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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 18342 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 37871 | . 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 37797 | . 2 β’ (π β π΄ β π β (BaseβπΎ)) |
5 | 2, 3 | atbase 37797 | . 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 18342 | . 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 5106 βcfv 6497 (class class class)co 7358 Basecbs 17088 lecple 17145 joincjn 18205 Latclat 18325 Atomscatm 37771 HLchlt 37858 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-lub 18240 df-join 18242 df-lat 18326 df-ats 37775 df-atl 37806 df-cvlat 37830 df-hlat 37859 |
This theorem is referenced by: hlatlej2 37884 cvratlem 37930 cvrat4 37952 ps-2 37987 lplnllnneN 38065 dalem1 38168 lnatexN 38288 lncmp 38292 2atm2atN 38294 2llnma3r 38297 dalawlem3 38382 dalawlem6 38385 dalawlem7 38386 dalawlem12 38391 trlval4 38697 cdlemc5 38704 cdlemc6 38705 cdlemd3 38709 cdleme0cp 38723 cdleme3h 38744 cdleme5 38749 cdleme9 38762 cdleme11c 38770 cdleme15b 38784 cdleme17b 38796 cdleme19a 38812 cdleme20c 38820 cdleme20j 38827 cdleme21c 38836 cdleme22b 38850 cdleme22d 38852 cdleme22e 38853 cdleme22eALTN 38854 cdleme35e 38962 cdleme35f 38963 cdleme42a 38980 cdleme17d2 39004 cdlemeg46req 39038 cdlemg13a 39160 cdlemg17a 39170 cdlemg18b 39188 cdlemg27a 39201 trlcoabs2N 39231 cdlemg42 39238 cdlemk4 39343 cdlemk1u 39368 cdlemk39 39425 dia2dimlem1 39573 dia2dimlem2 39574 dia2dimlem3 39575 cdlemm10N 39627 cdlemn10 39715 dihjatcclem1 39927 |
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