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Metamath Proof Explorer

Theorem ltrncnvnid

Description: If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013)

Ref Expression
Hypotheses ltrn1o.b 
|- B = ( Base ` K )
ltrn1o.h 
|- H = ( LHyp ` K )
ltrn1o.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrncnvnid 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> `' F =/= ( _I |` B ) )

Proof

Step Hyp Ref Expression
1 ltrn1o.b 
 |-  B = ( Base ` K )
2 ltrn1o.h 
 |-  H = ( LHyp ` K )
3 ltrn1o.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> F =/= ( _I |` B ) )
5 1 2 3 ltrn1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F : B -1-1-onto-> B )
6 5 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> F : B -1-1-onto-> B )
7 f1orel 
 |-  ( F : B -1-1-onto-> B -> Rel F )
8 6 7 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> Rel F )
9 dfrel2 
 |-  ( Rel F <-> `' `' F = F )
10 8 9 sylib 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> `' `' F = F )
11 cnveq 
 |-  ( `' F = ( _I |` B ) -> `' `' F = `' ( _I |` B ) )
12 10 11 sylan9req 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ `' F = ( _I |` B ) ) -> F = `' ( _I |` B ) )
13 cnvresid 
 |-  `' ( _I |` B ) = ( _I |` B )
14 12 13 eqtrdi 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ `' F = ( _I |` B ) ) -> F = ( _I |` B ) )
15 14 ex 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( `' F = ( _I |` B ) -> F = ( _I |` B ) ) )
16 15 necon3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( F =/= ( _I |` B ) -> `' F =/= ( _I |` B ) ) )
17 4 16 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> `' F =/= ( _I |` B ) )