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

Theorem trlle 30918
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 2435 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 eqid 2435 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 trlle.h . . . . 5  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4lhpocnel 30752 . . . 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 2435 . . . 4  |-  ( join `  K )  =  (
join `  K )
8 eqid 2435 . . . 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 30897 . . 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 30098 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  Lat )
15 hlop 30097 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  K  e.  OP )
17 eqid 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1817, 4lhpbase 30732 . . . . . 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 29929 . . . . 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 30859 . . . . 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 14471 . . . 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 14498 . . 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 4224 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   occoc 13529   joincjn 14393   meetcmee 14394   Latclat 14466   OPcops 29907   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892
This theorem is referenced by:  trlne  30919  cdlemc5  30929  cdlemg6c  31354  cdlemg10c  31373  cdlemg10  31375  cdlemg17dALTN  31398  cdlemg27a  31426  cdlemg31b0N  31428  cdlemg31b0a  31429  cdlemg27b  31430  cdlemg31c  31433  cdlemg35  31447  cdlemh2  31550  cdlemh  31551  cdlemk3  31567  cdlemk9  31573  cdlemk9bN  31574  cdlemk10  31577  cdlemk12  31584  cdlemk14  31588  cdlemk12u  31606  cdlemkfid1N  31655  cdlemk47  31683  dia1N  31788  dia1dim  31796  dia2dimlem1  31799  dia2dimlem10  31808  dib1dim  31900  cdlemn2a  31931  dih1dimb  31975  dihopelvalcpre  31983  dihwN  32024  dihglblem5apreN  32026  dih1dimatlem  32064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893
  Copyright terms: Public domain W3C validator