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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpmat | Structured version Visualization version GIF version |
Description: An element covered by the lattice unity, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.) |
Ref | Expression |
---|---|
lhpmat.l | β’ β€ = (leβπΎ) |
lhpmat.m | β’ β§ = (meetβπΎ) |
lhpmat.z | β’ 0 = (0.βπΎ) |
lhpmat.a | β’ π΄ = (AtomsβπΎ) |
lhpmat.h | β’ π» = (LHypβπΎ) |
Ref | Expression |
---|---|
lhpmat | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (π β§ π) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 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) | |
2 | hlatl 37851 | . . . 4 β’ (πΎ β HL β πΎ β AtLat) | |
3 | 2 | ad2antrr 725 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β πΎ β AtLat) |
4 | simprl 770 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β π β π΄) | |
5 | eqid 2737 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
6 | lhpmat.h | . . . . 5 β’ π» = (LHypβπΎ) | |
7 | 5, 6 | lhpbase 38490 | . . . 4 β’ (π β π» β π β (BaseβπΎ)) |
8 | 7 | ad2antlr 726 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β π β (BaseβπΎ)) |
9 | lhpmat.l | . . . 4 β’ β€ = (leβπΎ) | |
10 | lhpmat.m | . . . 4 β’ β§ = (meetβπΎ) | |
11 | lhpmat.z | . . . 4 β’ 0 = (0.βπΎ) | |
12 | lhpmat.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
13 | 5, 9, 10, 11, 12 | atnle 37808 | . . 3 β’ ((πΎ β AtLat β§ π β π΄ β§ π β (BaseβπΎ)) β (Β¬ π β€ π β (π β§ π) = 0 )) |
14 | 3, 4, 8, 13 | syl3anc 1372 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (Β¬ π β€ π β (π β§ π) = 0 )) |
15 | 1, 14 | mpbid 231 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (π β§ π) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5110 βcfv 6501 (class class class)co 7362 Basecbs 17090 lecple 17147 meetcmee 18208 0.cp0 18319 Atomscatm 37754 AtLatcal 37755 HLchlt 37841 LHypclh 38476 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-proset 18191 df-poset 18209 df-plt 18226 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-p0 18321 df-lat 18328 df-covers 37757 df-ats 37758 df-atl 37789 df-cvlat 37813 df-hlat 37842 df-lhyp 38480 |
This theorem is referenced by: lhpmatb 38523 lhp2at0 38524 lhpelim 38529 lhple 38534 idltrn 38642 ltrnmw 38643 trl0 38662 cdleme0e 38709 cdleme2 38720 cdleme7c 38737 cdleme22d 38835 cdlemefrs29pre00 38887 cdlemefrs29bpre0 38888 cdlemefrs29cpre1 38890 cdleme32fva 38929 cdleme35d 38944 cdleme42ke 38977 cdlemeg46frv 39017 cdleme50trn3 39045 cdlemg2fv2 39092 cdlemg8a 39119 cdlemg10bALTN 39128 cdlemh2 39308 cdlemk9 39331 cdlemk9bN 39332 dia2dimlem1 39556 dihvalcqat 39731 dihjatc1 39803 |
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