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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme7b | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 39631 and cdleme7 39632. (Contributed by NM, 7-Jun-2012.) |
Ref | Expression |
---|---|
cdleme4.l | β’ β€ = (leβπΎ) |
cdleme4.j | β’ β¨ = (joinβπΎ) |
cdleme4.m | β’ β§ = (meetβπΎ) |
cdleme4.a | β’ π΄ = (AtomsβπΎ) |
cdleme4.h | β’ π» = (LHypβπΎ) |
cdleme4.u | β’ π = ((π β¨ π) β§ π) |
cdleme4.f | β’ πΉ = ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) |
cdleme4.g | β’ πΊ = ((π β¨ π) β§ (πΉ β¨ ((π β¨ π) β§ π))) |
cdleme7.v | β’ π = ((π β¨ π) β§ π) |
Ref | Expression |
---|---|
cdleme7b | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ 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 | cdleme7.v | . 2 β’ π = ((π β¨ π) β§ π) | |
2 | simp1 1133 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ (π β¨ π) β§ π β€ (π β¨ π))) β (πΎ β HL β§ π β π»)) | |
3 | simp2 1134 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ 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§ Β¬ π β€ π)) | |
4 | simp31 1206 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ 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΄) | |
5 | simp33 1208 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ 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)) | |
6 | simp32 1207 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ (π β¨ π) β§ π β€ (π β¨ π))) β Β¬ π β€ (π β¨ π)) | |
7 | nbrne2 5161 | . . . 4 β’ ((π β€ (π β¨ π) β§ Β¬ π β€ (π β¨ π)) β π β π) | |
8 | 5, 6, 7 | syl2anc 583 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ 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) |
9 | cdleme4.l | . . . 4 β’ β€ = (leβπΎ) | |
10 | cdleme4.j | . . . 4 β’ β¨ = (joinβπΎ) | |
11 | cdleme4.m | . . . 4 β’ β§ = (meetβπΎ) | |
12 | cdleme4.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
13 | cdleme4.h | . . . 4 β’ π» = (LHypβπΎ) | |
14 | 9, 10, 11, 12, 13 | lhpat 39426 | . . 3 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΄) |
15 | 2, 3, 4, 8, 14 | syl112anc 1371 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ 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΄) |
16 | 1, 15 | eqeltrid 2831 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ 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 2934 class class class wbr 5141 βcfv 6536 (class class class)co 7404 lecple 17210 joincjn 18273 meetcmee 18274 Atomscatm 38645 HLchlt 38732 LHypclh 39367 |
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 7721 |
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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-lhyp 39371 |
This theorem is referenced by: cdleme7c 39628 cdleme7d 39629 cdleme7ga 39631 |
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