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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atnlej2 | Structured version Visualization version GIF version |
Description: If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.) |
Ref | Expression |
---|---|
atnlej.l | β’ β€ = (leβπΎ) |
atnlej.j | β’ β¨ = (joinβπΎ) |
atnlej.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
atnlej2 | β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π )) β π β π ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 38745 | . . 3 β’ (πΎ β HL β πΎ β Lat) | |
2 | 1 | 3ad2ant1 1130 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π )) β πΎ β Lat) |
3 | simp21 1203 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π )) β π β π΄) | |
4 | eqid 2726 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
5 | atnlej.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | 4, 5 | atbase 38671 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
7 | 3, 6 | syl 17 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π )) β π β (BaseβπΎ)) |
8 | simp22 1204 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π )) β π β π΄) | |
9 | 4, 5 | atbase 38671 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
10 | 8, 9 | syl 17 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π )) β π β (BaseβπΎ)) |
11 | simp23 1205 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π )) β π β π΄) | |
12 | 4, 5 | atbase 38671 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
13 | 11, 12 | syl 17 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π )) β π β (BaseβπΎ)) |
14 | simp3 1135 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π )) β Β¬ π β€ (π β¨ π )) | |
15 | atnlej.l | . . 3 β’ β€ = (leβπΎ) | |
16 | atnlej.j | . . 3 β’ β¨ = (joinβπΎ) | |
17 | 4, 15, 16 | latnlej1r 18420 | . 2 β’ ((πΎ β Lat β§ (π β (BaseβπΎ) β§ π β (BaseβπΎ) β§ π β (BaseβπΎ)) β§ Β¬ π β€ (π β¨ π )) β π β π ) |
18 | 2, 7, 10, 13, 14, 17 | syl131anc 1380 | 1 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π )) β π β π ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6536 (class class class)co 7404 Basecbs 17150 lecple 17210 joincjn 18273 Latclat 18393 Atomscatm 38645 HLchlt 38732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-lub 18308 df-join 18310 df-lat 18394 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 |
This theorem is referenced by: lplnri2N 38937 lplnri3N 38938 lplnexllnN 38947 dalem41 39096 paddasslem2 39204 4atexlemc 39452 |
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