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Metamath Proof Explorer

Theorem tendo0co2

Description: The additive identity trace-preserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 ? (Contributed by NM, 11-Jun-2013)

Ref Expression
Hypotheses tendo0.b 
|- B = ( Base ` K )
tendo0.h 
|- H = ( LHyp ` K )
tendo0.t 
|- T = ( ( LTrn ` K ) ` W )
tendo0.e 
|- E = ( ( TEndo ` K ) ` W )
tendo0.o 
|- O = ( f e. T |-> ( _I |` B ) )
Assertion tendo0co2 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( O ` ( F o. G ) ) = ( ( O ` F ) o. ( O ` G ) ) )

Proof

Step Hyp Ref Expression
1 tendo0.b 
 |-  B = ( Base ` K )
2 tendo0.h 
 |-  H = ( LHyp ` K )
3 tendo0.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 tendo0.e 
 |-  E = ( ( TEndo ` K ) ` W )
5 tendo0.o 
 |-  O = ( f e. T |-> ( _I |` B ) )
6 2 3 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. G ) e. T )
7 5 1 tendo02 
 |-  ( ( F o. G ) e. T -> ( O ` ( F o. G ) ) = ( _I |` B ) )
8 6 7 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( O ` ( F o. G ) ) = ( _I |` B ) )
9 5 1 tendo02 
 |-  ( F e. T -> ( O ` F ) = ( _I |` B ) )
10 9 3ad2ant2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( O ` F ) = ( _I |` B ) )
11 5 1 tendo02 
 |-  ( G e. T -> ( O ` G ) = ( _I |` B ) )
12 11 3ad2ant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( O ` G ) = ( _I |` B ) )
13 10 12 coeq12d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( O ` F ) o. ( O ` G ) ) = ( ( _I |` B ) o. ( _I |` B ) ) )
14 f1oi 
 |-  ( _I |` B ) : B -1-1-onto-> B
15 f1of 
 |-  ( ( _I |` B ) : B -1-1-onto-> B -> ( _I |` B ) : B --> B )
16 fcoi1 
 |-  ( ( _I |` B ) : B --> B -> ( ( _I |` B ) o. ( _I |` B ) ) = ( _I |` B ) )
17 14 15 16 mp2b 
 |-  ( ( _I |` B ) o. ( _I |` B ) ) = ( _I |` B )
18 13 17 eqtr2di 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( _I |` B ) = ( ( O ` F ) o. ( O ` G ) ) )
19 8 18 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( O ` ( F o. G ) ) = ( ( O ` F ) o. ( O ` G ) ) )