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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 30798 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 1136 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β πΎ β Lat) | |
| 5 | simp2 1137 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 6 | simp3 1138 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 7 | eqid 2732 | . . . 4 β’ (meetβπΎ) = (meetβπΎ) | |
| 8 | 1, 3, 7, 4, 5, 6 | latcl2 18391 | . . 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 495 | . 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 18339 | 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 1087 = wceq 1541 β wcel 2106 β¨cop 4634 class class class wbr 5148 dom cdm 5676 βcfv 6543 (class class class)co 7411 Basecbs 17146 lecple 17206 joincjn 18266 meetcmee 18267 Latclat 18386 |
| 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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
| 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-lub 18301 df-join 18303 df-lat 18387 |
| This theorem is referenced by: latjlej1 18408 latnlej 18411 latnlej2 18414 latjidm 18417 latnle 18428 latabs2 18431 latmlej11 18433 latjass 18438 mod1ile 18448 lubun 18470 oldmm1 38173 olj01 38181 omllaw5N 38203 cvlexchb1 38286 cvlsupr2 38299 cvlsupr7 38304 hlatlej1 38331 hlrelat5N 38358 2atjm 38402 2llnmj 38517 lplnexllnN 38521 2llnjaN 38523 2llnm2N 38525 4atlem3a 38554 2lplnja 38576 2lplnm2N 38578 2lplnmj 38579 dalemply 38611 dalemsly 38612 dalem10 38630 dalem13 38633 dalem21 38651 dalem55 38684 2llnma1b 38743 cdlema1N 38748 elpaddn0 38757 paddasslem12 38788 paddasslem13 38789 pmapjoin 38809 dalawlem2 38829 dalawlem7 38834 dalawlem11 38838 dalawlem12 38839 lhpmcvr3 38982 lhpmcvr5N 38984 lhpmcvr6N 38985 lautj 39050 trljat1 39123 cdlemc1 39148 cdlemc4 39151 cdleme1 39184 cdleme8 39207 cdleme11g 39222 cdleme22e 39301 cdleme22eALTN 39302 cdleme23b 39307 cdleme23c 39308 cdleme27N 39326 cdleme30a 39335 cdleme35fnpq 39406 cdleme35b 39407 cdleme35c 39408 cdleme42h 39439 cdleme42i 39440 cdleme48bw 39459 cdlemg2fv2 39557 cdlemg7fvbwN 39564 cdlemg8b 39585 cdlemg11b 39599 trlcolem 39683 trljco 39697 cdlemi1 39775 cdlemk48 39907 cdlemn2 40152 dihjustlem 40173 dihord1 40175 dihord5apre 40219 dihglbcpreN 40257 dihmeetlem3N 40262 dihmeetlem11N 40274 |
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