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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnn0N | Structured version Visualization version GIF version |
Description: A lattice plane is nonzero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lplnn0.z | β’ 0 = (0.βπΎ) |
lplnn0.p | β’ π = (LPlanesβπΎ) |
Ref | Expression |
---|---|
lplnn0N | β’ ((πΎ β HL β§ π β π) β π β 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . 5 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
2 | 1 | atex 38911 | . . . 4 β’ (πΎ β HL β (AtomsβπΎ) β β ) |
3 | n0 4350 | . . . 4 β’ ((AtomsβπΎ) β β β βπ π β (AtomsβπΎ)) | |
4 | 2, 3 | sylib 217 | . . 3 β’ (πΎ β HL β βπ π β (AtomsβπΎ)) |
5 | 4 | adantr 479 | . 2 β’ ((πΎ β HL β§ π β π) β βπ π β (AtomsβπΎ)) |
6 | eqid 2728 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
7 | lplnn0.p | . . . . 5 β’ π = (LPlanesβπΎ) | |
8 | 6, 1, 7 | lplnnleat 39047 | . . . 4 β’ ((πΎ β HL β§ π β π β§ π β (AtomsβπΎ)) β Β¬ π(leβπΎ)π) |
9 | 8 | 3expa 1115 | . . 3 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β Β¬ π(leβπΎ)π) |
10 | hlop 38866 | . . . . . . 7 β’ (πΎ β HL β πΎ β OP) | |
11 | 10 | ad2antrr 724 | . . . . . 6 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β πΎ β OP) |
12 | eqid 2728 | . . . . . . . 8 β’ (BaseβπΎ) = (BaseβπΎ) | |
13 | 12, 1 | atbase 38793 | . . . . . . 7 β’ (π β (AtomsβπΎ) β π β (BaseβπΎ)) |
14 | 13 | adantl 480 | . . . . . 6 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β π β (BaseβπΎ)) |
15 | lplnn0.z | . . . . . . 7 β’ 0 = (0.βπΎ) | |
16 | 12, 6, 15 | op0le 38690 | . . . . . 6 β’ ((πΎ β OP β§ π β (BaseβπΎ)) β 0 (leβπΎ)π) |
17 | 11, 14, 16 | syl2anc 582 | . . . . 5 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β 0 (leβπΎ)π) |
18 | breq1 5155 | . . . . 5 β’ (π = 0 β (π(leβπΎ)π β 0 (leβπΎ)π)) | |
19 | 17, 18 | syl5ibrcom 246 | . . . 4 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β (π = 0 β π(leβπΎ)π)) |
20 | 19 | necon3bd 2951 | . . 3 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β (Β¬ π(leβπΎ)π β π β 0 )) |
21 | 9, 20 | mpd 15 | . 2 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β π β 0 ) |
22 | 5, 21 | exlimddv 1930 | 1 β’ ((πΎ β HL β§ π β π) β π β 0 ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1533 βwex 1773 β wcel 2098 β wne 2937 β c0 4326 class class class wbr 5152 βcfv 6553 Basecbs 17187 lecple 17247 0.cp0 18422 OPcops 38676 Atomscatm 38767 HLchlt 38854 LPlanesclpl 38997 |
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 7746 |
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 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 |
This theorem is referenced by: (None) |
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