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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 1137 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β πΎ β Lat) | |
| 5 | simp2 1138 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 6 | simp3 1139 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 7 | eqid 2733 | . . . 4 β’ (joinβπΎ) = (joinβπΎ) | |
| 8 | 1, 7, 3, 4, 5, 6 | latcl2 18389 | . . 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 497 | . 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 18351 | 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 β¨cop 4635 class class class wbr 5149 dom cdm 5677 βcfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 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-glb 18300 df-meet 18302 df-lat 18385 |
| This theorem is referenced by: latleeqm1 18420 latmlem1 18422 latnlemlt 18425 latmidm 18427 latabs1 18428 latledi 18430 latmlej11 18431 oldmm1 38087 cmtbr3N 38124 cmtbr4N 38125 lecmtN 38126 cvrat4 38314 2llnmat 38395 llnmlplnN 38410 dalem3 38535 dalem27 38570 dalem54 38597 dalem55 38598 2lnat 38655 cdlema1N 38662 llnexchb2lem 38739 dalawlem1 38742 dalawlem6 38747 dalawlem11 38752 dalawlem12 38753 4atexlemunv 38937 4atexlemc 38940 4atexlemnclw 38941 4atexlemex2 38942 4atexlemcnd 38943 lautm 38965 trlval3 39058 cdlemeulpq 39091 cdleme3h 39106 cdleme4a 39110 cdleme9 39124 cdleme11g 39136 cdleme13 39143 cdleme16e 39153 cdlemednpq 39170 cdleme19b 39175 cdleme20e 39184 cdleme20j 39189 cdleme22cN 39213 cdleme22e 39215 cdleme22eALTN 39216 cdleme22g 39219 cdleme35b 39321 cdleme35f 39325 cdlemeg46vrg 39398 cdlemg11b 39513 cdlemg12f 39519 cdlemg19a 39554 cdlemg31a 39568 cdlemk12 39721 cdlemkole 39724 cdlemk12u 39743 cdlemk37 39785 dia2dimlem1 39935 dihopelvalcpre 40119 dihmeetlem1N 40161 dihglblem5apreN 40162 dihglblem2N 40165 dihmeetlem2N 40170 |
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