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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpjat1 | 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, 18-May-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 |
---|---|
lhpjat1 | 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 | simpll 763 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β πΎ β HL) | |
2 | eqid 2730 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | lhpjat.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | 2, 3 | lhpbase 39172 | . . 3 β’ (π β π» β π β (BaseβπΎ)) |
5 | 4 | ad2antlr 723 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β π β (BaseβπΎ)) |
6 | simprl 767 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β π β π΄) | |
7 | lhpjat.u | . . . 4 β’ 1 = (1.βπΎ) | |
8 | eqid 2730 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
9 | 7, 8, 3 | lhp1cvr 39173 | . . 3 β’ ((πΎ β HL β§ π β π») β π( β βπΎ) 1 ) |
10 | 9 | adantr 479 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β π( β βπΎ) 1 ) |
11 | simprr 769 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β Β¬ π β€ π) | |
12 | lhpjat.l | . . 3 β’ β€ = (leβπΎ) | |
13 | lhpjat.j | . . 3 β’ β¨ = (joinβπΎ) | |
14 | lhpjat.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
15 | 2, 12, 13, 7, 8, 14 | 1cvrjat 38649 | . 2 β’ (((πΎ β HL β§ π β (BaseβπΎ) β§ π β π΄) β§ (π( β βπΎ) 1 β§ Β¬ π β€ π)) β (π β¨ π) = 1 ) |
16 | 1, 5, 6, 10, 11, 15 | syl32anc 1376 | 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 1539 β wcel 2104 class class class wbr 5147 βcfv 6542 (class class class)co 7411 Basecbs 17148 lecple 17208 joincjn 18268 1.cp1 18381 β ccvr 38435 Atomscatm 38436 HLchlt 38523 LHypclh 39158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-lhyp 39162 |
This theorem is referenced by: lhpjat2 39195 lhpj1 39196 trljat1 39340 trljat2 39341 cdlemc1 39365 cdlemc6 39370 cdleme20c 39485 cdleme20j 39492 trlcolem 39900 |
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