Metamath Proof Explorer

Theorem trljco2

Description: Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013)

Ref Expression
Hypotheses trljco.j 
|- .\/ = ( join ` K )
trljco.h 
|- H = ( LHyp ` K )
trljco.t 
|- T = ( ( LTrn ` K ) ` W )
trljco.r 
|- R = ( ( trL ` K ) ` W )
Assertion trljco2 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` G ) .\/ ( R ` ( F o. G ) ) ) )

Proof

Step Hyp Ref Expression
1 trljco.j 
 |-  .\/ = ( join ` K )
2 trljco.h 
 |-  H = ( LHyp ` K )
3 trljco.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 trljco.r 
 |-  R = ( ( trL ` K ) ` W )
5 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> K e. HL )
6 5 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> K e. Lat )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 2 3 4 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
9 8 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` F ) e. ( Base ` K ) )
10 7 2 3 4 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) )
11 10 3adant2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) )
12 7 1 latjcom 
 |-  ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) = ( ( R ` G ) .\/ ( R ` F ) ) )
13 6 9 11 12 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` G ) ) = ( ( R ` G ) .\/ ( R ` F ) ) )
14 1 2 3 4 trljco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ F e. T ) -> ( ( R ` G ) .\/ ( R ` ( G o. F ) ) ) = ( ( R ` G ) .\/ ( R ` F ) ) )
15 14 3com23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` G ) .\/ ( R ` ( G o. F ) ) ) = ( ( R ` G ) .\/ ( R ` F ) ) )
16 13 15 eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` G ) ) = ( ( R ` G ) .\/ ( R ` ( G o. F ) ) ) )
17 1 2 3 4 trljco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` F ) .\/ ( R ` G ) ) )
18 2 3 ltrncom 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. G ) = ( G o. F ) )
19 18 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` ( F o. G ) ) = ( R ` ( G o. F ) ) )
20 19 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` G ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` G ) .\/ ( R ` ( G o. F ) ) ) )
21 16 17 20 3eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` G ) .\/ ( R ` ( F o. G ) ) ) )