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

Theorem ltrnnid 30870
Description: If a lattice translation is not the identity, then there is an atom not under the fiducial co-atom 
W and not equal to its translation. (Contributed by NM, 24-May-2012.)
Hypotheses
Ref Expression
ltrneq.b  |-  B  =  ( Base `  K
)
ltrneq.l  |-  .<_  =  ( le `  K )
ltrneq.a  |-  A  =  ( Atoms `  K )
ltrneq.h  |-  H  =  ( LHyp `  K
)
ltrneq.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnnid  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( F `  p )  =/=  p
) )
Distinct variable groups:    A, p    B, p    F, p    H, p    K, p    T, p    W, p
Allowed substitution hint:    .<_ ( p)

Proof of Theorem ltrnnid
StepHypRef Expression
1 ralinexa 2742 . . . . 5  |-  ( A. p  e.  A  ( -.  p  .<_  W  ->  -.  ( F `  p
)  =/=  p )  <->  -.  E. p  e.  A  ( -.  p  .<_  W  /\  ( F `  p )  =/=  p
) )
2 nne 2602 . . . . . . . 8  |-  ( -.  ( F `  p
)  =/=  p  <->  ( F `  p )  =  p )
32biimpi 187 . . . . . . 7  |-  ( -.  ( F `  p
)  =/=  p  -> 
( F `  p
)  =  p )
43imim2i 14 . . . . . 6  |-  ( ( -.  p  .<_  W  ->  -.  ( F `  p
)  =/=  p )  ->  ( -.  p  .<_  W  ->  ( F `  p )  =  p ) )
54ralimi 2773 . . . . 5  |-  ( A. p  e.  A  ( -.  p  .<_  W  ->  -.  ( F `  p
)  =/=  p )  ->  A. p  e.  A  ( -.  p  .<_  W  ->  ( F `  p )  =  p ) )
61, 5sylbir 205 . . . 4  |-  ( -. 
E. p  e.  A  ( -.  p  .<_  W  /\  ( F `  p )  =/=  p
)  ->  A. p  e.  A  ( -.  p  .<_  W  ->  ( F `  p )  =  p ) )
7 ltrneq.b . . . . 5  |-  B  =  ( Base `  K
)
8 ltrneq.l . . . . 5  |-  .<_  =  ( le `  K )
9 ltrneq.a . . . . 5  |-  A  =  ( Atoms `  K )
10 ltrneq.h . . . . 5  |-  H  =  ( LHyp `  K
)
11 ltrneq.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
127, 8, 9, 10, 11ltrnid 30869 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( A. p  e.  A  ( -.  p  .<_  W  -> 
( F `  p
)  =  p )  <-> 
F  =  (  _I  |`  B ) ) )
136, 12syl5ib 211 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( -.  E. p  e.  A  ( -.  p  .<_  W  /\  ( F `  p )  =/=  p )  ->  F  =  (  _I  |`  B ) ) )
1413necon1ad 2665 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( F `  p )  =/=  p
) ) )
15143impia 1150 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( F `  p )  =/=  p
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   class class class wbr 4204    _I cid 4485    |` cres 4872   ` cfv 5446   Basecbs 13461   lecple 13528   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835
This theorem is referenced by:  trlnidat  30907
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-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  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-laut 30723  df-ldil 30838  df-ltrn 30839
  Copyright terms: Public domain W3C validator