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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 18343 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 37883 | . . 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 37876 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) = (π β¨ π)) |
7 | 5, 6 | breqtrrd 5134 | 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 lecple 17145 joincjn 18205 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: 2llnne2N 37917 cvrat3 37951 cvrat4 37952 hlatexch3N 37989 hlatexch4 37990 dalem3 38173 dalem25 38207 lnatexN 38288 lncmp 38292 2llnma3r 38297 paddasslem5 38333 dalawlem3 38382 dalawlem6 38385 dalawlem7 38386 dalawlem12 38391 lhp2atne 38543 lhp2at0ne 38545 4atexlemunv 38575 cdlemc2 38701 cdlemc5 38704 cdleme3h 38744 cdleme7 38758 cdleme9 38762 cdleme11c 38770 cdleme11dN 38771 cdleme11j 38776 cdleme16b 38788 cdleme17b 38796 cdleme18a 38800 cdleme18b 38801 cdleme18c 38802 cdleme19a 38812 cdleme20d 38821 cdleme20j 38827 cdleme21ct 38838 cdleme22a 38849 cdleme22e 38853 cdleme22eALTN 38854 cdleme35b 38959 cdlemg9a 39141 cdlemg12a 39152 cdlemg13a 39160 cdlemg17a 39170 cdlemg17g 39176 cdlemg18c 39189 cdlemg33b0 39210 cdlemg46 39244 cdlemh1 39324 cdlemh 39326 cdlemk4 39343 cdlemki 39350 cdlemksv2 39356 cdlemk12 39359 cdlemk15 39364 cdlemk12u 39381 cdlemkid1 39431 dia2dimlem1 39573 dia2dimlem3 39575 cdlemn10 39715 dihjatcclem1 39927 |
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