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Mirrors > Home > HSE Home > Th. List > chjcli | Structured version Visualization version GIF version |
Description: Closure of Cℋ join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0le.1 | ⊢ 𝐴 ∈ Cℋ |
chjcl.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
chjcli | ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch0le.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | chshii 28609 | . 2 ⊢ 𝐴 ∈ Sℋ |
3 | chjcl.2 | . . 3 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | chshii 28609 | . 2 ⊢ 𝐵 ∈ Sℋ |
5 | 2, 4 | shjcli 28759 | 1 ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 (class class class)co 6878 Cℋ cch 28311 ∨ℋ chj 28315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304 ax-hilex 28381 ax-hfvadd 28382 ax-hvcom 28383 ax-hvass 28384 ax-hv0cl 28385 ax-hvaddid 28386 ax-hfvmul 28387 ax-hvmulid 28388 ax-hvmulass 28389 ax-hvdistr1 28390 ax-hvdistr2 28391 ax-hvmul0 28392 ax-hfi 28461 ax-his1 28464 ax-his2 28465 ax-his3 28466 ax-his4 28467 ax-hcompl 28584 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-icc 12431 df-fz 12581 df-fzo 12721 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-clim 14560 df-sum 14758 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-hom 16291 df-cco 16292 df-rest 16398 df-topn 16399 df-0g 16417 df-gsum 16418 df-topgen 16419 df-pt 16420 df-prds 16423 df-xrs 16477 df-qtop 16482 df-imas 16483 df-xps 16485 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-mulg 17857 df-cntz 18062 df-cmn 18510 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-cnfld 20069 df-top 21027 df-topon 21044 df-topsp 21066 df-bases 21079 df-cn 21360 df-cnp 21361 df-lm 21362 df-haus 21448 df-tx 21694 df-hmeo 21887 df-xms 22453 df-ms 22454 df-tms 22455 df-cau 23382 df-grpo 27873 df-gid 27874 df-ginv 27875 df-gdiv 27876 df-ablo 27925 df-vc 27939 df-nv 27972 df-va 27975 df-ba 27976 df-sm 27977 df-0v 27978 df-vs 27979 df-nmcv 27980 df-ims 27981 df-dip 28081 df-hnorm 28350 df-hvsub 28353 df-hlim 28354 df-hcau 28355 df-sh 28589 df-ch 28603 df-oc 28634 df-chj 28694 |
This theorem is referenced by: chdmm1i 28861 chjassi 28870 chj1i 28873 chj4i 28907 lejdii 28922 pjoml2i 28969 pjoml3i 28970 pjoml4i 28971 pjoml5i 28972 cmcmlem 28975 cmbr2i 28980 cmj1i 28988 cmj2i 28989 fh3i 29007 fh4i 29008 qlaxr3i 29020 osumcor2i 29028 spansnji 29030 5oai 29045 3oalem5 29050 3oalem6 29051 3oai 29052 pjdsi 29096 pjds3i 29097 mayete3i 29112 mayetes3i 29113 pjscji 29554 pjci 29584 stlei 29624 golem2 29656 goeqi 29657 stcltrlem2 29661 mdslle1i 29701 mdslj1i 29703 mdslj2i 29704 mdslmd1lem1 29709 mdslmd1lem2 29710 mdslmd1i 29713 mdslmd2i 29714 mdsldmd1i 29715 mdexchi 29719 cvexchi 29753 atabsi 29785 atabs2i 29786 mdsymlem6 29792 sumdmdlem2 29803 dmdbr5ati 29806 mdcompli 29813 dmdcompli 29814 |
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