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Theorem trlnidat 29629
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 2284 . . 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 29592 . 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 29625 . . . 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 2670 . 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 1628    e. wcel 1688    =/= wne 2447   E.wrex 2545   class class class wbr 4024    _I cid 4303    |` cres 4690   ` cfv 5221   Basecbs 13142   lecple 13209   Atomscatm 28720   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614
This theorem is referenced by:  ltrnnidn  29630  trlnidatb  29633  trlcone  30184  cdlemg46  30191  trljco  30196  cdlemh2  30272  cdlemh  30273  tendotr  30286  cdlemk3  30289  cdlemk12  30306  cdlemkole  30309  cdlemk14  30310  cdlemk15  30311  cdlemk1u  30315  cdlemk5u  30317  cdlemk12u  30328  cdlemk37  30370  cdlemk39  30372  cdlemkid1  30378  cdlemk47  30405  cdlemk51  30409  cdlemk52  30410  cdleml1N  30432
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615
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