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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatlej1 | Structured version Visualization version GIF version |
Description: A join's first argument is less than or equal to the join. Special case of latlej1 18401 to show an atom is on a line. (Contributed by NM, 15-May-2013.) |
Ref | Expression |
---|---|
hlatlej.l | β’ β€ = (leβπΎ) |
hlatlej.j | β’ β¨ = (joinβπΎ) |
hlatlej.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
hlatlej1 | β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β π β€ (π β¨ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 38233 | . 2 β’ (πΎ β HL β πΎ β Lat) | |
2 | eqid 2733 | . . 3 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | hlatlej.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | 2, 3 | atbase 38159 | . 2 β’ (π β π΄ β π β (BaseβπΎ)) |
5 | 2, 3 | atbase 38159 | . 2 β’ (π β π΄ β π β (BaseβπΎ)) |
6 | hlatlej.l | . . 3 β’ β€ = (leβπΎ) | |
7 | hlatlej.j | . . 3 β’ β¨ = (joinβπΎ) | |
8 | 2, 6, 7 | latlej1 18401 | . 2 β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π β (BaseβπΎ)) β π β€ (π β¨ π)) |
9 | 1, 4, 5, 8 | syl3an 1161 | 1 β’ ((πΎ β HL 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 class class class wbr 5149 βcfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 joincjn 18264 Latclat 18384 Atomscatm 38133 HLchlt 38220 |
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 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 |
This theorem is referenced by: hlatlej2 38246 cvratlem 38292 cvrat4 38314 ps-2 38349 lplnllnneN 38427 dalem1 38530 lnatexN 38650 lncmp 38654 2atm2atN 38656 2llnma3r 38659 dalawlem3 38744 dalawlem6 38747 dalawlem7 38748 dalawlem12 38753 trlval4 39059 cdlemc5 39066 cdlemc6 39067 cdlemd3 39071 cdleme0cp 39085 cdleme3h 39106 cdleme5 39111 cdleme9 39124 cdleme11c 39132 cdleme15b 39146 cdleme17b 39158 cdleme19a 39174 cdleme20c 39182 cdleme20j 39189 cdleme21c 39198 cdleme22b 39212 cdleme22d 39214 cdleme22e 39215 cdleme22eALTN 39216 cdleme35e 39324 cdleme35f 39325 cdleme42a 39342 cdleme17d2 39366 cdlemeg46req 39400 cdlemg13a 39522 cdlemg17a 39532 cdlemg18b 39550 cdlemg27a 39563 trlcoabs2N 39593 cdlemg42 39600 cdlemk4 39705 cdlemk1u 39730 cdlemk39 39787 dia2dimlem1 39935 dia2dimlem2 39936 dia2dimlem3 39937 cdlemm10N 39989 cdlemn10 40077 dihjatcclem1 40289 |
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