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| Mirrors > Home > MPE Home > Th. List > latmle2 | Structured version Visualization version GIF version | ||
| Description: A meet is less than or equal to its second argument. (Contributed by NM, 21-Oct-2011.) |
| Ref | Expression |
|---|---|
| latmle.b | β’ π΅ = (BaseβπΎ) |
| latmle.l | β’ β€ = (leβπΎ) |
| latmle.m | β’ β§ = (meetβπΎ) |
| Ref | Expression |
|---|---|
| latmle2 | β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) β€ π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latmle.b | . 2 β’ π΅ = (BaseβπΎ) | |
| 2 | latmle.l | . 2 β’ β€ = (leβπΎ) | |
| 3 | latmle.m | . 2 β’ β§ = (meetβπΎ) | |
| 4 | simp1 1135 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β πΎ β Lat) | |
| 5 | simp2 1136 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 6 | simp3 1137 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 7 | eqid 2731 | . . . 4 β’ (joinβπΎ) = (joinβπΎ) | |
| 8 | 1, 7, 3, 4, 5, 6 | latcl2 18394 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (β¨π, πβ© β dom (joinβπΎ) β§ β¨π, πβ© β dom β§ )) |
| 9 | 8 | simprd 495 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β β¨π, πβ© β dom β§ ) |
| 10 | 1, 2, 3, 4, 5, 6, 9 | lemeet2 18357 | 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 1086 = wceq 1540 β wcel 2105 β¨cop 4634 class class class wbr 5148 dom cdm 5676 βcfv 6543 (class class class)co 7412 Basecbs 17149 lecple 17209 joincjn 18269 meetcmee 18270 Latclat 18389 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-glb 18305 df-meet 18307 df-lat 18390 |
| This theorem is referenced by: latmlem1 18427 latledi 18435 mod1ile 18451 oldmm1 38391 olm01 38410 cmtcomlemN 38422 cmtbr4N 38429 meetat 38470 cvrexchlem 38594 cvrat4 38618 2llnmj 38735 2lplnmj 38797 dalem25 38873 dalem54 38901 dalem57 38904 cdlema1N 38966 cdlemb 38969 llnexchb2lem 39043 llnexch2N 39045 dalawlem1 39046 dalawlem3 39048 pl42lem1N 39154 lhpelim 39212 lhpat3 39221 4atexlemunv 39241 4atexlemtlw 39242 4atexlemnclw 39245 4atexlemex2 39246 lautm 39269 trlle 39359 cdlemc2 39367 cdlemc5 39370 cdlemd2 39374 cdleme0b 39387 cdleme0c 39388 cdleme0fN 39393 cdleme01N 39396 cdleme0ex1N 39398 cdleme2 39403 cdleme3b 39404 cdleme3c 39405 cdleme3g 39409 cdleme3h 39410 cdleme7aa 39417 cdleme7c 39420 cdleme7d 39421 cdleme7e 39422 cdleme7ga 39423 cdleme11fN 39439 cdleme11k 39443 cdleme15d 39452 cdleme16f 39458 cdlemednpq 39474 cdleme19c 39480 cdleme20aN 39484 cdleme20c 39486 cdleme20j 39493 cdleme21c 39502 cdleme21ct 39504 cdleme22cN 39517 cdleme22f 39521 cdleme23a 39524 cdleme28a 39545 cdleme35d 39627 cdleme35f 39629 cdlemeg46frv 39700 cdlemeg46rgv 39703 cdlemeg46req 39704 cdlemg2fv2 39775 cdlemg2m 39779 cdlemg4 39792 cdlemg10bALTN 39811 cdlemg31b 39873 trlcolem 39901 cdlemk14 40029 dia2dimlem1 40239 docaclN 40299 doca2N 40301 djajN 40312 dihjustlem 40391 dihord1 40393 dihord2a 40394 dihord2b 40395 dihord2cN 40396 dihord11b 40397 dihord11c 40399 dihord2pre 40400 dihlsscpre 40409 dihvalcq2 40422 dihopelvalcpre 40423 dihord6apre 40431 dihord5b 40434 dihord5apre 40437 dihmeetlem1N 40465 dihglblem5apreN 40466 dihglblem3N 40470 dihmeetbclemN 40479 dihmeetlem4preN 40481 dihmeetlem7N 40485 dihmeetlem9N 40490 dihjatcclem4 40596 |
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