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Mirrors > Home > MPE Home > Th. List > latjle12 | Structured version Visualization version GIF version |
Description: A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 30749 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 |
---|---|
latjle12 | β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) 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 | latpos 18387 | . . 3 β’ (πΎ β Lat β πΎ β Poset) | |
5 | 4 | adantr 481 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β Poset) |
6 | simpr1 1194 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
7 | simpr2 1195 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
8 | simpr3 1196 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
9 | eqid 2732 | . . . 4 β’ (meetβπΎ) = (meetβπΎ) | |
10 | simpl 483 | . . . 4 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β Lat) | |
11 | 1, 3, 9, 10, 6, 7 | latcl2 18385 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (β¨π, πβ© β dom β¨ β§ β¨π, πβ© β dom (meetβπΎ))) |
12 | 11 | simpld 495 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β β¨π, πβ© β dom β¨ ) |
13 | 1, 2, 3, 5, 6, 7, 8, 12 | joinle 18335 | 1 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) 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 wb 205 β§ wa 396 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 Posetcpo 18256 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-poset 18262 df-lub 18295 df-join 18297 df-lat 18381 |
This theorem is referenced by: latleeqj1 18400 latjlej1 18402 latjidm 18411 latledi 18426 latjass 18432 mod1ile 18442 lubun 18464 oldmm1 38075 olj01 38083 cvlexchb1 38188 cvlcvr1 38197 hlrelat 38261 hlrelat2 38262 exatleN 38263 hlrelat3 38271 cvrexchlem 38278 cvratlem 38280 cvrat 38281 atlelt 38297 ps-1 38336 hlatexch3N 38339 hlatexch4 38340 3atlem1 38342 3atlem2 38343 lplnexllnN 38423 2llnjaN 38425 4atlem3 38455 4atlem10 38465 4atlem11b 38467 4atlem11 38468 4atlem12b 38470 4atlem12 38471 2lplnja 38478 dalem1 38518 dalem3 38523 dalem8 38529 dalem16 38538 dalem17 38539 dalem21 38553 dalem25 38557 dalem39 38570 dalem54 38585 dalem60 38591 linepsubN 38611 pmapsub 38627 lneq2at 38637 2llnma3r 38647 cdlema1N 38650 cdlemblem 38652 paddasslem5 38683 paddasslem12 38690 paddasslem13 38691 llnexchb2 38728 dalawlem3 38732 dalawlem5 38734 dalawlem8 38737 dalawlem11 38740 dalawlem12 38741 lhp2lt 38860 lhpexle2lem 38868 lhpexle3lem 38870 4atexlemtlw 38926 4atexlemnclw 38929 lautj 38952 cdlemd3 39059 cdleme3g 39093 cdleme3h 39094 cdleme7d 39105 cdleme11c 39120 cdleme15d 39136 cdleme17b 39146 cdleme19a 39162 cdleme20j 39177 cdleme21c 39186 cdleme22b 39200 cdleme22d 39202 cdleme28a 39229 cdleme35a 39307 cdleme35fnpq 39308 cdleme35b 39309 cdleme35f 39313 cdleme42c 39331 cdleme42i 39342 cdlemf1 39420 cdlemg4c 39471 cdlemg6c 39479 cdlemg8b 39487 cdlemg10 39500 cdlemg11b 39501 cdlemg13a 39510 cdlemg17a 39520 cdlemg18b 39538 cdlemg27a 39551 cdlemg33b0 39560 cdlemg35 39572 cdlemg42 39588 cdlemg46 39594 trljco 39599 tendopltp 39639 cdlemk3 39692 cdlemk10 39702 cdlemk1u 39718 cdlemk39 39775 dialss 39905 dia2dimlem1 39923 dia2dimlem10 39932 dia2dimlem12 39934 cdlemm10N 39977 djajN 39996 diblss 40029 cdlemn2 40054 dihord2pre2 40085 dib2dim 40102 dih2dimb 40103 dih2dimbALTN 40104 dihmeetlem6 40168 dihjatcclem1 40277 |
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