Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > latjcl | Structured version Visualization version GIF version | ||
| Description: Closure of join operation in a lattice. (chjcom 31014 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 2732 | . . 3 β’ (meetβπΎ) = (meetβπΎ) | |
| 4 | 1, 2, 3 | latlem 18394 | . 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 495 | 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 1087 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7411 Basecbs 17148 joincjn 18268 meetcmee 18269 Latclat 18388 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-lat 18389 |
| This theorem is referenced by: latleeqj1 18408 latjlej1 18410 latjlej12 18412 latnlej2 18416 latjidm 18419 latnle 18430 latabs2 18433 latledi 18434 latmlej11 18435 latjass 18440 latj13 18443 latj31 18444 latj4 18446 mod1ile 18450 mod2ile 18451 latdisdlem 18453 lubun 18472 oldmm1 38390 olj01 38398 latmassOLD 38402 omllaw5N 38420 cmtcomlemN 38421 cmtbr2N 38426 cmtbr3N 38427 cmtbr4N 38428 lecmtN 38429 omlfh1N 38431 omlfh3N 38432 omlmod1i2N 38433 cvlexchb1 38503 cvlcvr1 38512 hlatjcl 38540 exatleN 38578 cvrval3 38587 cvrexchlem 38593 cvrexch 38594 cvratlem 38595 cvrat 38596 lnnat 38601 cvrat2 38603 atcvrj2b 38606 atltcvr 38609 atlelt 38612 2atlt 38613 atexchcvrN 38614 cvrat3 38616 cvrat4 38617 2atjm 38619 4noncolr3 38627 athgt 38630 3dim0 38631 3dimlem4a 38637 1cvratex 38647 1cvrjat 38649 1cvrat 38650 ps-2 38652 3atlem1 38657 3atlem2 38658 3at 38664 2atm 38701 lplni2 38711 lplnle 38714 2llnmj 38734 2atmat 38735 lplnexllnN 38738 2llnjaN 38740 lvoli3 38751 islvol5 38753 lvoli2 38755 lvolnle3at 38756 3atnelvolN 38760 islvol2aN 38766 4atlem3 38770 4atlem4d 38776 4atlem9 38777 4atlem10a 38778 4atlem10 38780 4atlem11a 38781 4atlem11b 38782 4atlem11 38783 4atlem12a 38784 4atlem12b 38785 4atlem12 38786 4at 38787 lplncvrlvol2 38789 2lplnja 38793 2lplnmj 38796 dalem5 38841 dalem8 38844 dalem-cly 38845 dalem38 38884 dalem39 38885 dalem44 38890 dalem54 38900 linepsubN 38926 pmapsub 38942 isline2 38948 linepmap 38949 isline3 38950 lncvrelatN 38955 2llnma1b 38960 cdlema1N 38965 cdlemblem 38967 cdlemb 38968 paddasslem5 38998 paddasslem12 39005 paddasslem13 39006 pmapjoin 39026 pmapjat1 39027 pmapjlln1 39029 hlmod1i 39030 llnmod1i2 39034 atmod2i1 39035 atmod2i2 39036 llnmod2i2 39037 atmod3i1 39038 atmod3i2 39039 dalawlem2 39046 dalawlem3 39047 dalawlem5 39049 dalawlem6 39050 dalawlem7 39051 dalawlem8 39052 dalawlem11 39055 dalawlem12 39056 pmapocjN 39104 paddatclN 39123 linepsubclN 39125 pl42lem1N 39153 pl42lem2N 39154 pl42N 39157 lhp2lt 39175 lhpj1 39196 lhpmod2i2 39212 lhpmod6i1 39213 4atexlemc 39243 lautj 39267 trlval2 39337 trlcl 39338 trljat1 39340 trljat2 39341 trlle 39358 cdlemc1 39365 cdlemc2 39366 cdlemc5 39369 cdlemd2 39373 cdlemd3 39374 cdleme0aa 39384 cdleme0b 39386 cdleme0c 39387 cdleme0cp 39388 cdleme0cq 39389 cdleme0fN 39392 cdleme1b 39400 cdleme1 39401 cdleme2 39402 cdleme3b 39403 cdleme3c 39404 cdleme4a 39413 cdleme5 39414 cdleme7e 39421 cdleme8 39424 cdleme9 39427 cdleme10 39428 cdleme11fN 39438 cdleme11g 39439 cdleme11k 39442 cdleme11 39444 cdleme15b 39449 cdleme15 39452 cdleme22gb 39468 cdleme19b 39478 cdleme20d 39486 cdleme20j 39492 cdleme20l 39496 cdleme20m 39497 cdleme22e 39518 cdleme22eALTN 39519 cdleme22f 39520 cdleme23b 39524 cdleme23c 39525 cdleme28a 39544 cdleme28b 39545 cdleme29ex 39548 cdleme30a 39552 cdlemefr29exN 39576 cdleme32e 39619 cdleme35fnpq 39623 cdleme35b 39624 cdleme35c 39625 cdleme42e 39653 cdleme42i 39657 cdleme42mgN 39662 cdlemg2fv2 39774 cdlemg7fvbwN 39781 cdlemg4c 39786 cdlemg6c 39794 cdlemg10 39815 cdlemg11b 39816 cdlemg31a 39871 cdlemg31b 39872 cdlemg35 39887 trlcolem 39900 cdlemg44a 39905 trljco 39914 tendopltp 39954 cdlemh1 39989 cdlemh2 39990 cdlemi1 39992 cdlemi 39994 cdlemk4 40008 cdlemkvcl 40016 cdlemk10 40017 cdlemk11 40023 cdlemk11u 40045 cdlemk37 40088 cdlemkid1 40096 cdlemk50 40126 cdlemk51 40127 cdlemk52 40128 dialss 40220 dia2dimlem2 40239 dia2dimlem3 40240 cdlemm10N 40292 docaclN 40298 doca2N 40300 djajN 40311 diblss 40344 cdlemn2 40369 cdlemn10 40380 dihord1 40392 dihord2pre2 40400 dihord5apre 40436 dihjatc1 40485 dihmeetlem10N 40490 dihmeetlem11N 40491 djhljjN 40576 djhj 40578 dihprrnlem1N 40598 dihprrnlem2 40599 dihjat6 40608 dihjat5N 40611 dvh4dimat 40612 |
| Copyright terms: Public domain | W3C validator |