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

Metamath Proof Explorer

Theorem dicopelval2

Description: Membership in value of the partial isomorphism C for a lattice K . (Contributed by NM, 20-Feb-2014)

Ref Expression
Hypotheses dicval.l 
|- .<_ = ( le ` K )
dicval.a 
|- A = ( Atoms ` K )
dicval.h 
|- H = ( LHyp ` K )
dicval.p 
|- P = ( ( oc ` K ) ` W )
dicval.t 
|- T = ( ( LTrn ` K ) ` W )
dicval.e 
|- E = ( ( TEndo ` K ) ` W )
dicval.i 
|- I = ( ( DIsoC ` K ) ` W )
dicval2.g 
|- G = ( iota_ g e. T ( g ` P ) = Q )
dicelval2.f 
|- F e. _V
dicelval2.s 
|- S e. _V
Assertion dicopelval2 
|- ( ( ( K e. V /\ 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 dicval.l 
 |-  .<_ = ( le ` K )
2 dicval.a 
 |-  A = ( Atoms ` K )
3 dicval.h 
 |-  H = ( LHyp ` K )
4 dicval.p 
 |-  P = ( ( oc ` K ) ` W )
5 dicval.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 dicval.e 
 |-  E = ( ( TEndo ` K ) ` W )
7 dicval.i 
 |-  I = ( ( DIsoC ` K ) ` W )
8 dicval2.g 
 |-  G = ( iota_ g e. T ( g ` P ) = Q )
9 dicelval2.f 
 |-  F e. _V
10 dicelval2.s 
 |-  S e. _V
11 1 2 3 4 5 6 7 9 10 dicopelval 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ S e. E ) ) )
12 8 fveq2i 
 |-  ( S ` G ) = ( S ` ( iota_ g e. T ( g ` P ) = Q ) )
13 12 eqeq2i 
 |-  ( F = ( S ` G ) <-> F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) )
14 13 anbi1i 
 |-  ( ( F = ( S ` G ) /\ S e. E ) <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ S e. E ) )
15 11 14 bitr4di 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> ( F = ( S ` G ) /\ S e. E ) ) )