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Metamath Proof Explorer

Theorem dibopelval3

Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014)

Ref Expression
Hypotheses dibval3.b 
|- B = ( Base ` K )
dibval3.l 
|- .<_ = ( le ` K )
dibval3.h 
|- H = ( LHyp ` K )
dibval3.t 
|- T = ( ( LTrn ` K ) ` W )
dibval3.r 
|- R = ( ( trL ` K ) ` W )
dibval3.o 
|- .0. = ( g e. T |-> ( _I |` B ) )
dibval3.i 
|- I = ( ( DIsoB ` K ) ` W )
Assertion dibopelval3 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> ( ( F e. T /\ ( R ` F ) .<_ X ) /\ S = .0. ) ) )

Proof

Step Hyp Ref Expression
1 dibval3.b 
 |-  B = ( Base ` K )
2 dibval3.l 
 |-  .<_ = ( le ` K )
3 dibval3.h 
 |-  H = ( LHyp ` K )
4 dibval3.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 dibval3.r 
 |-  R = ( ( trL ` K ) ` W )
6 dibval3.o 
 |-  .0. = ( g e. T |-> ( _I |` B ) )
7 dibval3.i 
 |-  I = ( ( DIsoB ` K ) ` W )
8 eqid 
 |-  ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
9 1 2 3 4 6 8 7 dibopelval2 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> ( F e. ( ( ( DIsoA ` K ) ` W ) ` X ) /\ S = .0. ) ) )
10 1 2 3 4 5 8 diaelval 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( F e. ( ( ( DIsoA ` K ) ` W ) ` X ) <-> ( F e. T /\ ( R ` F ) .<_ X ) ) )
11 10 anbi1d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( F e. ( ( ( DIsoA ` K ) ` W ) ` X ) /\ S = .0. ) <-> ( ( F e. T /\ ( R ` F ) .<_ X ) /\ S = .0. ) ) )
12 9 11 bitrd 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> ( ( F e. T /\ ( R ` F ) .<_ X ) /\ S = .0. ) ) )