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Theorem trlcl 30646
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 2404 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2404 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 eqid 2404 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 trlcl.h . . . . 5  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 30500 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
) ( le `  K ) W ) )
65adantr 452 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )  e.  ( Atoms `  K )  /\  -.  ( ( oc
`  K ) `  W ) ( le
`  K ) W ) )
7 eqid 2404 . . . 4  |-  ( join `  K )  =  (
join `  K )
8 eqid 2404 . . . 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 30645 . . 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 1280 . 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 29846 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  Lat )
15 hlop 29845 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  OP )
17 trlcl.b . . . . . . 7  |-  B  =  ( Base `  K
)
1817, 4lhpbase 30480 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
1918ad2antlr 708 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  W  e.  B )
2017, 2opoccl 29677 . . . . 5  |-  ( ( K  e.  OP  /\  W  e.  B )  ->  ( ( oc `  K ) `  W
)  e.  B )
2116, 19, 20syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( oc `  K ) `  W )  e.  B
)
2217, 4, 9ltrncl 30607 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( oc `  K ) `  W
)  e.  B )  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  B
)
2321, 22mpd3an3 1280 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  B
)
2417, 7latjcl 14434 . . . 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 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )
( join `  K )
( F `  (
( oc `  K
) `  W )
) )  e.  B
)
2617, 8latmcl 14435 . . 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 1184 . 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 2478 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 359    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   occoc 13492   joincjn 14356   meetcmee 14357   Latclat 14429   OPcops 29655   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem is referenced by:  trljat1  30648  trljat2  30649  trlval3  30669  cdlemc3  30675  cdlemc5  30677  trlord  31051  cdlemg4c  31094  cdlemg4  31099  cdlemg6c  31102  cdlemg10c  31121  cdlemg10  31123  cdlemg12e  31129  cdlemg17dALTN  31146  cdlemg31a  31179  cdlemg31b  31180  cdlemg35  31195  cdlemg44a  31213  trljco  31222  trljco2  31223  tendoidcl  31251  tendococl  31254  tendoid  31255  tendopltp  31262  tendo0tp  31271  cdlemh1  31297  cdlemh2  31298  cdlemi1  31300  cdlemi  31302  cdlemk9  31321  cdlemk9bN  31322  cdlemkvcl  31324  cdlemk10  31325  cdlemk11  31331  cdlemk11u  31353  cdlemk37  31396  cdlemkfid1N  31403  cdlemkid1  31404  cdlemkid2  31406  cdlemk39s-id  31422  cdlemk48  31432  cdlemk50  31434  cdlemk51  31435  cdlemk52  31436  cdlemk39u  31450  tendoex  31457  dialss  31529  dia0  31535  diaglbN  31538  dia1dim  31544  dia2dimlem2  31548  dia2dimlem3  31549  dia2dimlem10  31556  cdlemm10N  31601  dib1dim  31648  diblss  31653  cdlemn2a  31679  dih1dimb  31723  dihopelvalcpre  31731  dih1  31769  dihmeetlem1N  31773  dihglblem5apreN  31774  dihglbcpreN  31783  dih1dimatlem  31812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
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