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| Mirrors > Home > MPE Home > Th. List > latleeqm2 | Structured version Visualization version GIF version | ||
| Description: "Less than or equal to" in terms of meet. (Contributed by NM, 7-Nov-2011.) |
| Ref | Expression |
|---|---|
| latmle.b | β’ π΅ = (BaseβπΎ) |
| latmle.l | β’ β€ = (leβπΎ) |
| latmle.m | β’ β§ = (meetβπΎ) |
| Ref | Expression |
|---|---|
| latleeqm2 | β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β€ π β (π β§ π) = π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latmle.b | . . 3 β’ π΅ = (BaseβπΎ) | |
| 2 | latmle.l | . . 3 β’ β€ = (leβπΎ) | |
| 3 | latmle.m | . . 3 β’ β§ = (meetβπΎ) | |
| 4 | 1, 2, 3 | latleeqm1 18425 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β€ π β (π β§ π) = π)) |
| 5 | 1, 3 | latmcom 18421 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) = (π β§ π)) |
| 6 | 5 | eqeq1d 2733 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β ((π β§ π) = π β (π β§ π) = π)) |
| 7 | 4, 6 | bitrd 278 | 1 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β€ π β (π β§ π) = π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ w3a 1086 = wceq 1540 β wcel 2105 class class class wbr 5149 βcfv 6544 (class class class)co 7412 Basecbs 17149 lecple 17209 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 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 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 7368 df-ov 7415 df-oprab 7416 df-proset 18253 df-poset 18271 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-lat 18390 |
| This theorem is referenced by: cmtcomlemN 38422 omlmod1i2N 38434 2llnma3r 38963 dalawlem7 39052 dalawlem11 39056 dalawlem12 39057 lhp2at0 39207 lhp2atnle 39208 cdleme9 39428 cdleme11g 39440 cdleme35c 39626 cdlemh1 39990 dia2dimlem2 40240 dia2dimlem3 40241 dihmeetlem15N 40496 |
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