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. ) ) )