This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem trlconid

Description: The composition of two different translations is not the identity translation. (Contributed by NM, 22-Jul-2013)

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

Proof

Step Hyp Ref Expression
1 trlconid.b 
 |-  B = ( Base ` K )
2 trlconid.h 
 |-  H = ( LHyp ` K )
3 trlconid.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 trlconid.r 
 |-  R = ( ( trL ` K ) ` W )
5 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
6 5 2 3 4 trlcoat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( R ` ( F o. G ) ) e. ( Atoms ` K ) )
7 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( K e. HL /\ W e. H ) )
8 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> F e. T )
9 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> G e. T )
10 2 3 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. G ) e. T )
11 7 8 9 10 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( F o. G ) e. T )
12 1 5 2 3 4 trlnidatb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F o. G ) e. T ) -> ( ( F o. G ) =/= ( _I |` B ) <-> ( R ` ( F o. G ) ) e. ( Atoms ` K ) ) )
13 7 11 12 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( ( F o. G ) =/= ( _I |` B ) <-> ( R ` ( F o. G ) ) e. ( Atoms ` K ) ) )
14 6 13 mpbird 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( F o. G ) =/= ( _I |` B ) )