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Theorem ltrnnid 30325
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 2588 . . . . 5  |-  ( A. p  e.  A  ( -.  p  .<_  W  ->  -.  ( F `  p
)  =/=  p )  <->  -.  E. p  e.  A  ( -.  p  .<_  W  /\  ( F `  p )  =/=  p
) )
2 nne 2450 . . . . . . . 8  |-  ( -.  ( F `  p
)  =/=  p  <->  ( F `  p )  =  p )
32biimpi 186 . . . . . . 7  |-  ( -.  ( F `  p
)  =/=  p  -> 
( F `  p
)  =  p )
43imim2i 13 . . . . . 6  |-  ( ( -.  p  .<_  W  ->  -.  ( F `  p
)  =/=  p )  ->  ( -.  p  .<_  W  ->  ( F `  p )  =  p ) )
54ralimi 2618 . . . . 5  |-  ( A. p  e.  A  ( -.  p  .<_  W  ->  -.  ( F `  p
)  =/=  p )  ->  A. p  e.  A  ( -.  p  .<_  W  ->  ( F `  p )  =  p ) )
61, 5sylbir 204 . . . 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 30324 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( A. p  e.  A  ( -.  p  .<_  W  -> 
( F `  p
)  =  p )  <-> 
F  =  (  _I  |`  B ) ) )
136, 12syl5ib 210 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( -.  E. p  e.  A  ( -.  p  .<_  W  /\  ( F `  p )  =/=  p )  ->  F  =  (  _I  |`  B ) ) )
1413necon1ad 2513 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( F `  p )  =/=  p
) ) )
15143impia 1148 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   class class class wbr 4023    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   lecple 13215   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290
This theorem is referenced by:  trlnidat  30362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-laut 30178  df-ldil 30293  df-ltrn 30294
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