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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme3fa | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. See cdleme3 38703. (Contributed by NM, 6-Oct-2012.) |
Ref | Expression |
---|---|
cdleme1.l | β’ β€ = (leβπΎ) |
cdleme1.j | β’ β¨ = (joinβπΎ) |
cdleme1.m | β’ β§ = (meetβπΎ) |
cdleme1.a | β’ π΄ = (AtomsβπΎ) |
cdleme1.h | β’ π» = (LHypβπΎ) |
cdleme1.u | β’ π = ((π β¨ π) β§ π) |
cdleme1.f | β’ πΉ = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) |
Ref | Expression |
---|---|
cdleme3fa | 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§ Β¬ π β€ π)) β§ (π β π β§ Β¬ π β€ (π β¨ π))) β πΉ β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme1.l | . 2 β’ β€ = (leβπΎ) | |
2 | cdleme1.j | . 2 β’ β¨ = (joinβπΎ) | |
3 | cdleme1.m | . 2 β’ β§ = (meetβπΎ) | |
4 | cdleme1.a | . 2 β’ π΄ = (AtomsβπΎ) | |
5 | cdleme1.h | . 2 β’ π» = (LHypβπΎ) | |
6 | cdleme1.u | . 2 β’ π = ((π β¨ π) β§ π) | |
7 | cdleme1.f | . 2 β’ πΉ = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) | |
8 | eqid 2737 | . 2 β’ ((π β¨ π ) β§ π) = ((π β¨ π ) β§ π) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | cdleme3h 38701 | 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§ Β¬ π β€ π)) β§ (π β π β§ Β¬ π β€ (π β¨ π))) β πΉ β π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 class class class wbr 5106 βcfv 6497 (class class class)co 7358 lecple 17141 joincjn 18201 meetcmee 18202 Atomscatm 37728 HLchlt 37815 LHypclh 38450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-lines 37967 df-psubsp 37969 df-pmap 37970 df-padd 38262 df-lhyp 38454 |
This theorem is referenced by: cdleme3 38703 cdleme7d 38712 cdleme7ga 38714 cdleme11j 38733 cdleme11k 38734 cdleme11 38736 cdleme14 38739 cdleme15a 38740 cdleme16b 38745 cdleme16c 38746 cdleme16d 38747 cdleme16e 38748 cdleme16f 38749 cdleme19d 38772 cdleme20f 38780 cdleme20l1 38786 cdleme20l2 38787 cdleme22f2 38813 cdleme22g 38814 cdlemefr32sn2aw 38870 cdleme35a 38914 cdleme36m 38927 cdleme43bN 38956 |
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