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Metamath Proof Explorer

Theorem trlcocnv

Description: Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013)

Ref Expression
Hypotheses trlcocnv.h 
|- H = ( LHyp ` K )
trlcocnv.t 
|- T = ( ( LTrn ` K ) ` W )
trlcocnv.r 
|- R = ( ( trL ` K ) ` W )
Assertion trlcocnv 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` ( F o. `' G ) ) = ( R ` ( G o. `' F ) ) )

Proof

Step Hyp Ref Expression
1 trlcocnv.h 
 |-  H = ( LHyp ` K )
2 trlcocnv.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 trlcocnv.r 
 |-  R = ( ( trL ` K ) ` W )
4 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( K e. HL /\ W e. H ) )
5 1 2 ltrncnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> `' G e. T )
6 5 3adant2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> `' G e. T )
7 1 2 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ `' G e. T ) -> ( F o. `' G ) e. T )
8 6 7 syld3an3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. `' G ) e. T )
9 1 2 3 trlcnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F o. `' G ) e. T ) -> ( R ` `' ( F o. `' G ) ) = ( R ` ( F o. `' G ) ) )
10 4 8 9 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` `' ( F o. `' G ) ) = ( R ` ( F o. `' G ) ) )
11 cnvco 
 |-  `' ( F o. `' G ) = ( `' `' G o. `' F )
12 cocnvcnv1 
 |-  ( `' `' G o. `' F ) = ( G o. `' F )
13 11 12 eqtri 
 |-  `' ( F o. `' G ) = ( G o. `' F )
14 13 fveq2i 
 |-  ( R ` `' ( F o. `' G ) ) = ( R ` ( G o. `' F ) )
15 10 14 eqtr3di 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` ( F o. `' G ) ) = ( R ` ( G o. `' F ) ) )