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| Mirrors > Home > MPE Home > Th. List > latmle1 | Structured version Visualization version GIF version | ||
| Description: A meet is less than or equal to its first 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 |
|---|---|
| latmle1 | 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 1136 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β πΎ β Lat) | |
| 5 | simp2 1137 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 6 | simp3 1138 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 7 | eqid 2732 | . . . 4 β’ (joinβπΎ) = (joinβπΎ) | |
| 8 | 1, 7, 3, 4, 5, 6 | latcl2 18393 | . . 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 496 | . 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 | lemeet1 18355 | 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 β¨cop 4634 class class class wbr 5148 dom cdm 5676 βcfv 6543 (class class class)co 7411 Basecbs 17148 lecple 17208 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-glb 18304 df-meet 18306 df-lat 18389 |
| This theorem is referenced by: latleeqm1 18424 latmlem1 18426 latnlemlt 18429 latmidm 18431 latabs1 18432 latledi 18434 latmlej11 18435 oldmm1 38390 cmtbr3N 38427 cmtbr4N 38428 lecmtN 38429 cvrat4 38617 2llnmat 38698 llnmlplnN 38713 dalem3 38838 dalem27 38873 dalem54 38900 dalem55 38901 2lnat 38958 cdlema1N 38965 llnexchb2lem 39042 dalawlem1 39045 dalawlem6 39050 dalawlem11 39055 dalawlem12 39056 4atexlemunv 39240 4atexlemc 39243 4atexlemnclw 39244 4atexlemex2 39245 4atexlemcnd 39246 lautm 39268 trlval3 39361 cdlemeulpq 39394 cdleme3h 39409 cdleme4a 39413 cdleme9 39427 cdleme11g 39439 cdleme13 39446 cdleme16e 39456 cdlemednpq 39473 cdleme19b 39478 cdleme20e 39487 cdleme20j 39492 cdleme22cN 39516 cdleme22e 39518 cdleme22eALTN 39519 cdleme22g 39522 cdleme35b 39624 cdleme35f 39628 cdlemeg46vrg 39701 cdlemg11b 39816 cdlemg12f 39822 cdlemg19a 39857 cdlemg31a 39871 cdlemk12 40024 cdlemkole 40027 cdlemk12u 40046 cdlemk37 40088 dia2dimlem1 40238 dihopelvalcpre 40422 dihmeetlem1N 40464 dihglblem5apreN 40465 dihglblem2N 40468 dihmeetlem2N 40473 |
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