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Mirrors > Home > HSE Home > Th. List > choccli | Structured version Visualization version GIF version |
Description: Closure of Cℋ orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
choccl.1 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
choccli | ⊢ (⊥‘𝐴) ∈ Cℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | choccl.1 | . 2 ⊢ 𝐴 ∈ Cℋ | |
2 | choccl 29086 | . 2 ⊢ (𝐴 ∈ Cℋ → (⊥‘𝐴) ∈ Cℋ ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (⊥‘𝐴) ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 ‘cfv 6358 Cℋ cch 28709 ⊥cort 28710 |
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 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 ax-hilex 28779 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvmulass 28787 ax-hvdistr1 28788 ax-hvdistr2 28789 ax-hvmul0 28790 ax-hfi 28859 ax-his1 28862 ax-his2 28863 ax-his3 28864 ax-his4 28865 ax-hcompl 28982 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cn 21838 df-cnp 21839 df-lm 21840 df-haus 21926 df-tx 22173 df-hmeo 22366 df-xms 22933 df-ms 22934 df-tms 22935 df-cau 23862 df-grpo 28273 df-gid 28274 df-ginv 28275 df-gdiv 28276 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-vs 28379 df-nmcv 28380 df-ims 28381 df-dip 28481 df-hnorm 28748 df-hvsub 28751 df-hlim 28752 df-hcau 28753 df-sh 28987 df-ch 29001 df-oc 29032 |
This theorem is referenced by: pjoc1i 29211 pjoc2i 29218 chsscon3i 29241 chsscon1i 29242 chdmm1i 29257 chdmm2i 29258 chdmm3i 29259 chdmm4i 29260 chdmj1i 29261 chdmj2i 29262 chdmj3i 29263 chdmj4i 29264 sshhococi 29326 h1de2bi 29334 h1de2ctlem 29335 h1de2ci 29336 spanunsni 29359 pjoml2i 29365 pjoml3i 29366 pjoml4i 29367 pjoml6i 29369 cmcmlem 29371 cmcm2i 29373 cmcm3i 29374 cmcm4i 29375 cmbr2i 29376 cmbr3i 29380 cmbr4i 29381 cm0 29389 fh3i 29403 fh4i 29404 cm2mi 29406 qlax5i 29411 qlaxr3i 29416 osumcori 29423 osumcor2i 29424 spansnji 29426 3oalem5 29446 3oalem6 29447 3oai 29448 pjcompi 29452 pjadjii 29454 pjaddii 29455 pjmulii 29457 pjss2i 29460 pjssmii 29461 pjssge0ii 29462 pjcji 29464 pjocini 29478 pjds3i 29493 pjnormi 29501 pjpythi 29502 pjneli 29503 mayetes3i 29509 riesz3i 29842 pjnormssi 29948 pjssdif2i 29954 pjssdif1i 29955 pjimai 29956 pjoccoi 29958 pjtoi 29959 pjoci 29960 pjclem1 29975 pjci 29980 hst0 30013 sto1i 30016 sto2i 30017 stlei 30020 stji1i 30022 golem1 30051 golem2 30052 goeqi 30053 stcltrlem1 30056 stcltrlem2 30057 mdsldmd1i 30111 hatomistici 30142 cvexchi 30149 atomli 30162 atordi 30164 chirredlem4 30173 chirredi 30174 mdsymi 30191 cmmdi 30196 cmdmdi 30197 mdoc1i 30205 mdoc2i 30206 dmdoc1i 30207 dmdoc2i 30208 mdcompli 30209 dmdcompli 30210 mddmdin0i 30211 |
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