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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atlltn0 | Structured version Visualization version GIF version |
Description: A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.) |
Ref | Expression |
---|---|
atlltne0.b | β’ π΅ = (BaseβπΎ) |
atlltne0.s | β’ < = (ltβπΎ) |
atlltne0.z | β’ 0 = (0.βπΎ) |
Ref | Expression |
---|---|
atlltn0 | β’ ((πΎ β AtLat β§ π β π΅) β ( 0 < π β π β 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . 3 β’ ((πΎ β AtLat β§ π β π΅) β πΎ β AtLat) | |
2 | atlltne0.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
3 | atlltne0.z | . . . . 5 β’ 0 = (0.βπΎ) | |
4 | 2, 3 | atl0cl 38807 | . . . 4 β’ (πΎ β AtLat β 0 β π΅) |
5 | 4 | adantr 479 | . . 3 β’ ((πΎ β AtLat β§ π β π΅) β 0 β π΅) |
6 | simpr 483 | . . 3 β’ ((πΎ β AtLat β§ π β π΅) β π β π΅) | |
7 | eqid 2728 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
8 | atlltne0.s | . . . 4 β’ < = (ltβπΎ) | |
9 | 7, 8 | pltval 18331 | . . 3 β’ ((πΎ β AtLat β§ 0 β π΅ β§ π β π΅) β ( 0 < π β ( 0 (leβπΎ)π β§ 0 β π))) |
10 | 1, 5, 6, 9 | syl3anc 1368 | . 2 β’ ((πΎ β AtLat β§ π β π΅) β ( 0 < π β ( 0 (leβπΎ)π β§ 0 β π))) |
11 | necom 2991 | . . 3 β’ (π β 0 β 0 β π) | |
12 | 2, 7, 3 | atl0le 38808 | . . . 4 β’ ((πΎ β AtLat β§ π β π΅) β 0 (leβπΎ)π) |
13 | 12 | biantrurd 531 | . . 3 β’ ((πΎ β AtLat β§ π β π΅) β ( 0 β π β ( 0 (leβπΎ)π β§ 0 β π))) |
14 | 11, 13 | bitr2id 283 | . 2 β’ ((πΎ β AtLat β§ π β π΅) β (( 0 (leβπΎ)π β§ 0 β π) β π β 0 )) |
15 | 10, 14 | bitrd 278 | 1 β’ ((πΎ β AtLat β§ π β π΅) β ( 0 < π β π β 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 class class class wbr 5152 βcfv 6553 Basecbs 17187 lecple 17247 ltcplt 18307 0.cp0 18422 AtLatcal 38768 |
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 |
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-plt 18329 df-glb 18346 df-p0 18424 df-atl 38802 |
This theorem is referenced by: isat3 38811 |
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