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

Theorem trlle 30299
Description: The trace of a lattice translation is less than the fiducial co-atom  W. (Contributed by NM, 25-May-2012.)
Hypotheses
Ref Expression
trlle.l  |-  .<_  =  ( le `  K )
trlle.h  |-  H  =  ( LHyp `  K
)
trlle.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlle.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlle  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )

Proof of Theorem trlle
StepHypRef Expression
1 trlle.l . . . . 5  |-  .<_  =  ( le `  K )
2 eqid 2388 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 eqid 2388 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 trlle.h . . . . 5  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 30133 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
)  .<_  W ) )
65adantr 452 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )  e.  ( Atoms `  K )  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
7 eqid 2388 . . . 4  |-  ( join `  K )  =  (
join `  K )
8 eqid 2388 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
9 trlle.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
10 trlle.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
111, 7, 8, 3, 4, 9, 10trlval2 30278 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
)  .<_  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 29479 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  Lat )
15 hlop 29478 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  OP )
17 eqid 2388 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1817, 4lhpbase 30113 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1918ad2antlr 708 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  W  e.  ( Base `  K )
)
2017, 2opoccl 29310 . . . . 5  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
2116, 19, 20syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( oc `  K ) `  W )  e.  (
Base `  K )
)
2217, 4, 9ltrncl 30240 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( oc `  K ) `  W
)  e.  ( Base `  K ) )  -> 
( F `  (
( oc `  K
) `  W )
)  e.  ( Base `  K ) )
2321, 22mpd3an3 1280 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  (
Base `  K )
)
2417, 7latjcl 14407 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  W
)  e.  ( Base `  K )  /\  ( F `  ( ( oc `  K ) `  W ) )  e.  ( Base `  K
) )  ->  (
( ( oc `  K ) `  W
) ( join `  K
) ( F `  ( ( oc `  K ) `  W
) ) )  e.  ( Base `  K
) )
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.  (
Base `  K )
)
2617, 1, 8latmle2 14434 . . 3  |-  ( ( K  e.  Lat  /\  ( ( ( oc
`  K ) `  W ) ( join `  K ) ( F `
 ( ( oc
`  K ) `  W ) ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( (
( ( oc `  K ) `  W
) ( join `  K
) ( F `  ( ( oc `  K ) `  W
) ) ) (
meet `  K ) W )  .<_  W )
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 )  .<_  W )
2812, 27eqbrtrd 4174 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Basecbs 13397   lecple 13464   occoc 13465   joincjn 14329   meetcmee 14330   Latclat 14402   OPcops 29288   Atomscatm 29379   HLchlt 29466   LHypclh 30099   LTrncltrn 30216   trLctrl 30273
This theorem is referenced by:  trlne  30300  cdlemc5  30310  cdlemg6c  30735  cdlemg10c  30754  cdlemg10  30756  cdlemg17dALTN  30779  cdlemg27a  30807  cdlemg31b0N  30809  cdlemg31b0a  30810  cdlemg27b  30811  cdlemg31c  30814  cdlemg35  30828  cdlemh2  30931  cdlemh  30932  cdlemk3  30948  cdlemk9  30954  cdlemk9bN  30955  cdlemk10  30958  cdlemk12  30965  cdlemk14  30969  cdlemk12u  30987  cdlemkfid1N  31036  cdlemk47  31064  dia1N  31169  dia1dim  31177  dia2dimlem1  31180  dia2dimlem10  31189  dib1dim  31281  cdlemn2a  31312  dih1dimb  31356  dihopelvalcpre  31364  dihwN  31405  dihglblem5apreN  31407  dih1dimatlem  31445
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-map 6957  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274
  Copyright terms: Public domain W3C validator