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Metamath Proof Explorer

Theorem trlnid

Description: Different translations with the same trace cannot be the identity. (Contributed by NM, 26-Jul-2013)

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

Proof

Step Hyp Ref Expression
1 trlnid.b 
 |-  B = ( Base ` K )
2 trlnid.h 
 |-  H = ( LHyp ` K )
3 trlnid.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 trlnid.r 
 |-  R = ( ( trL ` K ) ` W )
5 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> F =/= G )
6 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( K e. HL /\ W e. H ) )
7 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> F e. T )
8 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
9 1 8 2 3 4 trlid0b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F = ( _I |` B ) <-> ( R ` F ) = ( 0. ` K ) ) )
10 6 7 9 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( F = ( _I |` B ) <-> ( R ` F ) = ( 0. ` K ) ) )
11 10 biimpar 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> F = ( _I |` B ) )
12 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( R ` F ) = ( R ` G ) )
13 12 eqeq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( ( R ` F ) = ( 0. ` K ) <-> ( R ` G ) = ( 0. ` K ) ) )
14 13 biimpa 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> ( R ` G ) = ( 0. ` K ) )
15 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> ( K e. HL /\ W e. H ) )
16 simpl2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> G e. T )
17 1 8 2 3 4 trlid0b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( G = ( _I |` B ) <-> ( R ` G ) = ( 0. ` K ) ) )
18 15 16 17 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> ( G = ( _I |` B ) <-> ( R ` G ) = ( 0. ` K ) ) )
19 14 18 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> G = ( _I |` B ) )
20 11 19 eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) /\ ( R ` F ) = ( 0. ` K ) ) -> F = G )
21 20 ex 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( ( R ` F ) = ( 0. ` K ) -> F = G ) )
22 10 21 sylbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( F = ( _I |` B ) -> F = G ) )
23 22 necon3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> ( F =/= G -> F =/= ( _I |` B ) ) )
24 5 23 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= G /\ ( R ` F ) = ( R ` G ) ) ) -> F =/= ( _I |` B ) )