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Theorem trlnidat 35999
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
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 trlnidat.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2457 . . 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 35961 . 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 1026 . . . 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 997 . . . 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 1024 . . . 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 1027 . . . 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 1025 . . . 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 35995 . . . 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 1237 . . 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 2951 . 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 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   class class class wbr 4456    _I cid 4799    |` cres 5010   ` cfv 5594   Basecbs 14643   lecple 14718   Atomscatm 35089   HLchlt 35176   LHypclh 35809   LTrncltrn 35926   trLctrl 35984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985
This theorem is referenced by:  ltrnnidn  36000  trlnidatb  36003  trlcone  36555  cdlemg46  36562  trljco  36567  cdlemh2  36643  cdlemh  36644  tendotr  36657  cdlemk3  36660  cdlemk12  36677  cdlemkole  36680  cdlemk14  36681  cdlemk15  36682  cdlemk1u  36686  cdlemk5u  36688  cdlemk12u  36699  cdlemk37  36741  cdlemk39  36743  cdlemkid1  36749  cdlemk47  36776  cdlemk51  36780  cdlemk52  36781  cdleml1N  36803
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