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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 38867 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
2 | 1 | ad2antrr 724 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β πΎ β Lat) |
3 | eqid 2728 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | lhpjat.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | atbase 38793 | . . . 4 β’ (π β π΄ β π β (BaseβπΎ)) |
6 | 5 | ad2antrl 726 | . . 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 39503 | . . . 4 β’ (π β π» β π β (BaseβπΎ)) |
9 | 8 | ad2antlr 725 | . . 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 18446 | . . 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 39525 | . 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 2768 | 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 394 = wceq 1533 β wcel 2098 class class class wbr 5152 βcfv 6553 (class class class)co 7426 Basecbs 17187 lecple 17247 joincjn 18310 1.cp1 18423 Latclat 18430 Atomscatm 38767 HLchlt 38854 LHypclh 39489 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-lhyp 39493 |
This theorem is referenced by: lhpmcvr3 39530 cdleme0cp 39719 cdleme0cq 39720 cdleme1 39732 cdleme4 39743 cdleme5 39745 cdleme8 39755 cdleme9 39758 cdleme10 39759 cdleme22e 39849 cdleme22eALTN 39850 cdleme35b 39955 cdleme35e 39958 cdleme42a 39976 trlcoabs2N 40227 cdlemi1 40323 cdlemk4 40339 dia2dimlem1 40569 cdlemn10 40711 dihglbcpreN 40805 |
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