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Mirrors > Home > MPE Home > Th. List > latjcl | Structured version Visualization version GIF version |
Description: Closure of join operation in a lattice. (chjcom 30490 analog.) (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
latjcl.b | β’ π΅ = (BaseβπΎ) |
latjcl.j | β’ β¨ = (joinβπΎ) |
Ref | Expression |
---|---|
latjcl | β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latjcl.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | latjcl.j | . . 3 β’ β¨ = (joinβπΎ) | |
3 | eqid 2733 | . . 3 β’ (meetβπΎ) = (meetβπΎ) | |
4 | 1, 2, 3 | latlem 18331 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β ((π β¨ π) β π΅ β§ (π(meetβπΎ)π) β π΅)) |
5 | 4 | simpld 496 | 1 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17088 joincjn 18205 meetcmee 18206 Latclat 18325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-lub 18240 df-glb 18241 df-join 18242 df-meet 18243 df-lat 18326 |
This theorem is referenced by: latleeqj1 18345 latjlej1 18347 latjlej12 18349 latnlej2 18353 latjidm 18356 latnle 18367 latabs2 18370 latledi 18371 latmlej11 18372 latjass 18377 latj13 18380 latj31 18381 latj4 18383 mod1ile 18387 mod2ile 18388 latdisdlem 18390 lubun 18409 oldmm1 37725 olj01 37733 latmassOLD 37737 omllaw5N 37755 cmtcomlemN 37756 cmtbr2N 37761 cmtbr3N 37762 cmtbr4N 37763 lecmtN 37764 omlfh1N 37766 omlfh3N 37767 omlmod1i2N 37768 cvlexchb1 37838 cvlcvr1 37847 hlatjcl 37875 exatleN 37913 cvrval3 37922 cvrexchlem 37928 cvrexch 37929 cvratlem 37930 cvrat 37931 lnnat 37936 cvrat2 37938 atcvrj2b 37941 atltcvr 37944 atlelt 37947 2atlt 37948 atexchcvrN 37949 cvrat3 37951 cvrat4 37952 2atjm 37954 4noncolr3 37962 athgt 37965 3dim0 37966 3dimlem4a 37972 1cvratex 37982 1cvrjat 37984 1cvrat 37985 ps-2 37987 3atlem1 37992 3atlem2 37993 3at 37999 2atm 38036 lplni2 38046 lplnle 38049 2llnmj 38069 2atmat 38070 lplnexllnN 38073 2llnjaN 38075 lvoli3 38086 islvol5 38088 lvoli2 38090 lvolnle3at 38091 3atnelvolN 38095 islvol2aN 38101 4atlem3 38105 4atlem4d 38111 4atlem9 38112 4atlem10a 38113 4atlem10 38115 4atlem11a 38116 4atlem11b 38117 4atlem11 38118 4atlem12a 38119 4atlem12b 38120 4atlem12 38121 4at 38122 lplncvrlvol2 38124 2lplnja 38128 2lplnmj 38131 dalem5 38176 dalem8 38179 dalem-cly 38180 dalem38 38219 dalem39 38220 dalem44 38225 dalem54 38235 linepsubN 38261 pmapsub 38277 isline2 38283 linepmap 38284 isline3 38285 lncvrelatN 38290 2llnma1b 38295 cdlema1N 38300 cdlemblem 38302 cdlemb 38303 paddasslem5 38333 paddasslem12 38340 paddasslem13 38341 pmapjoin 38361 pmapjat1 38362 pmapjlln1 38364 hlmod1i 38365 llnmod1i2 38369 atmod2i1 38370 atmod2i2 38371 llnmod2i2 38372 atmod3i1 38373 atmod3i2 38374 dalawlem2 38381 dalawlem3 38382 dalawlem5 38384 dalawlem6 38385 dalawlem7 38386 dalawlem8 38387 dalawlem11 38390 dalawlem12 38391 pmapocjN 38439 paddatclN 38458 linepsubclN 38460 pl42lem1N 38488 pl42lem2N 38489 pl42N 38492 lhp2lt 38510 lhpj1 38531 lhpmod2i2 38547 lhpmod6i1 38548 4atexlemc 38578 lautj 38602 trlval2 38672 trlcl 38673 trljat1 38675 trljat2 38676 trlle 38693 cdlemc1 38700 cdlemc2 38701 cdlemc5 38704 cdlemd2 38708 cdlemd3 38709 cdleme0aa 38719 cdleme0b 38721 cdleme0c 38722 cdleme0cp 38723 cdleme0cq 38724 cdleme0fN 38727 cdleme1b 38735 cdleme1 38736 cdleme2 38737 cdleme3b 38738 cdleme3c 38739 cdleme4a 38748 cdleme5 38749 cdleme7e 38756 cdleme8 38759 cdleme9 38762 cdleme10 38763 cdleme11fN 38773 cdleme11g 38774 cdleme11k 38777 cdleme11 38779 cdleme15b 38784 cdleme15 38787 cdleme22gb 38803 cdleme19b 38813 cdleme20d 38821 cdleme20j 38827 cdleme20l 38831 cdleme20m 38832 cdleme22e 38853 cdleme22eALTN 38854 cdleme22f 38855 cdleme23b 38859 cdleme23c 38860 cdleme28a 38879 cdleme28b 38880 cdleme29ex 38883 cdleme30a 38887 cdlemefr29exN 38911 cdleme32e 38954 cdleme35fnpq 38958 cdleme35b 38959 cdleme35c 38960 cdleme42e 38988 cdleme42i 38992 cdleme42mgN 38997 cdlemg2fv2 39109 cdlemg7fvbwN 39116 cdlemg4c 39121 cdlemg6c 39129 cdlemg10 39150 cdlemg11b 39151 cdlemg31a 39206 cdlemg31b 39207 cdlemg35 39222 trlcolem 39235 cdlemg44a 39240 trljco 39249 tendopltp 39289 cdlemh1 39324 cdlemh2 39325 cdlemi1 39327 cdlemi 39329 cdlemk4 39343 cdlemkvcl 39351 cdlemk10 39352 cdlemk11 39358 cdlemk11u 39380 cdlemk37 39423 cdlemkid1 39431 cdlemk50 39461 cdlemk51 39462 cdlemk52 39463 dialss 39555 dia2dimlem2 39574 dia2dimlem3 39575 cdlemm10N 39627 docaclN 39633 doca2N 39635 djajN 39646 diblss 39679 cdlemn2 39704 cdlemn10 39715 dihord1 39727 dihord2pre2 39735 dihord5apre 39771 dihjatc1 39820 dihmeetlem10N 39825 dihmeetlem11N 39826 djhljjN 39911 djhj 39913 dihprrnlem1N 39933 dihprrnlem2 39934 dihjat6 39943 dihjat5N 39946 dvh4dimat 39947 |
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