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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 30091 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 2736 | . 2 β’ (glbβπΎ) = (glbβπΎ) | |
3 | op0le.l | . 2 β’ β€ = (leβπΎ) | |
4 | op0le.z | . 2 β’ 0 = (0.βπΎ) | |
5 | simpl 483 | . 2 β’ ((πΎ β OP β§ π β π΅) β πΎ β OP) | |
6 | simpr 485 | . 2 β’ ((πΎ β OP β§ π β π΅) β π β π΅) | |
7 | eqid 2736 | . . . . 5 β’ (lubβπΎ) = (lubβπΎ) | |
8 | 1, 7, 2 | op01dm 37450 | . . . 4 β’ (πΎ β OP β (π΅ β dom (lubβπΎ) β§ π΅ β dom (glbβπΎ))) |
9 | 8 | simprd 496 | . . 3 β’ (πΎ β OP β π΅ β dom (glbβπΎ)) |
10 | 9 | adantr 481 | . 2 β’ ((πΎ β OP β§ π β π΅) β π΅ β dom (glbβπΎ)) |
11 | 1, 2, 3, 4, 5, 6, 10 | p0le 18244 | 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 396 = wceq 1540 β wcel 2105 class class class wbr 5092 dom cdm 5620 βcfv 6479 Basecbs 17009 lecple 17066 lubclub 18124 glbcglb 18125 0.cp0 18238 OPcops 37439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-glb 18162 df-p0 18240 df-oposet 37443 |
This theorem is referenced by: ople0 37454 opnlen0 37455 lub0N 37456 opltn0 37457 olj01 37492 olm01 37503 leatb 37559 1cvratex 37741 llnn0 37784 lplnn0N 37815 lvoln0N 37859 dalemcea 37928 ltrnatb 38405 tendo0tp 39057 cdlemk39s-id 39208 dia0eldmN 39308 dib0 39432 dih0 39548 dihmeetlem18N 39592 |
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