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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 2725 | . . 3 β’ (ocβπΎ) = (ocβπΎ) | |
3 | lhpocnel2.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | lhpocnel2.h | . . 3 β’ π» = (LHypβπΎ) | |
5 | 1, 2, 3, 4 | lhpocnel 39547 | . 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 2816 | . . 3 β’ (π β π΄ β ((ocβπΎ)βπ) β π΄) |
8 | 6 | breq1i 5150 | . . . 4 β’ (π β€ π β ((ocβπΎ)βπ) β€ π) |
9 | 8 | notbii 319 | . . 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 394 = wceq 1533 β wcel 2098 class class class wbr 5143 βcfv 6543 lecple 17239 occoc 17240 Atomscatm 38791 HLchlt 38878 LHypclh 39513 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-oposet 38704 df-ol 38706 df-oml 38707 df-covers 38794 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 df-lhyp 39517 |
This theorem is referenced by: cdlemk56w 40502 diclspsn 40723 cdlemn3 40726 cdlemn4 40727 cdlemn4a 40728 cdlemn6 40731 cdlemn8 40733 cdlemn9 40734 cdlemn11a 40736 dihordlem7b 40744 dihopelvalcpre 40777 dihmeetlem1N 40819 dihglblem5apreN 40820 dihglbcpreN 40829 dihmeetlem4preN 40835 dihmeetlem13N 40848 dih1dimatlem0 40857 dih1dimatlem 40858 dihpN 40865 dihatexv 40867 dihjatcclem3 40949 dihjatcclem4 40950 |
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