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| Mirrors > Home > MPE Home > Th. List > latlej2 | Structured version Visualization version GIF version | ||
| Description: A join's second argument is less than or equal to the join. (chub2 31028 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 |
|---|---|
| latlej2 | 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 1134 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β πΎ β Lat) | |
| 5 | simp2 1135 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 6 | simp3 1136 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 7 | eqid 2730 | . . . 4 β’ (meetβπΎ) = (meetβπΎ) | |
| 8 | 1, 3, 7, 4, 5, 6 | latcl2 18393 | . . 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 493 | . 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 | lejoin2 18342 | 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 1085 = wceq 1539 β wcel 2104 β¨cop 4633 class class class wbr 5147 dom cdm 5675 βcfv 6542 (class class class)co 7411 Basecbs 17148 lecple 17208 joincjn 18268 meetcmee 18269 Latclat 18388 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-lub 18303 df-join 18305 df-lat 18389 |
| This theorem is referenced by: latleeqj1 18408 latjlej1 18410 latnlej 18413 latnlej2 18416 latjass 18440 lubun 18472 oldmm1 38390 cmtcomlemN 38421 cmtbr4N 38428 cvlexchb1 38503 cvlatexch1 38509 cvrval5 38589 2llnjaN 38740 4atlem3b 38772 2lplnja 38793 dalem5 38841 dalem17 38854 dalem39 38885 dalem43 38889 elpaddn0 38974 pmapjoin 39026 dalawlem2 39046 dalawlem11 39055 dalawlem12 39056 lautj 39267 trljat2 39341 cdleme0cq 39389 cdleme1 39401 cdleme3 39411 cdleme5 39414 cdleme7ga 39422 cdleme10 39428 cdleme15b 39449 cdleme16b 39453 cdleme20k 39493 cdleme22e 39518 cdleme22eALTN 39519 cdleme23c 39525 cdleme28a 39544 cdleme32e 39619 cdleme35a 39622 cdlemg4c 39786 cdlemg6c 39794 trlcolem 39900 cdlemi1 39992 dia2dimlem2 40239 cdlemm10N 40292 dihord2pre2 40400 dihord5apre 40436 dihjatc1 40485 |
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