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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpat2 | Structured version Visualization version GIF version |
Description: Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.) |
Ref | Expression |
---|---|
lhpat.l | β’ β€ = (leβπΎ) |
lhpat.j | β’ β¨ = (joinβπΎ) |
lhpat.m | β’ β§ = (meetβπΎ) |
lhpat.a | β’ π΄ = (AtomsβπΎ) |
lhpat.h | β’ π» = (LHypβπΎ) |
lhpat2.r | β’ π = ((π β¨ π) β§ π) |
Ref | Expression |
---|---|
lhpat2 | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ π β π)) β π β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpat2.r | . 2 β’ π = ((π β¨ π) β§ π) | |
2 | lhpat.l | . . 3 β’ β€ = (leβπΎ) | |
3 | lhpat.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | lhpat.m | . . 3 β’ β§ = (meetβπΎ) | |
5 | lhpat.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
6 | lhpat.h | . . 3 β’ π» = (LHypβπΎ) | |
7 | 2, 3, 4, 5, 6 | lhpat 39418 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ π β π)) β ((π β¨ π) β§ π) β π΄) |
8 | 1, 7 | eqeltrid 2829 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ π β π)) β π β π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 class class class wbr 5139 βcfv 6534 (class class class)co 7402 lecple 17209 joincjn 18272 meetcmee 18273 Atomscatm 38637 HLchlt 38724 LHypclh 39359 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-clat 18460 df-oposet 38550 df-ol 38552 df-oml 38553 df-covers 38640 df-ats 38641 df-atl 38672 df-cvlat 38696 df-hlat 38725 df-lhyp 39363 |
This theorem is referenced by: lhpat3 39421 4atexlemu 39439 4atexlemv 39440 cdleme0a 39586 cdleme0dN 39591 cdleme0e 39592 cdleme02N 39597 cdleme0ex1N 39598 cdleme0moN 39600 cdleme3b 39604 cdleme3c 39605 cdleme3g 39609 cdleme3h 39610 cdleme3 39612 cdleme7aa 39617 cdleme7c 39620 cdleme7d 39621 cdleme7e 39622 cdleme7ga 39623 cdleme7 39624 cdleme9a 39626 cdleme16aN 39634 cdleme11a 39635 cdleme11c 39636 cdleme12 39646 cdleme16b 39654 cdleme16c 39655 cdleme16d 39656 cdleme20h 39691 cdleme20j 39693 cdleme20l2 39696 cdlemeg46rgv 39903 cdlemeg46req 39904 |
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