Nearby theorems
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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > trlcl | Structured version Visualization version GIF version | ||
| Description: Closure of the trace of a lattice translation. (Contributed by NM, 22-May-2012.) |
| Ref | Expression |
|---|---|
| trlcl.b | β’ π΅ = (BaseβπΎ) |
| trlcl.h | β’ π» = (LHypβπΎ) |
| trlcl.t | β’ π = ((LTrnβπΎ)βπ) |
| trlcl.r | β’ π = ((trLβπΎ)βπ) |
| Ref | Expression |
|---|---|
| trlcl | β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π βπΉ) β π΅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2733 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
| 2 | eqid 2733 | . . . . 5 β’ (ocβπΎ) = (ocβπΎ) | |
| 3 | eqid 2733 | . . . . 5 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
| 4 | trlcl.h | . . . . 5 β’ π» = (LHypβπΎ) | |
| 5 | 1, 2, 3, 4 | lhpocnel 38889 | . . . 4 β’ ((πΎ β HL β§ π β π») β (((ocβπΎ)βπ) β (AtomsβπΎ) β§ Β¬ ((ocβπΎ)βπ)(leβπΎ)π)) |
| 6 | 5 | adantr 482 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (((ocβπΎ)βπ) β (AtomsβπΎ) β§ Β¬ ((ocβπΎ)βπ)(leβπΎ)π)) |
| 7 | eqid 2733 | . . . 4 β’ (joinβπΎ) = (joinβπΎ) | |
| 8 | eqid 2733 | . . . 4 β’ (meetβπΎ) = (meetβπΎ) | |
| 9 | trlcl.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
| 10 | trlcl.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
| 11 | 1, 7, 8, 3, 4, 9, 10 | trlval2 39034 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (((ocβπΎ)βπ) β (AtomsβπΎ) β§ Β¬ ((ocβπΎ)βπ)(leβπΎ)π)) β (π βπΉ) = ((((ocβπΎ)βπ)(joinβπΎ)(πΉβ((ocβπΎ)βπ)))(meetβπΎ)π)) |
| 12 | 6, 11 | mpd3an3 1463 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π βπΉ) = ((((ocβπΎ)βπ)(joinβπΎ)(πΉβ((ocβπΎ)βπ)))(meetβπΎ)π)) |
| 13 | hllat 38233 | . . . 4 β’ (πΎ β HL β πΎ β Lat) | |
| 14 | 13 | ad2antrr 725 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β πΎ β Lat) |
| 15 | hlop 38232 | . . . . . 6 β’ (πΎ β HL β πΎ β OP) | |
| 16 | 15 | ad2antrr 725 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β πΎ β OP) |
| 17 | trlcl.b | . . . . . . 7 β’ π΅ = (BaseβπΎ) | |
| 18 | 17, 4 | lhpbase 38869 | . . . . . 6 β’ (π β π» β π β π΅) |
| 19 | 18 | ad2antlr 726 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β π β π΅) |
| 20 | 17, 2 | opoccl 38064 | . . . . 5 β’ ((πΎ β OP β§ π β π΅) β ((ocβπΎ)βπ) β π΅) |
| 21 | 16, 19, 20 | syl2anc 585 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β ((ocβπΎ)βπ) β π΅) |
| 22 | 17, 4, 9 | ltrncl 38996 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ ((ocβπΎ)βπ) β π΅) β (πΉβ((ocβπΎ)βπ)) β π΅) |
| 23 | 21, 22 | mpd3an3 1463 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΉβ((ocβπΎ)βπ)) β π΅) |
| 24 | 17, 7 | latjcl 18392 | . . . 4 β’ ((πΎ β Lat β§ ((ocβπΎ)βπ) β π΅ β§ (πΉβ((ocβπΎ)βπ)) β π΅) β (((ocβπΎ)βπ)(joinβπΎ)(πΉβ((ocβπΎ)βπ))) β π΅) |
| 25 | 14, 21, 23, 24 | syl3anc 1372 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (((ocβπΎ)βπ)(joinβπΎ)(πΉβ((ocβπΎ)βπ))) β π΅) |
| 26 | 17, 8 | latmcl 18393 | . . 3 β’ ((πΎ β Lat β§ (((ocβπΎ)βπ)(joinβπΎ)(πΉβ((ocβπΎ)βπ))) β π΅ β§ π β π΅) β ((((ocβπΎ)βπ)(joinβπΎ)(πΉβ((ocβπΎ)βπ)))(meetβπΎ)π) β π΅) |
| 27 | 14, 25, 19, 26 | syl3anc 1372 | . 2 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β ((((ocβπΎ)βπ)(joinβπΎ)(πΉβ((ocβπΎ)βπ)))(meetβπΎ)π) β π΅) |
| 28 | 12, 27 | eqeltrd 2834 | 1 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π βπΉ) β π΅) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 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 occoc 17205 joincjn 18264 meetcmee 18265 Latclat 18384 OPcops 38042 Atomscatm 38133 HLchlt 38220 LHypclh 38855 LTrncltrn 38972 trLctrl 39029 |
| 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-mpo 7414 df-map 8822 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-p1 18379 df-lat 18385 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-lhyp 38859 df-laut 38860 df-ldil 38975 df-ltrn 38976 df-trl 39030 |
| This theorem is referenced by: trljat1 39037 trljat2 39038 trlval3 39058 cdlemc3 39064 cdlemc5 39066 trlord 39440 cdlemg4c 39483 cdlemg4 39488 cdlemg6c 39491 cdlemg10c 39510 cdlemg10 39512 cdlemg12e 39518 cdlemg17dALTN 39535 cdlemg31a 39568 cdlemg31b 39569 cdlemg35 39584 cdlemg44a 39602 trljco 39611 trljco2 39612 tendoidcl 39640 tendococl 39643 tendoid 39644 tendopltp 39651 tendo0tp 39660 cdlemh1 39686 cdlemh2 39687 cdlemi1 39689 cdlemi 39691 cdlemk9 39710 cdlemk9bN 39711 cdlemkvcl 39713 cdlemk10 39714 cdlemk11 39720 cdlemk11u 39742 cdlemk37 39785 cdlemkfid1N 39792 cdlemkid1 39793 cdlemkid2 39795 cdlemk39s-id 39811 cdlemk48 39821 cdlemk50 39823 cdlemk51 39824 cdlemk52 39825 cdlemk39u 39839 tendoex 39846 dialss 39917 dia0 39923 diaglbN 39926 dia1dim 39932 dia2dimlem2 39936 dia2dimlem3 39937 dia2dimlem10 39944 cdlemm10N 39989 dib1dim 40036 diblss 40041 cdlemn2a 40067 dih1dimb 40111 dihopelvalcpre 40119 dih1 40157 dihmeetlem1N 40161 dihglblem5apreN 40162 dihglbcpreN 40171 dih1dimatlem 40200 |
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