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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemeulpq | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Dec-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 |
---|---|
cdlemeulpq | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β π β€ (π β¨ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme0.u | . 2 β’ π = ((π β¨ π) β§ π) | |
2 | simpll 766 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β πΎ β HL) | |
3 | 2 | hllatd 38234 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β πΎ β Lat) |
4 | simprl 770 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β π β π΄) | |
5 | simprr 772 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β π β π΄) | |
6 | eqid 2733 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
7 | cdleme0.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
8 | cdleme0.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
9 | 6, 7, 8 | hlatjcl 38237 | . . . 4 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) β (BaseβπΎ)) |
10 | 2, 4, 5, 9 | syl3anc 1372 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β (π β¨ π) β (BaseβπΎ)) |
11 | cdleme0.h | . . . . 5 β’ π» = (LHypβπΎ) | |
12 | 6, 11 | lhpbase 38869 | . . . 4 β’ (π β π» β π β (BaseβπΎ)) |
13 | 12 | ad2antlr 726 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β π β (BaseβπΎ)) |
14 | cdleme0.l | . . . 4 β’ β€ = (leβπΎ) | |
15 | cdleme0.m | . . . 4 β’ β§ = (meetβπΎ) | |
16 | 6, 14, 15 | latmle1 18417 | . . 3 β’ ((πΎ β Lat β§ (π β¨ π) β (BaseβπΎ) β§ π β (BaseβπΎ)) β ((π β¨ π) β§ π) β€ (π β¨ π)) |
17 | 3, 10, 13, 16 | syl3anc 1372 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ π β π΄)) β ((π β¨ π) β§ π) β€ (π β¨ π)) |
18 | 1, 17 | eqbrtrid 5184 | 1 β’ (((πΎ β HL β§ π 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 wi 4 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5149 βcfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 joincjn 18264 meetcmee 18265 Latclat 18384 Atomscatm 38133 HLchlt 38220 LHypclh 38855 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-lat 18385 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-lhyp 38859 |
This theorem is referenced by: cdleme01N 39092 cdleme0ex1N 39094 cdleme1 39098 cdlemednuN 39171 cdleme21c 39198 cdleme22e 39215 cdleme22eALTN 39216 cdleme35fnpq 39320 |
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