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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atl0le | Structured version Visualization version GIF version |
Description: Orthoposet zero is less than or equal to any element. (ch0le 31199 analog.) (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
atl0le.b | β’ π΅ = (BaseβπΎ) |
atl0le.l | β’ β€ = (leβπΎ) |
atl0le.z | β’ 0 = (0.βπΎ) |
Ref | Expression |
---|---|
atl0le | β’ ((πΎ β AtLat β§ π β π΅) β 0 β€ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atl0le.b | . 2 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2726 | . 2 β’ (glbβπΎ) = (glbβπΎ) | |
3 | atl0le.l | . 2 β’ β€ = (leβπΎ) | |
4 | atl0le.z | . 2 β’ 0 = (0.βπΎ) | |
5 | simpl 482 | . 2 β’ ((πΎ β AtLat β§ π β π΅) β πΎ β AtLat) | |
6 | simpr 484 | . 2 β’ ((πΎ β AtLat β§ π β π΅) β π β π΅) | |
7 | eqid 2726 | . . . 4 β’ (lubβπΎ) = (lubβπΎ) | |
8 | 1, 7, 2 | atl0dm 38683 | . . 3 β’ (πΎ β AtLat β π΅ β dom (glbβπΎ)) |
9 | 8 | adantr 480 | . 2 β’ ((πΎ β AtLat β§ π β π΅) β π΅ β dom (glbβπΎ)) |
10 | 1, 2, 3, 4, 5, 6, 9 | p0le 18392 | 1 β’ ((πΎ β AtLat β§ π β π΅) β 0 β€ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5141 dom cdm 5669 βcfv 6536 Basecbs 17151 lecple 17211 lubclub 18272 glbcglb 18273 0.cp0 18386 AtLatcal 38645 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-glb 18310 df-p0 18388 df-atl 38679 |
This theorem is referenced by: atlle0 38686 atlltn0 38687 atcvreq0 38695 trlval4 39570 dian0 40421 dia0 40434 dihmeetlem4preN 40688 |
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