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Metamath Proof Explorer

Theorem trlid0b

Description: A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013)

Ref Expression
Hypotheses trlid0b.b 
|- B = ( Base ` K )
trlid0b.z 
|- .0. = ( 0. ` K )
trlid0b.h 
|- H = ( LHyp ` K )
trlid0b.t 
|- T = ( ( LTrn ` K ) ` W )
trlid0b.r 
|- R = ( ( trL ` K ) ` W )
Assertion trlid0b 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F = ( _I |` B ) <-> ( R ` F ) = .0. ) )

Proof

Step Hyp Ref Expression
1 trlid0b.b 
 |-  B = ( Base ` K )
2 trlid0b.z 
 |-  .0. = ( 0. ` K )
3 trlid0b.h 
 |-  H = ( LHyp ` K )
4 trlid0b.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 trlid0b.r 
 |-  R = ( ( trL ` K ) ` W )
6 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
7 1 6 3 4 5 trlnidatb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F =/= ( _I |` B ) <-> ( R ` F ) e. ( Atoms ` K ) ) )
8 2 6 3 4 5 trlatn0 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( ( R ` F ) e. ( Atoms ` K ) <-> ( R ` F ) =/= .0. ) )
9 7 8 bitrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F =/= ( _I |` B ) <-> ( R ` F ) =/= .0. ) )
10 9 necon4bid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F = ( _I |` B ) <-> ( R ` F ) = .0. ) )