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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > llnn0 | Structured version Visualization version GIF version |
Description: A lattice line is nonzero. (Contributed by NM, 15-Jul-2012.) |
Ref | Expression |
---|---|
llnn0.z | β’ 0 = (0.βπΎ) |
llnn0.n | β’ π = (LLinesβπΎ) |
Ref | Expression |
---|---|
llnn0 | β’ ((πΎ β HL β§ π β π) β π β 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . . . 5 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
2 | 1 | atex 38781 | . . . 4 β’ (πΎ β HL β (AtomsβπΎ) β β ) |
3 | n0 4339 | . . . 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 480 | . 2 β’ ((πΎ β HL β§ π β π) β βπ π β (AtomsβπΎ)) |
6 | eqid 2724 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
7 | llnn0.n | . . . . 5 β’ π = (LLinesβπΎ) | |
8 | 6, 1, 7 | llnnleat 38888 | . . . 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 38736 | . . . . . . 7 β’ (πΎ β HL β πΎ β OP) | |
11 | 10 | ad2antrr 723 | . . . . . 6 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β πΎ β OP) |
12 | eqid 2724 | . . . . . . . 8 β’ (BaseβπΎ) = (BaseβπΎ) | |
13 | 12, 1 | atbase 38663 | . . . . . . 7 β’ (π β (AtomsβπΎ) β π β (BaseβπΎ)) |
14 | 13 | adantl 481 | . . . . . 6 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β π β (BaseβπΎ)) |
15 | llnn0.z | . . . . . . 7 β’ 0 = (0.βπΎ) | |
16 | 12, 6, 15 | op0le 38560 | . . . . . 6 β’ ((πΎ β OP β§ π β (BaseβπΎ)) β 0 (leβπΎ)π) |
17 | 11, 14, 16 | syl2anc 583 | . . . . 5 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β 0 (leβπΎ)π) |
18 | breq1 5142 | . . . . 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 2946 | . . 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 395 = wceq 1533 βwex 1773 β wcel 2098 β wne 2932 β c0 4315 class class class wbr 5139 βcfv 6534 Basecbs 17149 lecple 17209 0.cp0 18384 OPcops 38546 Atomscatm 38637 HLchlt 38724 LLinesclln 38866 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-clat 18460 df-oposet 38550 df-ol 38552 df-oml 38553 df-covers 38640 df-ats 38641 df-atl 38672 df-cvlat 38696 df-hlat 38725 df-llines 38873 |
This theorem is referenced by: 2llnm3N 38944 cdleme22b 39716 |
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