![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpmcvr2 | Structured version Visualization version GIF version |
Description: Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013.) |
Ref | Expression |
---|---|
lhpmcvr2.b | β’ π΅ = (BaseβπΎ) |
lhpmcvr2.l | β’ β€ = (leβπΎ) |
lhpmcvr2.j | β’ β¨ = (joinβπΎ) |
lhpmcvr2.m | β’ β§ = (meetβπΎ) |
lhpmcvr2.a | β’ π΄ = (AtomsβπΎ) |
lhpmcvr2.h | β’ π» = (LHypβπΎ) |
Ref | Expression |
---|---|
lhpmcvr2 | β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β βπ β π΄ (Β¬ π β€ π β§ (π β¨ (π β§ π)) = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpmcvr2.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | lhpmcvr2.l | . . 3 β’ β€ = (leβπΎ) | |
3 | lhpmcvr2.m | . . 3 β’ β§ = (meetβπΎ) | |
4 | eqid 2733 | . . 3 β’ ( β βπΎ) = ( β βπΎ) | |
5 | lhpmcvr2.h | . . 3 β’ π» = (LHypβπΎ) | |
6 | 1, 2, 3, 4, 5 | lhpmcvr 38532 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β (π β§ π)( β βπΎ)π) |
7 | simpll 766 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β πΎ β HL) | |
8 | simprl 770 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β π β π΅) | |
9 | 1, 5 | lhpbase 38507 | . . . 4 β’ (π β π» β π β π΅) |
10 | 9 | ad2antlr 726 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β π β π΅) |
11 | lhpmcvr2.j | . . . 4 β’ β¨ = (joinβπΎ) | |
12 | lhpmcvr2.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
13 | 1, 2, 11, 3, 4, 12 | cvrval5 37924 | . . 3 β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β ((π β§ π)( β βπΎ)π β βπ β π΄ (Β¬ π β€ π β§ (π β¨ (π β§ π)) = π))) |
14 | 7, 8, 10, 13 | syl3anc 1372 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β ((π β§ π)( β βπΎ)π β βπ β π΄ (Β¬ π β€ π β§ (π β¨ (π β§ π)) = π))) |
15 | 6, 14 | mpbid 231 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β βπ β π΄ (Β¬ π β€ π β§ (π β¨ (π β§ π)) = π)) |
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 βwrex 3070 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17088 lecple 17145 joincjn 18205 meetcmee 18206 β ccvr 37770 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: lhpmcvr5N 38536 cdleme29ex 38883 cdleme29c 38885 cdlemefrs29cpre1 38907 cdlemefr29exN 38911 cdleme32d 38953 cdleme32f 38955 cdleme48gfv1 39045 cdlemg7fvbwN 39116 cdlemg7aN 39134 dihlsscpre 39743 dihvalcqpre 39744 dihord6apre 39765 dihord4 39767 dihord5b 39768 dihord5apre 39771 dihmeetlem1N 39799 dihglblem5apreN 39800 dihglbcpreN 39809 |
Copyright terms: Public domain | W3C validator |