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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 37871 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
2 | 1 | ad2antrr 725 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β πΎ β Lat) |
3 | eqid 2733 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | lhpjat.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | atbase 37797 | . . . 4 β’ (π β π΄ β π β (BaseβπΎ)) |
6 | 5 | ad2antrl 727 | . . 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 38507 | . . . 4 β’ (π β π» β π β (BaseβπΎ)) |
9 | 8 | ad2antlr 726 | . . 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 18341 | . . 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 1372 | . 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 38529 | . 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 2773 | 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 5106 βcfv 6497 (class class class)co 7358 Basecbs 17088 lecple 17145 joincjn 18205 1.cp1 18318 Latclat 18325 Atomscatm 37771 HLchlt 37858 LHypclh 38493 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-proset 18189 df-poset 18207 df-plt 18224 df-lub 18240 df-glb 18241 df-join 18242 df-meet 18243 df-p0 18319 df-p1 18320 df-lat 18326 df-clat 18393 df-oposet 37684 df-ol 37686 df-oml 37687 df-covers 37774 df-ats 37775 df-atl 37806 df-cvlat 37830 df-hlat 37859 df-lhyp 38497 |
This theorem is referenced by: lhpmcvr3 38534 cdleme0cp 38723 cdleme0cq 38724 cdleme1 38736 cdleme4 38747 cdleme5 38749 cdleme8 38759 cdleme9 38762 cdleme10 38763 cdleme22e 38853 cdleme22eALTN 38854 cdleme35b 38959 cdleme35e 38962 cdleme42a 38980 trlcoabs2N 39231 cdlemi1 39327 cdlemk4 39343 dia2dimlem1 39573 cdlemn10 39715 dihglbcpreN 39809 |
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