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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 2732 | . . 3 β’ (ocβπΎ) = (ocβπΎ) | |
3 | lhpocnel2.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | lhpocnel2.h | . . 3 β’ π» = (LHypβπΎ) | |
5 | 1, 2, 3, 4 | lhpocnel 38884 | . 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 2824 | . . 3 β’ (π β π΄ β ((ocβπΎ)βπ) β π΄) |
8 | 6 | breq1i 5155 | . . . 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 627 | . 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 396 = wceq 1541 β wcel 2106 class class class wbr 5148 βcfv 6543 lecple 17203 occoc 17204 Atomscatm 38128 HLchlt 38215 LHypclh 38850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7364 df-ov 7411 df-oprab 7412 df-proset 18247 df-poset 18265 df-plt 18282 df-lub 18298 df-glb 18299 df-meet 18301 df-p0 18377 df-p1 18378 df-lat 18384 df-oposet 38041 df-ol 38043 df-oml 38044 df-covers 38131 df-ats 38132 df-atl 38163 df-cvlat 38187 df-hlat 38216 df-lhyp 38854 |
This theorem is referenced by: cdlemk56w 39839 diclspsn 40060 cdlemn3 40063 cdlemn4 40064 cdlemn4a 40065 cdlemn6 40068 cdlemn8 40070 cdlemn9 40071 cdlemn11a 40073 dihordlem7b 40081 dihopelvalcpre 40114 dihmeetlem1N 40156 dihglblem5apreN 40157 dihglbcpreN 40166 dihmeetlem4preN 40172 dihmeetlem13N 40185 dih1dimatlem0 40194 dih1dimatlem 40195 dihpN 40202 dihatexv 40204 dihjatcclem3 40286 dihjatcclem4 40287 |
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