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

Metamath Proof Explorer

Theorem dihopelvalbN

Description: Ordered pair member of the partial isomorphism H for argument under W . (Contributed by NM, 21-Mar-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihval3.b 
|- B = ( Base ` K )
dihval3.l 
|- .<_ = ( le ` K )
dihval3.h 
|- H = ( LHyp ` K )
dihval3.t 
|- T = ( ( LTrn ` K ) ` W )
dihval3.r 
|- R = ( ( trL ` K ) ` W )
dihval3.o 
|- O = ( g e. T |-> ( _I |` B ) )
dihval3.i 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihopelvalbN 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> ( ( F e. T /\ ( R ` F ) .<_ X ) /\ S = O ) ) )

Proof

Step Hyp Ref Expression
1 dihval3.b 
 |-  B = ( Base ` K )
2 dihval3.l 
 |-  .<_ = ( le ` K )
3 dihval3.h 
 |-  H = ( LHyp ` K )
4 dihval3.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 dihval3.r 
 |-  R = ( ( trL ` K ) ` W )
6 dihval3.o 
 |-  O = ( g e. T |-> ( _I |` B ) )
7 dihval3.i 
 |-  I = ( ( DIsoH ` K ) ` W )
8 eqid 
 |-  ( ( DIsoB ` K ) ` W ) = ( ( DIsoB ` K ) ` W )
9 1 2 3 7 8 dihvalb 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( DIsoB ` K ) ` W ) ` X ) )
10 9 eleq2d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> <. F , S >. e. ( ( ( DIsoB ` K ) ` W ) ` X ) ) )
11 1 2 3 4 5 6 8 dibopelval3 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( ( ( DIsoB ` K ) ` W ) ` X ) <-> ( ( F e. T /\ ( R ` F ) .<_ X ) /\ S = O ) ) )
12 10 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 = O ) ) )