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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 18385 | . . 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 18347 | 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 4633 class class class wbr 5147 dom cdm 5675 βcfv 6540 (class class class)co 7405 Basecbs 17140 lecple 17200 joincjn 18260 meetcmee 18261 Latclat 18380 |
| 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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
| 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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-glb 18296 df-meet 18298 df-lat 18381 |
| This theorem is referenced by: latleeqm1 18416 latmlem1 18418 latnlemlt 18421 latmidm 18423 latabs1 18424 latledi 18426 latmlej11 18427 oldmm1 38075 cmtbr3N 38112 cmtbr4N 38113 lecmtN 38114 cvrat4 38302 2llnmat 38383 llnmlplnN 38398 dalem3 38523 dalem27 38558 dalem54 38585 dalem55 38586 2lnat 38643 cdlema1N 38650 llnexchb2lem 38727 dalawlem1 38730 dalawlem6 38735 dalawlem11 38740 dalawlem12 38741 4atexlemunv 38925 4atexlemc 38928 4atexlemnclw 38929 4atexlemex2 38930 4atexlemcnd 38931 lautm 38953 trlval3 39046 cdlemeulpq 39079 cdleme3h 39094 cdleme4a 39098 cdleme9 39112 cdleme11g 39124 cdleme13 39131 cdleme16e 39141 cdlemednpq 39158 cdleme19b 39163 cdleme20e 39172 cdleme20j 39177 cdleme22cN 39201 cdleme22e 39203 cdleme22eALTN 39204 cdleme22g 39207 cdleme35b 39309 cdleme35f 39313 cdlemeg46vrg 39386 cdlemg11b 39501 cdlemg12f 39507 cdlemg19a 39542 cdlemg31a 39556 cdlemk12 39709 cdlemkole 39712 cdlemk12u 39731 cdlemk37 39773 dia2dimlem1 39923 dihopelvalcpre 40107 dihmeetlem1N 40149 dihglblem5apreN 40150 dihglblem2N 40153 dihmeetlem2N 40158 |
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