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Theorem trlle 34980
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 2467 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 eqid 2467 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 trlle.h . . . . 5  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 34814 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  (
Atoms `  K )  /\  -.  ( ( oc `  K ) `  W
)  .<_  W ) )
65adantr 465 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( oc `  K
) `  W )  e.  ( Atoms `  K )  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
7 eqid 2467 . . . 4  |-  ( join `  K )  =  (
join `  K )
8 eqid 2467 . . . 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 34959 . . 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 1325 . 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 34160 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 725 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  Lat )
15 hlop 34159 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 725 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  OP )
17 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1817, 4lhpbase 34794 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1918ad2antlr 726 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  W  e.  ( Base `  K )
)
2017, 2opoccl 33991 . . . . 5  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
2116, 19, 20syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( oc `  K ) `  W )  e.  (
Base `  K )
)
2217, 4, 9ltrncl 34921 . . . . 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 1325 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F `  ( ( oc `  K ) `  W
) )  e.  (
Base `  K )
)
2417, 7latjcl 15531 . . . 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 1228 . . 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 15557 . . 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 1228 . 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 4467 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   lecple 14555   occoc 14556   joincjn 15424   meetcmee 15425   Latclat 15525   OPcops 33969   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   trLctrl 34954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955
This theorem is referenced by:  trlne  34981  cdlemc5  34991  cdlemg6c  35416  cdlemg10c  35435  cdlemg10  35437  cdlemg17dALTN  35460  cdlemg27a  35488  cdlemg31b0N  35490  cdlemg31b0a  35491  cdlemg27b  35492  cdlemg31c  35495  cdlemg35  35509  cdlemh2  35612  cdlemh  35613  cdlemk3  35629  cdlemk9  35635  cdlemk9bN  35636  cdlemk10  35639  cdlemk12  35646  cdlemk14  35650  cdlemk12u  35668  cdlemkfid1N  35717  cdlemk47  35745  dia1N  35850  dia1dim  35858  dia2dimlem1  35861  dia2dimlem10  35870  dib1dim  35962  cdlemn2a  35993  dih1dimb  36037  dihopelvalcpre  36045  dihwN  36086  dihglblem5apreN  36088  dih1dimatlem  36126
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