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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 30759 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 18390 | . 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 6544 (class class class)co 7409 Basecbs 17144 joincjn 18264 meetcmee 18265 Latclat 18384 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| 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 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-lat 18385 |
| This theorem is referenced by: latleeqj1 18404 latjlej1 18406 latjlej12 18408 latnlej2 18412 latjidm 18415 latnle 18426 latabs2 18429 latledi 18430 latmlej11 18431 latjass 18436 latj13 18439 latj31 18440 latj4 18442 mod1ile 18446 mod2ile 18447 latdisdlem 18449 lubun 18468 oldmm1 38087 olj01 38095 latmassOLD 38099 omllaw5N 38117 cmtcomlemN 38118 cmtbr2N 38123 cmtbr3N 38124 cmtbr4N 38125 lecmtN 38126 omlfh1N 38128 omlfh3N 38129 omlmod1i2N 38130 cvlexchb1 38200 cvlcvr1 38209 hlatjcl 38237 exatleN 38275 cvrval3 38284 cvrexchlem 38290 cvrexch 38291 cvratlem 38292 cvrat 38293 lnnat 38298 cvrat2 38300 atcvrj2b 38303 atltcvr 38306 atlelt 38309 2atlt 38310 atexchcvrN 38311 cvrat3 38313 cvrat4 38314 2atjm 38316 4noncolr3 38324 athgt 38327 3dim0 38328 3dimlem4a 38334 1cvratex 38344 1cvrjat 38346 1cvrat 38347 ps-2 38349 3atlem1 38354 3atlem2 38355 3at 38361 2atm 38398 lplni2 38408 lplnle 38411 2llnmj 38431 2atmat 38432 lplnexllnN 38435 2llnjaN 38437 lvoli3 38448 islvol5 38450 lvoli2 38452 lvolnle3at 38453 3atnelvolN 38457 islvol2aN 38463 4atlem3 38467 4atlem4d 38473 4atlem9 38474 4atlem10a 38475 4atlem10 38477 4atlem11a 38478 4atlem11b 38479 4atlem11 38480 4atlem12a 38481 4atlem12b 38482 4atlem12 38483 4at 38484 lplncvrlvol2 38486 2lplnja 38490 2lplnmj 38493 dalem5 38538 dalem8 38541 dalem-cly 38542 dalem38 38581 dalem39 38582 dalem44 38587 dalem54 38597 linepsubN 38623 pmapsub 38639 isline2 38645 linepmap 38646 isline3 38647 lncvrelatN 38652 2llnma1b 38657 cdlema1N 38662 cdlemblem 38664 cdlemb 38665 paddasslem5 38695 paddasslem12 38702 paddasslem13 38703 pmapjoin 38723 pmapjat1 38724 pmapjlln1 38726 hlmod1i 38727 llnmod1i2 38731 atmod2i1 38732 atmod2i2 38733 llnmod2i2 38734 atmod3i1 38735 atmod3i2 38736 dalawlem2 38743 dalawlem3 38744 dalawlem5 38746 dalawlem6 38747 dalawlem7 38748 dalawlem8 38749 dalawlem11 38752 dalawlem12 38753 pmapocjN 38801 paddatclN 38820 linepsubclN 38822 pl42lem1N 38850 pl42lem2N 38851 pl42N 38854 lhp2lt 38872 lhpj1 38893 lhpmod2i2 38909 lhpmod6i1 38910 4atexlemc 38940 lautj 38964 trlval2 39034 trlcl 39035 trljat1 39037 trljat2 39038 trlle 39055 cdlemc1 39062 cdlemc2 39063 cdlemc5 39066 cdlemd2 39070 cdlemd3 39071 cdleme0aa 39081 cdleme0b 39083 cdleme0c 39084 cdleme0cp 39085 cdleme0cq 39086 cdleme0fN 39089 cdleme1b 39097 cdleme1 39098 cdleme2 39099 cdleme3b 39100 cdleme3c 39101 cdleme4a 39110 cdleme5 39111 cdleme7e 39118 cdleme8 39121 cdleme9 39124 cdleme10 39125 cdleme11fN 39135 cdleme11g 39136 cdleme11k 39139 cdleme11 39141 cdleme15b 39146 cdleme15 39149 cdleme22gb 39165 cdleme19b 39175 cdleme20d 39183 cdleme20j 39189 cdleme20l 39193 cdleme20m 39194 cdleme22e 39215 cdleme22eALTN 39216 cdleme22f 39217 cdleme23b 39221 cdleme23c 39222 cdleme28a 39241 cdleme28b 39242 cdleme29ex 39245 cdleme30a 39249 cdlemefr29exN 39273 cdleme32e 39316 cdleme35fnpq 39320 cdleme35b 39321 cdleme35c 39322 cdleme42e 39350 cdleme42i 39354 cdleme42mgN 39359 cdlemg2fv2 39471 cdlemg7fvbwN 39478 cdlemg4c 39483 cdlemg6c 39491 cdlemg10 39512 cdlemg11b 39513 cdlemg31a 39568 cdlemg31b 39569 cdlemg35 39584 trlcolem 39597 cdlemg44a 39602 trljco 39611 tendopltp 39651 cdlemh1 39686 cdlemh2 39687 cdlemi1 39689 cdlemi 39691 cdlemk4 39705 cdlemkvcl 39713 cdlemk10 39714 cdlemk11 39720 cdlemk11u 39742 cdlemk37 39785 cdlemkid1 39793 cdlemk50 39823 cdlemk51 39824 cdlemk52 39825 dialss 39917 dia2dimlem2 39936 dia2dimlem3 39937 cdlemm10N 39989 docaclN 39995 doca2N 39997 djajN 40008 diblss 40041 cdlemn2 40066 cdlemn10 40077 dihord1 40089 dihord2pre2 40097 dihord5apre 40133 dihjatc1 40182 dihmeetlem10N 40187 dihmeetlem11N 40188 djhljjN 40273 djhj 40275 dihprrnlem1N 40295 dihprrnlem2 40296 dihjat6 40305 dihjat5N 40308 dvh4dimat 40309 |
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