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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0ltat | Structured version Visualization version GIF version |
Description: An atom is greater than zero. (Contributed by NM, 4-Jul-2012.) |
Ref | Expression |
---|---|
0ltat.z | β’ 0 = (0.βπΎ) |
0ltat.s | β’ < = (ltβπΎ) |
0ltat.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
0ltat | β’ ((πΎ β OP β§ π β π΄) β 0 < π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . 2 β’ ((πΎ β OP β§ π β π΄) β πΎ β OP) | |
2 | eqid 2724 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | 0ltat.z | . . . 4 β’ 0 = (0.βπΎ) | |
4 | 2, 3 | op0cl 38557 | . . 3 β’ (πΎ β OP β 0 β (BaseβπΎ)) |
5 | 4 | adantr 480 | . 2 β’ ((πΎ β OP β§ π β π΄) β 0 β (BaseβπΎ)) |
6 | 0ltat.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
7 | 2, 6 | atbase 38662 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
8 | 7 | adantl 481 | . 2 β’ ((πΎ β OP β§ π β π΄) β π β (BaseβπΎ)) |
9 | eqid 2724 | . . 3 β’ ( β βπΎ) = ( β βπΎ) | |
10 | 3, 9, 6 | atcvr0 38661 | . 2 β’ ((πΎ β OP β§ π β π΄) β 0 ( β βπΎ)π) |
11 | 0ltat.s | . . 3 β’ < = (ltβπΎ) | |
12 | 2, 11, 9 | cvrlt 38643 | . 2 β’ (((πΎ β OP β§ 0 β (BaseβπΎ) β§ π β (BaseβπΎ)) β§ 0 ( β βπΎ)π) β 0 < π) |
13 | 1, 5, 8, 10, 12 | syl31anc 1370 | 1 β’ ((πΎ β OP β§ π β π΄) β 0 < π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5139 βcfv 6534 Basecbs 17149 ltcplt 18269 0.cp0 18384 OPcops 38545 β ccvr 38635 Atomscatm 38636 |
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-glb 18308 df-p0 18386 df-oposet 38549 df-covers 38639 df-ats 38640 |
This theorem is referenced by: 2atm2atN 39159 dia2dimlem2 40439 dia2dimlem3 40440 |
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