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 ) ) )