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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 2736 | . . . . 5 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
2 | 1 | atex 37682 | . . . 4 β’ (πΎ β HL β (AtomsβπΎ) β β ) |
3 | n0 4293 | . . . 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 481 | . 2 β’ ((πΎ β HL β§ π β π) β βπ π β (AtomsβπΎ)) |
6 | eqid 2736 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
7 | llnn0.n | . . . . 5 β’ π = (LLinesβπΎ) | |
8 | 6, 1, 7 | llnnleat 37789 | . . . 4 β’ ((πΎ β HL β§ π β π β§ π β (AtomsβπΎ)) β Β¬ π(leβπΎ)π) |
9 | 8 | 3expa 1117 | . . 3 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β Β¬ π(leβπΎ)π) |
10 | hlop 37637 | . . . . . . 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 2736 | . . . . . . . 8 β’ (BaseβπΎ) = (BaseβπΎ) | |
13 | 12, 1 | atbase 37564 | . . . . . . 7 β’ (π β (AtomsβπΎ) β π β (BaseβπΎ)) |
14 | 13 | adantl 482 | . . . . . 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 37461 | . . . . . 6 β’ ((πΎ β OP β§ π β (BaseβπΎ)) β 0 (leβπΎ)π) |
17 | 11, 14, 16 | syl2anc 584 | . . . . 5 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β 0 (leβπΎ)π) |
18 | breq1 5095 | . . . . 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 2954 | . . 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 1937 | 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 396 = wceq 1540 βwex 1780 β wcel 2105 β wne 2940 β c0 4269 class class class wbr 5092 βcfv 6479 Basecbs 17009 lecple 17066 0.cp0 18238 OPcops 37447 Atomscatm 37538 HLchlt 37625 LLinesclln 37767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-proset 18110 df-poset 18128 df-plt 18145 df-lub 18161 df-glb 18162 df-join 18163 df-meet 18164 df-p0 18240 df-p1 18241 df-lat 18247 df-clat 18314 df-oposet 37451 df-ol 37453 df-oml 37454 df-covers 37541 df-ats 37542 df-atl 37573 df-cvlat 37597 df-hlat 37626 df-llines 37774 |
This theorem is referenced by: 2llnm3N 37845 cdleme22b 38617 |
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