Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 27549 | . 2 ⊢ (𝐴 ∈ Cℋ → (⊥‘𝐴) ∈ Cℋ ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (⊥‘𝐴) ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ‘cfv 5804 Cℋ cch 27170 ⊥cort 27171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326 ax-hcompl 27443 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-lm 20843 df-haus 20929 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-cau 22862 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-dip 26940 df-hnorm 27209 df-hvsub 27212 df-hlim 27213 df-hcau 27214 df-sh 27448 df-ch 27462 df-oc 27493 |
This theorem is referenced by: pjoc1i 27674 pjoc2i 27681 chsscon3i 27704 chsscon1i 27705 chdmm1i 27720 chdmm2i 27721 chdmm3i 27722 chdmm4i 27723 chdmj1i 27724 chdmj2i 27725 chdmj3i 27726 chdmj4i 27727 sshhococi 27789 h1de2bi 27797 h1de2ctlem 27798 h1de2ci 27799 spanunsni 27822 pjoml2i 27828 pjoml3i 27829 pjoml4i 27830 pjoml6i 27832 cmcmlem 27834 cmcm2i 27836 cmcm3i 27837 cmcm4i 27838 cmbr2i 27839 cmbr3i 27843 cmbr4i 27844 cm0 27852 fh3i 27866 fh4i 27867 cm2mi 27869 qlax5i 27874 qlaxr3i 27879 osumcori 27886 osumcor2i 27887 spansnji 27889 3oalem5 27909 3oalem6 27910 3oai 27911 pjcompi 27915 pjadjii 27917 pjaddii 27918 pjmulii 27920 pjss2i 27923 pjssmii 27924 pjssge0ii 27925 pjcji 27927 pjocini 27941 pjds3i 27956 pjnormi 27964 pjpythi 27965 pjneli 27966 mayetes3i 27972 riesz3i 28305 pjnormssi 28411 pjssdif2i 28417 pjssdif1i 28418 pjimai 28419 pjoccoi 28421 pjtoi 28422 pjoci 28423 pjclem1 28438 pjci 28443 hst0 28476 sto1i 28479 sto2i 28480 stlei 28483 stji1i 28485 golem1 28514 golem2 28515 goeqi 28516 stcltrlem1 28519 stcltrlem2 28520 mdsldmd1i 28574 hatomistici 28605 cvexchi 28612 atomli 28625 atordi 28627 chirredlem4 28636 chirredi 28637 mdsymi 28654 cmmdi 28659 cmdmdi 28660 mdoc1i 28668 mdoc2i 28669 dmdoc1i 28670 dmdoc2i 28671 mdcompli 28672 dmdcompli 28673 mddmdin0i 28674 |
Copyright terms: Public domain | W3C validator |