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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpocnel2 | Structured version Visualization version GIF version |
Description: The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014.) |
Ref | Expression |
---|---|
lhpocnel2.l | β’ β€ = (leβπΎ) |
lhpocnel2.a | β’ π΄ = (AtomsβπΎ) |
lhpocnel2.h | β’ π» = (LHypβπΎ) |
lhpocnel2.p | β’ π = ((ocβπΎ)βπ) |
Ref | Expression |
---|---|
lhpocnel2 | β’ ((πΎ β HL β§ π β π») β (π β π΄ β§ Β¬ π β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpocnel2.l | . . 3 β’ β€ = (leβπΎ) | |
2 | eqid 2726 | . . 3 β’ (ocβπΎ) = (ocβπΎ) | |
3 | lhpocnel2.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | lhpocnel2.h | . . 3 β’ π» = (LHypβπΎ) | |
5 | 1, 2, 3, 4 | lhpocnel 39402 | . 2 β’ ((πΎ β HL β§ π β π») β (((ocβπΎ)βπ) β π΄ β§ Β¬ ((ocβπΎ)βπ) β€ π)) |
6 | lhpocnel2.p | . . . 4 β’ π = ((ocβπΎ)βπ) | |
7 | 6 | eleq1i 2818 | . . 3 β’ (π β π΄ β ((ocβπΎ)βπ) β π΄) |
8 | 6 | breq1i 5148 | . . . 4 β’ (π β€ π β ((ocβπΎ)βπ) β€ π) |
9 | 8 | notbii 320 | . . 3 β’ (Β¬ π β€ π β Β¬ ((ocβπΎ)βπ) β€ π) |
10 | 7, 9 | anbi12i 626 | . 2 β’ ((π β π΄ β§ Β¬ π β€ π) β (((ocβπΎ)βπ) β π΄ β§ Β¬ ((ocβπΎ)βπ) β€ π)) |
11 | 5, 10 | sylibr 233 | 1 β’ ((πΎ β HL β§ π β π») β (π β π΄ β§ Β¬ π β€ π)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6537 lecple 17213 occoc 17214 Atomscatm 38646 HLchlt 38733 LHypclh 39368 |
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 ax-un 7722 |
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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-lhyp 39372 |
This theorem is referenced by: cdlemk56w 40357 diclspsn 40578 cdlemn3 40581 cdlemn4 40582 cdlemn4a 40583 cdlemn6 40586 cdlemn8 40588 cdlemn9 40589 cdlemn11a 40591 dihordlem7b 40599 dihopelvalcpre 40632 dihmeetlem1N 40674 dihglblem5apreN 40675 dihglbcpreN 40684 dihmeetlem4preN 40690 dihmeetlem13N 40703 dih1dimatlem0 40712 dih1dimatlem 40713 dihpN 40720 dihatexv 40722 dihjatcclem3 40804 dihjatcclem4 40805 |
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