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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atnlej1 | 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 |
---|---|
atnlej1 | 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 38319 | . . 3 β’ (πΎ β HL β πΎ β Lat) | |
2 | 1 | 3ad2ant1 1133 | . 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 1206 | . . 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 2732 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
5 | atnlej.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | 4, 5 | atbase 38245 | . . 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 1207 | . . 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 38245 | . . 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 1208 | . . 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 38245 | . . 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 1138 | . 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 | latnlej1l 18412 | . 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 1383 | 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 1087 = wceq 1541 β wcel 2106 β wne 2940 class class class wbr 5148 βcfv 6543 (class class class)co 7411 Basecbs 17146 lecple 17206 joincjn 18266 Latclat 18386 Atomscatm 38219 HLchlt 38306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-lub 18301 df-join 18303 df-lat 18387 df-ats 38223 df-atl 38254 df-cvlat 38278 df-hlat 38307 |
This theorem is referenced by: 4atlem0be 38552 dalem5 38624 dalem-cly 38628 4atexlemex6 39031 cdleme00a 39166 cdleme21a 39282 cdleme21b 39283 cdleme21c 39284 |
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