Theorem tendotp

Description: Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013)

	Ref	Expression
Hypotheses	tendoset.l	|- .<_ = ( le ` K )
	tendoset.h	|- H = ( LHyp ` K )
	tendoset.t	|- T = ( ( LTrn ` K ) ` W )
	tendoset.r	|- R = ( ( trL ` K ) ` W )
	tendoset.e	|- E = ( ( TEndo ` K ) ` W )
Assertion	tendotp	|- ( ( ( K e. V /\ W e. H ) /\ S e. E /\ F e. T ) -> ( R ` ( S ` F ) ) .<_ ( R ` F ) )

Proof

Step	Hyp	Ref	Expression
1		tendoset.l	|- .<_ = ( le ` K )
2		tendoset.h	|- H = ( LHyp ` K )
3		tendoset.t	|- T = ( ( LTrn ` K ) ` W )
4		tendoset.r	|- R = ( ( trL ` K ) ` W )
5		tendoset.e	|- E = ( ( TEndo ` K ) ` W )
6	1 2 3 4 5	istendo	|- ( ( K e. V /\ W e. H ) -> ( S e. E <-> ( S : T --> T /\ A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) /\ A. f e. T ( R ` ( S ` f ) ) .<_ ( R ` f ) ) ) )
7		2fveq3	|- ( f = F -> ( R ` ( S ` f ) ) = ( R ` ( S ` F ) ) )
8		fveq2	|- ( f = F -> ( R ` f ) = ( R ` F ) )
9	7 8	breq12d	|- ( f = F -> ( ( R ` ( S ` f ) ) .<_ ( R ` f ) <-> ( R ` ( S ` F ) ) .<_ ( R ` F ) ) )
10	9	rspccv	|- ( A. f e. T ( R ` ( S ` f ) ) .<_ ( R ` f ) -> ( F e. T -> ( R ` ( S ` F ) ) .<_ ( R ` F ) ) )
11	10	3ad2ant3	|- ( ( S : T --> T /\ A. f e. T A. g e. T ( S ` ( f o. g ) ) = ( ( S ` f ) o. ( S ` g ) ) /\ A. f e. T ( R ` ( S ` f ) ) .<_ ( R ` f ) ) -> ( F e. T -> ( R ` ( S ` F ) ) .<_ ( R ` F ) ) )
12	6 11	biimtrdi	|- ( ( K e. V /\ W e. H ) -> ( S e. E -> ( F e. T -> ( R ` ( S ` F ) ) .<_ ( R ` F ) ) ) )
13	12	3imp	|- ( ( ( K e. V /\ W e. H ) /\ S e. E /\ F e. T ) -> ( R ` ( S ` F ) ) .<_ ( R ` F ) )