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Theorem trlle 30995
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 2296 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 eqid 2296 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 trlle.h . . . . 5  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 30829 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
)  .<_  W ) )
65adantr 451 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )  e.  ( Atoms `  K )  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
7 eqid 2296 . . . 4  |-  ( join `  K )  =  (
join `  K )
8 eqid 2296 . . . 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 30974 . . 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 1278 . 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 30175 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 706 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  Lat )
15 hlop 30174 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  OP )
17 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1817, 4lhpbase 30809 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1918ad2antlr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  W  e.  ( Base `  K )
)
2017, 2opoccl 30006 . . . . 5  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
2116, 19, 20syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( oc `  K ) `  W )  e.  (
Base `  K )
)
2217, 4, 9ltrncl 30936 . . . . 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 1278 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  (
Base `  K )
)
2417, 7latjcl 14172 . . . 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 1182 . . 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 14199 . . 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 1182 . 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 4059 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 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   occoc 13232   joincjn 14094   meetcmee 14095   Latclat 14167   OPcops 29984   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969
This theorem is referenced by:  trlne  30996  cdlemc5  31006  cdlemg6c  31431  cdlemg10c  31450  cdlemg10  31452  cdlemg17dALTN  31475  cdlemg27a  31503  cdlemg31b0N  31505  cdlemg31b0a  31506  cdlemg27b  31507  cdlemg31c  31510  cdlemg35  31524  cdlemh2  31627  cdlemh  31628  cdlemk3  31644  cdlemk9  31650  cdlemk9bN  31651  cdlemk10  31654  cdlemk12  31661  cdlemk14  31665  cdlemk12u  31683  cdlemkfid1N  31732  cdlemk47  31760  dia1N  31865  dia1dim  31873  dia2dimlem1  31876  dia2dimlem10  31885  dib1dim  31977  cdlemn2a  32008  dih1dimb  32052  dihopelvalcpre  32060  dihwN  32101  dihglblem5apreN  32103  dih1dimatlem  32141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970
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