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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatlej2 | Structured version Visualization version GIF version |
Description: A join's second argument is less than or equal to the join. Special case of latlej2 18402 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 |
---|---|
hlatlej2 | β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β π β€ (π β¨ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlatlej.l | . . . 4 β’ β€ = (leβπΎ) | |
2 | hlatlej.j | . . . 4 β’ β¨ = (joinβπΎ) | |
3 | hlatlej.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
4 | 1, 2, 3 | hlatlej1 38245 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β π β€ (π β¨ π)) |
5 | 4 | 3com23 1127 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β π β€ (π β¨ π)) |
6 | 2, 3 | hlatjcom 38238 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) = (π β¨ π)) |
7 | 5, 6 | breqtrrd 5177 | 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 lecple 17204 joincjn 18264 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: 2llnne2N 38279 cvrat3 38313 cvrat4 38314 hlatexch3N 38351 hlatexch4 38352 dalem3 38535 dalem25 38569 lnatexN 38650 lncmp 38654 2llnma3r 38659 paddasslem5 38695 dalawlem3 38744 dalawlem6 38747 dalawlem7 38748 dalawlem12 38753 lhp2atne 38905 lhp2at0ne 38907 4atexlemunv 38937 cdlemc2 39063 cdlemc5 39066 cdleme3h 39106 cdleme7 39120 cdleme9 39124 cdleme11c 39132 cdleme11dN 39133 cdleme11j 39138 cdleme16b 39150 cdleme17b 39158 cdleme18a 39162 cdleme18b 39163 cdleme18c 39164 cdleme19a 39174 cdleme20d 39183 cdleme20j 39189 cdleme21ct 39200 cdleme22a 39211 cdleme22e 39215 cdleme22eALTN 39216 cdleme35b 39321 cdlemg9a 39503 cdlemg12a 39514 cdlemg13a 39522 cdlemg17a 39532 cdlemg17g 39538 cdlemg18c 39551 cdlemg33b0 39572 cdlemg46 39606 cdlemh1 39686 cdlemh 39688 cdlemk4 39705 cdlemki 39712 cdlemksv2 39718 cdlemk12 39721 cdlemk15 39726 cdlemk12u 39743 cdlemkid1 39793 dia2dimlem1 39935 dia2dimlem3 39937 cdlemn10 40077 dihjatcclem1 40289 |
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