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Theorem trlcl 30353
Description: Closure of the trace of a lattice translation. (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
trlcl.b  |-  B  =  ( Base `  K
)
trlcl.h  |-  H  =  ( LHyp `  K
)
trlcl.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlcl.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlcl  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  B
)

Proof of Theorem trlcl
StepHypRef Expression
1 eqid 2283 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2283 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 eqid 2283 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 trlcl.h . . . . 5  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 30207 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
) ( le `  K ) W ) )
65adantr 451 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )  e.  ( Atoms `  K )  /\  -.  ( ( oc
`  K ) `  W ) ( le
`  K ) W ) )
7 eqid 2283 . . . 4  |-  ( join `  K )  =  (
join `  K )
8 eqid 2283 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
9 trlcl.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
10 trlcl.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
111, 7, 8, 3, 4, 9, 10trlval2 30352 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
) ( le `  K ) W ) )  ->  ( R `  F )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( F `
 ( ( oc
`  K ) `  W ) ) ) ( meet `  K
) W ) )
126, 11mpd3an3 1278 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( F `
 ( ( oc
`  K ) `  W ) ) ) ( meet `  K
) W ) )
13 hllat 29553 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 706 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  Lat )
15 hlop 29552 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  OP )
17 trlcl.b . . . . . . 7  |-  B  =  ( Base `  K
)
1817, 4lhpbase 30187 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
1918ad2antlr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  W  e.  B )
2017, 2opoccl 29384 . . . . 5  |-  ( ( K  e.  OP  /\  W  e.  B )  ->  ( ( oc `  K ) `  W
)  e.  B )
2116, 19, 20syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( oc `  K ) `  W )  e.  B
)
2217, 4, 9ltrncl 30314 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( oc `  K ) `  W
)  e.  B )  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  B
)
2321, 22mpd3an3 1278 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  B
)
2417, 7latjcl 14156 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  W
)  e.  B  /\  ( F `  ( ( oc `  K ) `
 W ) )  e.  B )  -> 
( ( ( oc
`  K ) `  W ) ( join `  K ) ( F `
 ( ( oc
`  K ) `  W ) ) )  e.  B )
2514, 21, 23, 24syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )
( join `  K )
( F `  (
( oc `  K
) `  W )
) )  e.  B
)
2617, 8latmcl 14157 . . 3  |-  ( ( K  e.  Lat  /\  ( ( ( oc
`  K ) `  W ) ( join `  K ) ( F `
 ( ( oc
`  K ) `  W ) ) )  e.  B  /\  W  e.  B )  ->  (
( ( ( oc
`  K ) `  W ) ( join `  K ) ( F `
 ( ( oc
`  K ) `  W ) ) ) ( meet `  K
) W )  e.  B )
2714, 25, 19, 26syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( ( oc `  K ) `  W
) ( join `  K
) ( F `  ( ( oc `  K ) `  W
) ) ) (
meet `  K ) W )  e.  B
)
2812, 27eqeltrd 2357 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   meetcmee 14079   Latclat 14151   OPcops 29362   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  trljat1  30355  trljat2  30356  trlval3  30376  cdlemc3  30382  cdlemc5  30384  trlord  30758  cdlemg4c  30801  cdlemg4  30806  cdlemg6c  30809  cdlemg10c  30828  cdlemg10  30830  cdlemg12e  30836  cdlemg17dALTN  30853  cdlemg31a  30886  cdlemg31b  30887  cdlemg35  30902  cdlemg44a  30920  trljco  30929  trljco2  30930  tendoidcl  30958  tendococl  30961  tendoid  30962  tendopltp  30969  tendo0tp  30978  cdlemh1  31004  cdlemh2  31005  cdlemi1  31007  cdlemi  31009  cdlemk9  31028  cdlemk9bN  31029  cdlemkvcl  31031  cdlemk10  31032  cdlemk11  31038  cdlemk11u  31060  cdlemk37  31103  cdlemkfid1N  31110  cdlemkid1  31111  cdlemkid2  31113  cdlemk39s-id  31129  cdlemk48  31139  cdlemk50  31141  cdlemk51  31142  cdlemk52  31143  cdlemk39u  31157  tendoex  31164  dialss  31236  dia0  31242  diaglbN  31245  dia1dim  31251  dia2dimlem2  31255  dia2dimlem3  31256  dia2dimlem10  31263  cdlemm10N  31308  dib1dim  31355  diblss  31360  cdlemn2a  31386  dih1dimb  31430  dihopelvalcpre  31438  dih1  31476  dihmeetlem1N  31480  dihglblem5apreN  31481  dihglbcpreN  31490  dih1dimatlem  31519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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