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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpjat2 | Structured version Visualization version GIF version |
Description: The join of a co-atom (hyperplane) and an atom not under it is the lattice unity. (Contributed by NM, 4-Jun-2012.) |
Ref | Expression |
---|---|
lhpjat.l | β’ β€ = (leβπΎ) |
lhpjat.j | β’ β¨ = (joinβπΎ) |
lhpjat.u | β’ 1 = (1.βπΎ) |
lhpjat.a | β’ π΄ = (AtomsβπΎ) |
lhpjat.h | β’ π» = (LHypβπΎ) |
Ref | Expression |
---|---|
lhpjat2 | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (π β¨ π) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 38745 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
2 | 1 | ad2antrr 723 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β πΎ β Lat) |
3 | eqid 2726 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | lhpjat.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | atbase 38671 | . . . 4 β’ (π β π΄ β π β (BaseβπΎ)) |
6 | 5 | ad2antrl 725 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β π β (BaseβπΎ)) |
7 | lhpjat.h | . . . . 5 β’ π» = (LHypβπΎ) | |
8 | 3, 7 | lhpbase 39381 | . . . 4 β’ (π β π» β π β (BaseβπΎ)) |
9 | 8 | ad2antlr 724 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β π β (BaseβπΎ)) |
10 | lhpjat.j | . . . 4 β’ β¨ = (joinβπΎ) | |
11 | 3, 10 | latjcom 18409 | . . 3 β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π β (BaseβπΎ)) β (π β¨ π) = (π β¨ π)) |
12 | 2, 6, 9, 11 | syl3anc 1368 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (π β¨ π) = (π β¨ π)) |
13 | lhpjat.l | . . 3 β’ β€ = (leβπΎ) | |
14 | lhpjat.u | . . 3 β’ 1 = (1.βπΎ) | |
15 | 13, 10, 14, 4, 7 | lhpjat1 39403 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (π β¨ π) = 1 ) |
16 | 12, 15 | eqtrd 2766 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (π β¨ π) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6536 (class class class)co 7404 Basecbs 17150 lecple 17210 joincjn 18273 1.cp1 18386 Latclat 18393 Atomscatm 38645 HLchlt 38732 LHypclh 39367 |
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-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-lhyp 39371 |
This theorem is referenced by: lhpmcvr3 39408 cdleme0cp 39597 cdleme0cq 39598 cdleme1 39610 cdleme4 39621 cdleme5 39623 cdleme8 39633 cdleme9 39636 cdleme10 39637 cdleme22e 39727 cdleme22eALTN 39728 cdleme35b 39833 cdleme35e 39836 cdleme42a 39854 trlcoabs2N 40105 cdlemi1 40201 cdlemk4 40217 dia2dimlem1 40447 cdlemn10 40589 dihglbcpreN 40683 |
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