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| Mirrors > Home > MPE Home > Th. List > latlej1 | Structured version Visualization version GIF version | ||
| Description: A join's first argument is less than or equal to the join. (chub1 30760 analog.) (Contributed by NM, 17-Sep-2011.) |
| Ref | Expression |
|---|---|
| latlej.b | β’ π΅ = (BaseβπΎ) |
| latlej.l | β’ β€ = (leβπΎ) |
| latlej.j | β’ β¨ = (joinβπΎ) |
| Ref | Expression |
|---|---|
| latlej1 | β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β€ (π β¨ π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latlej.b | . 2 β’ π΅ = (BaseβπΎ) | |
| 2 | latlej.l | . 2 β’ β€ = (leβπΎ) | |
| 3 | latlej.j | . 2 β’ β¨ = (joinβπΎ) | |
| 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’ (meetβπΎ) = (meetβπΎ) | |
| 8 | 1, 3, 7, 4, 5, 6 | latcl2 18389 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (β¨π, πβ© β dom β¨ β§ β¨π, πβ© β dom (meetβπΎ))) |
| 9 | 8 | simpld 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 | lejoin1 18337 | 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-lub 18299 df-join 18301 df-lat 18385 |
| This theorem is referenced by: latjlej1 18406 latnlej 18409 latnlej2 18412 latjidm 18415 latnle 18426 latabs2 18429 latmlej11 18431 latjass 18436 mod1ile 18446 lubun 18468 oldmm1 38087 olj01 38095 omllaw5N 38117 cvlexchb1 38200 cvlsupr2 38213 cvlsupr7 38218 hlatlej1 38245 hlrelat5N 38272 2atjm 38316 2llnmj 38431 lplnexllnN 38435 2llnjaN 38437 2llnm2N 38439 4atlem3a 38468 2lplnja 38490 2lplnm2N 38492 2lplnmj 38493 dalemply 38525 dalemsly 38526 dalem10 38544 dalem13 38547 dalem21 38565 dalem55 38598 2llnma1b 38657 cdlema1N 38662 elpaddn0 38671 paddasslem12 38702 paddasslem13 38703 pmapjoin 38723 dalawlem2 38743 dalawlem7 38748 dalawlem11 38752 dalawlem12 38753 lhpmcvr3 38896 lhpmcvr5N 38898 lhpmcvr6N 38899 lautj 38964 trljat1 39037 cdlemc1 39062 cdlemc4 39065 cdleme1 39098 cdleme8 39121 cdleme11g 39136 cdleme22e 39215 cdleme22eALTN 39216 cdleme23b 39221 cdleme23c 39222 cdleme27N 39240 cdleme30a 39249 cdleme35fnpq 39320 cdleme35b 39321 cdleme35c 39322 cdleme42h 39353 cdleme42i 39354 cdleme48bw 39373 cdlemg2fv2 39471 cdlemg7fvbwN 39478 cdlemg8b 39499 cdlemg11b 39513 trlcolem 39597 trljco 39611 cdlemi1 39689 cdlemk48 39821 cdlemn2 40066 dihjustlem 40087 dihord1 40089 dihord5apre 40133 dihglbcpreN 40171 dihmeetlem3N 40176 dihmeetlem11N 40188 |
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