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Metamath Proof Explorer

Theorem ltrnnid

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)

Ref Expression
Hypotheses 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 ltrnnid 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> E. p e. A ( -. p .<_ W /\ ( F ` p ) =/= p ) )

Proof

Step Hyp Ref Expression
1 ltrneq.b 
 |-  B = ( Base ` K )
2 ltrneq.l 
 |-  .<_ = ( le ` K )
3 ltrneq.a 
 |-  A = ( Atoms ` K )
4 ltrneq.h 
 |-  H = ( LHyp ` K )
5 ltrneq.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 ralinexa 
 |-  ( A. p e. A ( -. p .<_ W -> -. ( F ` p ) =/= p ) <-> -. E. p e. A ( -. p .<_ W /\ ( F ` p ) =/= p ) )
7 nne 
 |-  ( -. ( F ` p ) =/= p <-> ( F ` p ) = p )
8 7 biimpi 
 |-  ( -. ( F ` p ) =/= p -> ( F ` p ) = p )
9 8 imim2i 
 |-  ( ( -. p .<_ W -> -. ( F ` p ) =/= p ) -> ( -. p .<_ W -> ( F ` p ) = p ) )
10 9 ralimi 
 |-  ( A. p e. A ( -. p .<_ W -> -. ( F ` p ) =/= p ) -> A. p e. A ( -. p .<_ W -> ( F ` p ) = p ) )
11 6 10 sylbir 
 |-  ( -. E. p e. A ( -. p .<_ W /\ ( F ` p ) =/= p ) -> A. p e. A ( -. p .<_ W -> ( F ` p ) = p ) )
12 1 2 3 4 5 ltrnid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( A. p e. A ( -. p .<_ W -> ( F ` p ) = p ) <-> F = ( _I |` B ) ) )
13 11 12 imbitrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( -. E. p e. A ( -. p .<_ W /\ ( F ` p ) =/= p ) -> F = ( _I |` B ) ) )
14 13 necon1ad 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F =/= ( _I |` B ) -> E. p e. A ( -. p .<_ W /\ ( F ` p ) =/= p ) ) )
15 14 3impia 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> E. p e. A ( -. p .<_ W /\ ( F ` p ) =/= p ) )