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Theorem trlcl 33813
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 2443 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2443 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 eqid 2443 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 trlcl.h . . . . 5  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 33667 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
) ( le `  K ) W ) )
65adantr 465 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )  e.  ( Atoms `  K )  /\  -.  ( ( oc
`  K ) `  W ) ( le
`  K ) W ) )
7 eqid 2443 . . . 4  |-  ( join `  K )  =  (
join `  K )
8 eqid 2443 . . . 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 33812 . . 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 1315 . 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 33013 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 725 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  Lat )
15 hlop 33012 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 725 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  OP )
17 trlcl.b . . . . . . 7  |-  B  =  ( Base `  K
)
1817, 4lhpbase 33647 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
1918ad2antlr 726 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  W  e.  B )
2017, 2opoccl 32844 . . . . 5  |-  ( ( K  e.  OP  /\  W  e.  B )  ->  ( ( oc `  K ) `  W
)  e.  B )
2116, 19, 20syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( oc `  K ) `  W )  e.  B
)
2217, 4, 9ltrncl 33774 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( oc `  K ) `  W
)  e.  B )  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  B
)
2321, 22mpd3an3 1315 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  B
)
2417, 7latjcl 15226 . . . 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 1218 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )
( join `  K )
( F `  (
( oc `  K
) `  W )
) )  e.  B
)
2617, 8latmcl 15227 . . 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 1218 . 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 2517 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   occoc 14251   joincjn 15119   meetcmee 15120   Latclat 15220   OPcops 32822   Atomscatm 32913   HLchlt 33000   LHypclh 33633   LTrncltrn 33750   trLctrl 33807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-lhyp 33637  df-laut 33638  df-ldil 33753  df-ltrn 33754  df-trl 33808
This theorem is referenced by:  trljat1  33815  trljat2  33816  trlval3  33836  cdlemc3  33842  cdlemc5  33844  trlord  34218  cdlemg4c  34261  cdlemg4  34266  cdlemg6c  34269  cdlemg10c  34288  cdlemg10  34290  cdlemg12e  34296  cdlemg17dALTN  34313  cdlemg31a  34346  cdlemg31b  34347  cdlemg35  34362  cdlemg44a  34380  trljco  34389  trljco2  34390  tendoidcl  34418  tendococl  34421  tendoid  34422  tendopltp  34429  tendo0tp  34438  cdlemh1  34464  cdlemh2  34465  cdlemi1  34467  cdlemi  34469  cdlemk9  34488  cdlemk9bN  34489  cdlemkvcl  34491  cdlemk10  34492  cdlemk11  34498  cdlemk11u  34520  cdlemk37  34563  cdlemkfid1N  34570  cdlemkid1  34571  cdlemkid2  34573  cdlemk39s-id  34589  cdlemk48  34599  cdlemk50  34601  cdlemk51  34602  cdlemk52  34603  cdlemk39u  34617  tendoex  34624  dialss  34696  dia0  34702  diaglbN  34705  dia1dim  34711  dia2dimlem2  34715  dia2dimlem3  34716  dia2dimlem10  34723  cdlemm10N  34768  dib1dim  34815  diblss  34820  cdlemn2a  34846  dih1dimb  34890  dihopelvalcpre  34898  dih1  34936  dihmeetlem1N  34940  dihglblem5apreN  34941  dihglbcpreN  34950  dih1dimatlem  34979
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