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Theorem trlnidat 29513
Description: The trace of a lattice translation other than the identity is an atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
Hypotheses
Ref Expression
trlnidat.b  |-  B  =  ( Base `  K
)
trlnidat.a  |-  A  =  ( Atoms `  K )
trlnidat.h  |-  H  =  ( LHyp `  K
)
trlnidat.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlnidat.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlnidat  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  A
)

Proof of Theorem trlnidat
StepHypRef Expression
1 trlnidat.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2256 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 trlnidat.a . . 3  |-  A  =  ( Atoms `  K )
4 trlnidat.h . . 3  |-  H  =  ( LHyp `  K
)
5 trlnidat.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
61, 2, 3, 4, 5ltrnnid 29476 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  E. p  e.  A  ( -.  p ( le `  K ) W  /\  ( F `  p )  =/=  p
) )
7 simp11 990 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  p  e.  A  /\  ( -.  p ( le `  K ) W  /\  ( F `  p )  =/=  p ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
8 simp2 961 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  p  e.  A  /\  ( -.  p ( le `  K ) W  /\  ( F `  p )  =/=  p ) )  ->  p  e.  A
)
9 simp3l 988 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  p  e.  A  /\  ( -.  p ( le `  K ) W  /\  ( F `  p )  =/=  p ) )  ->  -.  p ( le `  K ) W )
10 simp12 991 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  p  e.  A  /\  ( -.  p ( le `  K ) W  /\  ( F `  p )  =/=  p ) )  ->  F  e.  T
)
11 simp3r 989 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  p  e.  A  /\  ( -.  p ( le `  K ) W  /\  ( F `  p )  =/=  p ) )  ->  ( F `  p )  =/=  p
)
12 trlnidat.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
132, 3, 4, 5, 12trlat 29509 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  -.  p
( le `  K
) W )  /\  ( F  e.  T  /\  ( F `  p
)  =/=  p ) )  ->  ( R `  F )  e.  A
)
147, 8, 9, 10, 11, 13syl122anc 1196 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  p  e.  A  /\  ( -.  p ( le `  K ) W  /\  ( F `  p )  =/=  p ) )  ->  ( R `  F )  e.  A
)
1514rexlimdv3a 2642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( E. p  e.  A  ( -.  p ( le `  K ) W  /\  ( F `  p )  =/=  p )  -> 
( R `  F
)  e.  A ) )
166, 15mpd 16 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517   class class class wbr 3983    _I cid 4262    |` cres 4649   ` cfv 4659   Basecbs 13096   lecple 13163   Atomscatm 28604   HLchlt 28691   LHypclh 29324   LTrncltrn 29441   trLctrl 29498
This theorem is referenced by:  ltrnnidn  29514  trlnidatb  29517  trlcone  30068  cdlemg46  30075  trljco  30080  cdlemh2  30156  cdlemh  30157  tendotr  30170  cdlemk3  30173  cdlemk12  30190  cdlemkole  30193  cdlemk14  30194  cdlemk15  30195  cdlemk1u  30199  cdlemk5u  30201  cdlemk12u  30212  cdlemk37  30254  cdlemk39  30256  cdlemkid1  30262  cdlemk47  30289  cdlemk51  30293  cdlemk52  30294  cdleml1N  30316
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-map 6728  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499
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