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Metamath Proof Explorer

Theorem diatrl

Description: Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014)

Ref Expression
Hypotheses diatrl.b 
|- B = ( Base ` K )
diatrl.l 
|- .<_ = ( le ` K )
diatrl.h 
|- H = ( LHyp ` K )
diatrl.t 
|- T = ( ( LTrn ` K ) ` W )
diatrl.r 
|- R = ( ( trL ` K ) ` W )
diatrl.i 
|- I = ( ( DIsoA ` K ) ` W )
Assertion diatrl 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ F e. ( I ` X ) ) -> ( R ` F ) .<_ X )

Proof

Step Hyp Ref Expression
1 diatrl.b 
 |-  B = ( Base ` K )
2 diatrl.l 
 |-  .<_ = ( le ` K )
3 diatrl.h 
 |-  H = ( LHyp ` K )
4 diatrl.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 diatrl.r 
 |-  R = ( ( trL ` K ) ` W )
6 diatrl.i 
 |-  I = ( ( DIsoA ` K ) ` W )
7 1 2 3 4 5 6 diaelval 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( F e. ( I ` X ) <-> ( F e. T /\ ( R ` F ) .<_ X ) ) )
8 simpr 
 |-  ( ( F e. T /\ ( R ` F ) .<_ X ) -> ( R ` F ) .<_ X )
9 7 8 biimtrdi 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( F e. ( I ` X ) -> ( R ` F ) .<_ X ) )
10 9 3impia 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ F e. ( I ` X ) ) -> ( R ` F ) .<_ X )