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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme0a | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.) |
Ref | Expression |
---|---|
cdleme0.l | β’ β€ = (leβπΎ) |
cdleme0.j | β’ β¨ = (joinβπΎ) |
cdleme0.m | β’ β§ = (meetβπΎ) |
cdleme0.a | β’ π΄ = (AtomsβπΎ) |
cdleme0.h | β’ π» = (LHypβπΎ) |
cdleme0.u | β’ π = ((π β¨ π) β§ π) |
Ref | Expression |
---|---|
cdleme0a | 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 | cdleme0.l | . 2 β’ β€ = (leβπΎ) | |
2 | cdleme0.j | . 2 β’ β¨ = (joinβπΎ) | |
3 | cdleme0.m | . 2 β’ β§ = (meetβπΎ) | |
4 | cdleme0.a | . 2 β’ π΄ = (AtomsβπΎ) | |
5 | cdleme0.h | . 2 β’ π» = (LHypβπΎ) | |
6 | cdleme0.u | . 2 β’ π = ((π β¨ π) β§ π) | |
7 | 1, 2, 3, 4, 5, 6 | lhpat2 38904 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 class class class wbr 5147 βcfv 6540 (class class class)co 7405 lecple 17200 joincjn 18260 meetcmee 18261 Atomscatm 38121 HLchlt 38208 LHypclh 38843 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-lhyp 38847 |
This theorem is referenced by: cdleme21c 39186 cdleme21ct 39188 cdleme22aa 39198 cdleme22e 39203 cdleme22eALTN 39204 cdleme35a 39307 cdleme35b 39309 cdleme35c 39310 cdleme35f 39313 cdleme36a 39319 cdleme42k 39343 cdlemg9a 39491 cdlemg12a 39502 cdlemg18b 39538 cdlemg18c 39539 |
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