Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trlcl Structured version   Unicode version

Theorem trlcl 31035
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 2438 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2438 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 eqid 2438 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 trlcl.h . . . . 5  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 30889 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
) ( le `  K ) W ) )
65adantr 453 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )  e.  ( Atoms `  K )  /\  -.  ( ( oc
`  K ) `  W ) ( le
`  K ) W ) )
7 eqid 2438 . . . 4  |-  ( join `  K )  =  (
join `  K )
8 eqid 2438 . . . 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 31034 . . 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 1281 . 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 30235 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 708 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  Lat )
15 hlop 30234 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 708 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  OP )
17 trlcl.b . . . . . . 7  |-  B  =  ( Base `  K
)
1817, 4lhpbase 30869 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
1918ad2antlr 709 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  W  e.  B )
2017, 2opoccl 30066 . . . . 5  |-  ( ( K  e.  OP  /\  W  e.  B )  ->  ( ( oc `  K ) `  W
)  e.  B )
2116, 19, 20syl2anc 644 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( oc `  K ) `  W )  e.  B
)
2217, 4, 9ltrncl 30996 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( oc `  K ) `  W
)  e.  B )  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  B
)
2321, 22mpd3an3 1281 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  B
)
2417, 7latjcl 14484 . . . 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 1185 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )
( join `  K )
( F `  (
( oc `  K
) `  W )
) )  e.  B
)
2617, 8latmcl 14485 . . 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 1185 . 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 2512 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 360    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   occoc 13542   joincjn 14406   meetcmee 14407   Latclat 14479   OPcops 30044   Atomscatm 30135   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   trLctrl 31029
This theorem is referenced by:  trljat1  31037  trljat2  31038  trlval3  31058  cdlemc3  31064  cdlemc5  31066  trlord  31440  cdlemg4c  31483  cdlemg4  31488  cdlemg6c  31491  cdlemg10c  31510  cdlemg10  31512  cdlemg12e  31518  cdlemg17dALTN  31535  cdlemg31a  31568  cdlemg31b  31569  cdlemg35  31584  cdlemg44a  31602  trljco  31611  trljco2  31612  tendoidcl  31640  tendococl  31643  tendoid  31644  tendopltp  31651  tendo0tp  31660  cdlemh1  31686  cdlemh2  31687  cdlemi1  31689  cdlemi  31691  cdlemk9  31710  cdlemk9bN  31711  cdlemkvcl  31713  cdlemk10  31714  cdlemk11  31720  cdlemk11u  31742  cdlemk37  31785  cdlemkfid1N  31792  cdlemkid1  31793  cdlemkid2  31795  cdlemk39s-id  31811  cdlemk48  31821  cdlemk50  31823  cdlemk51  31824  cdlemk52  31825  cdlemk39u  31839  tendoex  31846  dialss  31918  dia0  31924  diaglbN  31927  dia1dim  31933  dia2dimlem2  31937  dia2dimlem3  31938  dia2dimlem10  31945  cdlemm10N  31990  dib1dim  32037  diblss  32042  cdlemn2a  32068  dih1dimb  32112  dihopelvalcpre  32120  dih1  32158  dihmeetlem1N  32162  dihglblem5apreN  32163  dihglbcpreN  32172  dih1dimatlem  32201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-map 7023  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030
  Copyright terms: Public domain W3C validator