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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpocnel | 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, 25-May-2012.) |
Ref | Expression |
---|---|
lhpocnel.l | β’ β€ = (leβπΎ) |
lhpocnel.o | β’ β₯ = (ocβπΎ) |
lhpocnel.a | β’ π΄ = (AtomsβπΎ) |
lhpocnel.h | β’ π» = (LHypβπΎ) |
Ref | Expression |
---|---|
lhpocnel | β’ ((πΎ β HL β§ π β π») β (( β₯ βπ) β π΄ β§ Β¬ ( β₯ βπ) β€ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpocnel.o | . . 3 β’ β₯ = (ocβπΎ) | |
2 | lhpocnel.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
3 | lhpocnel.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | 1, 2, 3 | lhpocat 39401 | . 2 β’ ((πΎ β HL β§ π β π») β ( β₯ βπ) β π΄) |
5 | lhpocnel.l | . . 3 β’ β€ = (leβπΎ) | |
6 | 5, 1, 3 | lhpocnle 39400 | . 2 β’ ((πΎ β HL β§ π β π») β Β¬ ( β₯ βπ) β€ π) |
7 | 4, 6 | jca 511 | 1 β’ ((πΎ β HL β§ π β π») β (( β₯ βπ) β π΄ β§ Β¬ ( β₯ βπ) 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: lhpocnel2 39403 trlcl 39548 trlle 39568 cdlemk19w 40356 dia2dimlem8 40455 dicssdvh 40570 dicvaddcl 40574 dicvscacl 40575 dicn0 40576 dih1 40670 dihatlat 40718 |
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