![]() |
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 2728 | . . 3 β’ ( β βπΎ) = ( β βπΎ) | |
5 | lhpmcvr2.h | . . 3 β’ π» = (LHypβπΎ) | |
6 | 1, 2, 3, 4, 5 | lhpmcvr 39536 | . 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 765 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π)) β πΎ β HL) | |
8 | simprl 769 | . . 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 39511 | . . . 4 β’ (π β π» β π β π΅) |
10 | 9 | ad2antlr 725 | . . 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 38928 | . . 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 1368 | . 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 394 = wceq 1533 β wcel 2098 βwrex 3067 class class class wbr 5152 βcfv 6553 (class class class)co 7426 Basecbs 17189 lecple 17249 joincjn 18312 meetcmee 18313 β ccvr 38774 Atomscatm 38775 HLchlt 38862 LHypclh 39497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-proset 18296 df-poset 18314 df-plt 18331 df-lub 18347 df-glb 18348 df-join 18349 df-meet 18350 df-p0 18426 df-p1 18427 df-lat 18433 df-clat 18500 df-oposet 38688 df-ol 38690 df-oml 38691 df-covers 38778 df-ats 38779 df-atl 38810 df-cvlat 38834 df-hlat 38863 df-lhyp 39501 |
This theorem is referenced by: lhpmcvr5N 39540 cdleme29ex 39887 cdleme29c 39889 cdlemefrs29cpre1 39911 cdlemefr29exN 39915 cdleme32d 39957 cdleme32f 39959 cdleme48gfv1 40049 cdlemg7fvbwN 40120 cdlemg7aN 40138 dihlsscpre 40747 dihvalcqpre 40748 dihord6apre 40769 dihord4 40771 dihord5b 40772 dihord5apre 40775 dihmeetlem1N 40803 dihglblem5apreN 40804 dihglbcpreN 40813 |
Copyright terms: Public domain | W3C validator |