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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 2728 | . . . . 5 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
2 | 1 | atex 38879 | . . . 4 β’ (πΎ β HL β (AtomsβπΎ) β β ) |
3 | n0 4347 | . . . 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 2728 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
7 | llnn0.n | . . . . 5 β’ π = (LLinesβπΎ) | |
8 | 6, 1, 7 | llnnleat 38986 | . . . 4 β’ ((πΎ β HL β§ π β π β§ π β (AtomsβπΎ)) β Β¬ π(leβπΎ)π) |
9 | 8 | 3expa 1116 | . . 3 β’ (((πΎ β HL β§ π β π) β§ π β (AtomsβπΎ)) β Β¬ π(leβπΎ)π) |
10 | hlop 38834 | . . . . . . 7 β’ (πΎ β HL β πΎ β OP) | |
11 | 10 | ad2antrr 725 | . . . . . 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 38761 | . . . . . . 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 38658 | . . . . . 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 5151 | . . . . 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 1931 | 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 1534 βwex 1774 β wcel 2099 β wne 2937 β c0 4323 class class class wbr 5148 βcfv 6548 Basecbs 17180 lecple 17240 0.cp0 18415 OPcops 38644 Atomscatm 38735 HLchlt 38822 LLinesclln 38964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 |
This theorem is referenced by: 2llnm3N 39042 cdleme22b 39814 |
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