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Theorem trlle 30373
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 2283 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 eqid 2283 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 trlle.h . . . . 5  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 30207 . . . 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 2283 . . . 4  |-  ( join `  K )  =  (
join `  K )
8 eqid 2283 . . . 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 30352 . . 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 29553 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 706 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  Lat )
15 hlop 29552 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  OP )
17 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1817, 4lhpbase 30187 . . . . . 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 29384 . . . . 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 30314 . . . . 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 14156 . . . 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 14183 . . 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 4043 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   meetcmee 14079   Latclat 14151   OPcops 29362   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  trlne  30374  cdlemc5  30384  cdlemg6c  30809  cdlemg10c  30828  cdlemg10  30830  cdlemg17dALTN  30853  cdlemg27a  30881  cdlemg31b0N  30883  cdlemg31b0a  30884  cdlemg27b  30885  cdlemg31c  30888  cdlemg35  30902  cdlemh2  31005  cdlemh  31006  cdlemk3  31022  cdlemk9  31028  cdlemk9bN  31029  cdlemk10  31032  cdlemk12  31039  cdlemk14  31043  cdlemk12u  31061  cdlemkfid1N  31110  cdlemk47  31138  dia1N  31243  dia1dim  31251  dia2dimlem1  31254  dia2dimlem10  31263  dib1dim  31355  cdlemn2a  31386  dih1dimb  31430  dihopelvalcpre  31438  dihwN  31479  dihglblem5apreN  31481  dih1dimatlem  31519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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