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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > op0le | Structured version Visualization version GIF version |
Description: Orthoposet zero is less than or equal to any element. (ch0le 30694 analog.) (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
op0le.b | β’ π΅ = (BaseβπΎ) |
op0le.l | β’ β€ = (leβπΎ) |
op0le.z | β’ 0 = (0.βπΎ) |
Ref | Expression |
---|---|
op0le | β’ ((πΎ β OP β§ π β π΅) β 0 β€ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op0le.b | . 2 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2733 | . 2 β’ (glbβπΎ) = (glbβπΎ) | |
3 | op0le.l | . 2 β’ β€ = (leβπΎ) | |
4 | op0le.z | . 2 β’ 0 = (0.βπΎ) | |
5 | simpl 484 | . 2 β’ ((πΎ β OP β§ π β π΅) β πΎ β OP) | |
6 | simpr 486 | . 2 β’ ((πΎ β OP β§ π β π΅) β π β π΅) | |
7 | eqid 2733 | . . . . 5 β’ (lubβπΎ) = (lubβπΎ) | |
8 | 1, 7, 2 | op01dm 38053 | . . . 4 β’ (πΎ β OP β (π΅ β dom (lubβπΎ) β§ π΅ β dom (glbβπΎ))) |
9 | 8 | simprd 497 | . . 3 β’ (πΎ β OP β π΅ β dom (glbβπΎ)) |
10 | 9 | adantr 482 | . 2 β’ ((πΎ β OP β§ π β π΅) β π΅ β dom (glbβπΎ)) |
11 | 1, 2, 3, 4, 5, 6, 10 | p0le 18382 | 1 β’ ((πΎ β OP β§ π β π΅) β 0 β€ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5149 dom cdm 5677 βcfv 6544 Basecbs 17144 lecple 17204 lubclub 18262 glbcglb 18263 0.cp0 18376 OPcops 38042 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
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 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-glb 18300 df-p0 18378 df-oposet 38046 |
This theorem is referenced by: ople0 38057 opnlen0 38058 lub0N 38059 opltn0 38060 olj01 38095 olm01 38106 leatb 38162 1cvratex 38344 llnn0 38387 lplnn0N 38418 lvoln0N 38462 dalemcea 38531 ltrnatb 39008 tendo0tp 39660 cdlemk39s-id 39811 dia0eldmN 39911 dib0 40035 dih0 40151 dihmeetlem18N 40195 |
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