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

Metamath Proof Explorer

Theorem tendo0tp

Description: Trace-preserving property of endomorphism additive identity. (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 ) )
tendo0tp.l 
|- .<_ = ( le ` K )
tendo0tp.r 
|- R = ( ( trL ` K ) ` W )
Assertion tendo0tp 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` ( O ` F ) ) .<_ ( R ` F ) )

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 tendo0tp.l 
 |-  .<_ = ( le ` K )
7 tendo0tp.r 
 |-  R = ( ( trL ` K ) ` W )
8 5 1 tendo02 
 |-  ( F e. T -> ( O ` F ) = ( _I |` B ) )
9 8 adantl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( O ` F ) = ( _I |` B ) )
10 9 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` ( O ` F ) ) = ( R ` ( _I |` B ) ) )
11 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
12 1 11 2 7 trlid0 
 |-  ( ( K e. HL /\ W e. H ) -> ( R ` ( _I |` B ) ) = ( 0. ` K ) )
13 12 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` ( _I |` B ) ) = ( 0. ` K ) )
14 10 13 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` ( O ` F ) ) = ( 0. ` K ) )
15 hlop 
 |-  ( K e. HL -> K e. OP )
16 15 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> K e. OP )
17 1 2 3 7 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. B )
18 1 6 11 op0le 
 |-  ( ( K e. OP /\ ( R ` F ) e. B ) -> ( 0. ` K ) .<_ ( R ` F ) )
19 16 17 18 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( 0. ` K ) .<_ ( R ` F ) )
20 14 19 eqbrtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` ( O ` F ) ) .<_ ( R ` F ) )