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Theorem trlcl 30329
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 2380 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2380 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 eqid 2380 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 trlcl.h . . . . 5  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 30183 . . . 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 2380 . . . 4  |-  ( join `  K )  =  (
join `  K )
8 eqid 2380 . . . 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 30328 . . 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 29529 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  Lat )
15 hlop 29528 . . . . . 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 30163 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
1918ad2antlr 708 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  W  e.  B )
2017, 2opoccl 29360 . . . . 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 30290 . . . . 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 14399 . . . 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 14400 . . 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 2454 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 1717   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   occoc 13457   joincjn 14321   meetcmee 14322   Latclat 14394   OPcops 29338   Atomscatm 29429   HLchlt 29516   LHypclh 30149   LTrncltrn 30266   trLctrl 30323
This theorem is referenced by:  trljat1  30331  trljat2  30332  trlval3  30352  cdlemc3  30358  cdlemc5  30360  trlord  30734  cdlemg4c  30777  cdlemg4  30782  cdlemg6c  30785  cdlemg10c  30804  cdlemg10  30806  cdlemg12e  30812  cdlemg17dALTN  30829  cdlemg31a  30862  cdlemg31b  30863  cdlemg35  30878  cdlemg44a  30896  trljco  30905  trljco2  30906  tendoidcl  30934  tendococl  30937  tendoid  30938  tendopltp  30945  tendo0tp  30954  cdlemh1  30980  cdlemh2  30981  cdlemi1  30983  cdlemi  30985  cdlemk9  31004  cdlemk9bN  31005  cdlemkvcl  31007  cdlemk10  31008  cdlemk11  31014  cdlemk11u  31036  cdlemk37  31079  cdlemkfid1N  31086  cdlemkid1  31087  cdlemkid2  31089  cdlemk39s-id  31105  cdlemk48  31115  cdlemk50  31117  cdlemk51  31118  cdlemk52  31119  cdlemk39u  31133  tendoex  31140  dialss  31212  dia0  31218  diaglbN  31221  dia1dim  31227  dia2dimlem2  31231  dia2dimlem3  31232  dia2dimlem10  31239  cdlemm10N  31284  dib1dim  31331  diblss  31336  cdlemn2a  31362  dih1dimb  31406  dihopelvalcpre  31414  dih1  31452  dihmeetlem1N  31456  dihglblem5apreN  31457  dihglbcpreN  31466  dih1dimatlem  31495
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-map 6949  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324
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