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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 484 | . . 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 37811 | . . . 4 β’ (πΎ β AtLat β 0 β π΅) |
5 | 4 | adantr 482 | . . 3 β’ ((πΎ β AtLat β§ π β π΅) β 0 β π΅) |
6 | simpr 486 | . . 3 β’ ((πΎ β AtLat β§ π β π΅) β π β π΅) | |
7 | eqid 2733 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
8 | atlltne0.s | . . . 4 β’ < = (ltβπΎ) | |
9 | 7, 8 | pltval 18226 | . . 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 1372 | . 2 β’ ((πΎ β AtLat β§ π β π΅) β ( 0 < π β ( 0 (leβπΎ)π β§ 0 β π))) |
11 | necom 2994 | . . 3 β’ (π β 0 β 0 β π) | |
12 | 2, 7, 3 | atl0le 37812 | . . . 4 β’ ((πΎ β AtLat β§ π β π΅) β 0 (leβπΎ)π) |
13 | 12 | biantrurd 534 | . . 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 284 | . 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 279 | 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 397 = wceq 1542 β wcel 2107 β wne 2940 class class class wbr 5106 βcfv 6497 Basecbs 17088 lecple 17145 ltcplt 18202 0.cp0 18317 AtLatcal 37772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-plt 18224 df-glb 18241 df-p0 18319 df-atl 37806 |
This theorem is referenced by: isat3 37815 |
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