ Cℋ → (⊥‘𝐴) ∈ Cℋ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chsh 28999 | . 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
2 | shoccl 29080 | . 2 ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Cℋ ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ Cℋ → (⊥‘𝐴) ∈ Cℋ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ‘cfv 6348 Sℋ csh 28703 Cℋ cch 28704 ⊥cort 28705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 ax-hilex 28774 ax-hfvadd 28775 ax-hvcom 28776 ax-hvass 28777 ax-hv0cl 28778 ax-hvaddid 28779 ax-hfvmul 28780 ax-hvmulid 28781 ax-hvmulass 28782 ax-hvdistr1 28783 ax-hvdistr2 28784 ax-hvmul0 28785 ax-hfi 28854 ax-his1 28857 ax-his2 28858 ax-his3 28859 ax-his4 28860 ax-hcompl 28977 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-fi 8868 df-sup 8899 df-inf 8900 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-icc 12739 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-clim 14840 df-sum 15038 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-hom 16584 df-cco 16585 df-rest 16691 df-topn 16692 df-0g 16710 df-gsum 16711 df-topgen 16712 df-pt 16713 df-prds 16716 df-xrs 16770 df-qtop 16775 df-imas 16776 df-xps 16778 df-mre 16852 df-mrc 16853 df-acs 16855 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-submnd 17952 df-mulg 18220 df-cntz 18442 df-cmn 18903 df-psmet 20532 df-xmet 20533 df-met 20534 df-bl 20535 df-mopn 20536 df-cnfld 20541 df-top 21497 df-topon 21514 df-topsp 21536 df-bases 21549 df-cn 21830 df-cnp 21831 df-lm 21832 df-haus 21918 df-tx 22165 df-hmeo 22358 df-xms 22925 df-ms 22926 df-tms 22927 df-cau 23854 df-grpo 28268 df-gid 28269 df-ginv 28270 df-gdiv 28271 df-ablo 28320 df-vc 28334 df-nv 28367 df-va 28370 df-ba 28371 df-sm 28372 df-0v 28373 df-vs 28374 df-nmcv 28375 df-ims 28376 df-dip 28476 df-hnorm 28743 df-hvsub 28746 df-hlim 28747 df-hcau 28748 df-sh 28982 df-ch 28996 df-oc 29027 |
This theorem is referenced by: choccli 29082 pjhtheu2 29191 pjpjpre 29194 pjpjhth 29200 pjop 29202 pjpo 29203 pjoccl 29208 chssoc 29271 chsscon1 29276 chpsscon1 29279 chpsscon2 29280 chdmm2 29301 chdmm3 29302 chdmm4 29303 chdmj1 29304 chdmj2 29305 chdmj3 29306 chdmj4 29307 spansnch 29335 pjspansn 29352 cmcm2 29391 fh1 29393 fh2 29394 cm2j 29395 pjorthi 29444 pjo 29446 pjocvec 29472 hstoc 29997 hstnmoc 29998 hstle1 30001 hst1h 30002 hstle 30005 hstoh 30007 cvcon3 30059 dmdmd 30075 mddmd 30076 ssdmd1 30088 ssdmd2 30089 cvdmd 30112 h1da 30124 atom1d 30128 chirredlem1 30165 chirredlem2 30166 dmdsym 30188 |
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