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Metamath Proof Explorer

Theorem dihopelvalcqat

Description: Ordered pair member of the partial isomorphism H for atom argument not under W . TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014)

Ref Expression
Hypotheses dihelval2.l 
|- .<_ = ( le ` K )
dihelval2.a 
|- A = ( Atoms ` K )
dihelval2.h 
|- H = ( LHyp ` K )
dihelval2.p 
|- P = ( ( oc ` K ) ` W )
dihelval2.t 
|- T = ( ( LTrn ` K ) ` W )
dihelval2.e 
|- E = ( ( TEndo ` K ) ` W )
dihelval2.i 
|- I = ( ( DIsoH ` K ) ` W )
dihelval2.g 
|- G = ( iota_ g e. T ( g ` P ) = Q )
dihelval2.f 
|- F e. _V
dihelval2.s 
|- S e. _V
Assertion dihopelvalcqat 
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> ( F = ( S ` G ) /\ S e. E ) ) )

Proof

Step Hyp Ref Expression
1 dihelval2.l 
 |-  .<_ = ( le ` K )
2 dihelval2.a 
 |-  A = ( Atoms ` K )
3 dihelval2.h 
 |-  H = ( LHyp ` K )
4 dihelval2.p 
 |-  P = ( ( oc ` K ) ` W )
5 dihelval2.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 dihelval2.e 
 |-  E = ( ( TEndo ` K ) ` W )
7 dihelval2.i 
 |-  I = ( ( DIsoH ` K ) ` W )
8 dihelval2.g 
 |-  G = ( iota_ g e. T ( g ` P ) = Q )
9 dihelval2.f 
 |-  F e. _V
10 dihelval2.s 
 |-  S e. _V
11 eqid 
 |-  ( ( DIsoC ` K ) ` W ) = ( ( DIsoC ` K ) ` W )
12 1 2 3 11 7 dihvalcqat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = ( ( ( DIsoC ` K ) ` W ) ` Q ) )
13 12 eleq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> <. F , S >. e. ( ( ( DIsoC ` K ) ` W ) ` Q ) ) )
14 1 2 3 4 5 6 11 8 9 10 dicopelval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( ( ( DIsoC ` K ) ` W ) ` Q ) <-> ( F = ( S ` G ) /\ S e. E ) ) )
15 13 14 bitrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> ( F = ( S ` G ) /\ S e. E ) ) )