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 766 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β πΎ β HL) | |
2 | eqid 2738 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | lhpjat.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | 2, 3 | lhpbase 38347 | . . 3 β’ (π β π» β π β (BaseβπΎ)) |
5 | 4 | ad2antlr 726 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β π β (BaseβπΎ)) |
6 | simprl 770 | . 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 2738 | . . . 4 β’ ( β βπΎ) = ( β βπΎ) | |
9 | 7, 8, 3 | lhp1cvr 38348 | . . 3 β’ ((πΎ β HL β§ π β π») β π( β βπΎ) 1 ) |
10 | 9 | adantr 482 | . 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 772 | . 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 37824 | . 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 1379 | 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 397 = wceq 1542 β wcel 2107 class class class wbr 5104 βcfv 6492 (class class class)co 7350 Basecbs 17018 lecple 17075 joincjn 18135 1.cp1 18248 β ccvr 37610 Atomscatm 37611 HLchlt 37698 LHypclh 38333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-proset 18119 df-poset 18137 df-plt 18154 df-lub 18170 df-glb 18171 df-join 18172 df-meet 18173 df-p0 18249 df-p1 18250 df-lat 18256 df-clat 18323 df-oposet 37524 df-ol 37526 df-oml 37527 df-covers 37614 df-ats 37615 df-atl 37646 df-cvlat 37670 df-hlat 37699 df-lhyp 38337 |
This theorem is referenced by: lhpjat2 38370 lhpj1 38371 trljat1 38515 trljat2 38516 cdlemc1 38540 cdlemc6 38545 cdleme20c 38660 cdleme20j 38667 trlcolem 39075 |
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