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Metamath Proof Explorer

Theorem dihopelvalc

Description: Member of value of isomorphism H for a lattice K when -. X .<_ W , given auxiliary atom Q . (Contributed by NM, 13-Mar-2014)

Ref Expression
Hypotheses dihopelvalcp.b 
|- B = ( Base ` K )
dihopelvalcp.l 
|- .<_ = ( le ` K )
dihopelvalcp.j 
|- .\/ = ( join ` K )
dihopelvalcp.m 
|- ./\ = ( meet ` K )
dihopelvalcp.a 
|- A = ( Atoms ` K )
dihopelvalcp.h 
|- H = ( LHyp ` K )
dihopelvalcp.p 
|- P = ( ( oc ` K ) ` W )
dihopelvalcp.t 
|- T = ( ( LTrn ` K ) ` W )
dihopelvalcp.r 
|- R = ( ( trL ` K ) ` W )
dihopelvalcp.e 
|- E = ( ( TEndo ` K ) ` W )
dihopelvalcp.i 
|- I = ( ( DIsoH ` K ) ` W )
dihopelvalcp.g 
|- G = ( iota_ g e. T ( g ` P ) = Q )
dihopelvalcp.f 
|- F e. _V
dihopelvalcp.s 
|- S e. _V
Assertion dihopelvalc 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) ) -> ( <. F , S >. e. ( I ` X ) <-> ( ( F e. T /\ S e. E ) /\ ( R ` ( F o. `' ( S ` G ) ) ) .<_ X ) ) )

Proof

Step Hyp Ref Expression
1 dihopelvalcp.b 
 |-  B = ( Base ` K )
2 dihopelvalcp.l 
 |-  .<_ = ( le ` K )
3 dihopelvalcp.j 
 |-  .\/ = ( join ` K )
4 dihopelvalcp.m 
 |-  ./\ = ( meet ` K )
5 dihopelvalcp.a 
 |-  A = ( Atoms ` K )
6 dihopelvalcp.h 
 |-  H = ( LHyp ` K )
7 dihopelvalcp.p 
 |-  P = ( ( oc ` K ) ` W )
8 dihopelvalcp.t 
 |-  T = ( ( LTrn ` K ) ` W )
9 dihopelvalcp.r 
 |-  R = ( ( trL ` K ) ` W )
10 dihopelvalcp.e 
 |-  E = ( ( TEndo ` K ) ` W )
11 dihopelvalcp.i 
 |-  I = ( ( DIsoH ` K ) ` W )
12 dihopelvalcp.g 
 |-  G = ( iota_ g e. T ( g ` P ) = Q )
13 dihopelvalcp.f 
 |-  F e. _V
14 dihopelvalcp.s 
 |-  S e. _V
15 eqid 
 |-  ( h e. T |-> ( _I |` B ) ) = ( h e. T |-> ( _I |` B ) )
16 eqid 
 |-  ( ( DIsoB ` K ) ` W ) = ( ( DIsoB ` K ) ` W )
17 eqid 
 |-  ( ( DIsoC ` K ) ` W ) = ( ( DIsoC ` K ) ` W )
18 eqid 
 |-  ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
19 eqid 
 |-  ( +g ` ( ( DVecH ` K ) ` W ) ) = ( +g ` ( ( DVecH ` K ) ` W ) )
20 eqid 
 |-  ( LSubSp ` ( ( DVecH ` K ) ` W ) ) = ( LSubSp ` ( ( DVecH ` K ) ` W ) )
21 eqid 
 |-  ( LSSum ` ( ( DVecH ` K ) ` W ) ) = ( LSSum ` ( ( DVecH ` K ) ` W ) )
22 eqid 
 |-  ( a e. E , b e. E |-> ( h e. T |-> ( ( a ` h ) o. ( b ` h ) ) ) ) = ( a e. E , b e. E |-> ( h e. T |-> ( ( a ` h ) o. ( b ` h ) ) ) )
23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 dihopelvalcpre 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( Q .\/ ( X ./\ W ) ) = X ) ) -> ( <. F , S >. e. ( I ` X ) <-> ( ( F e. T /\ S e. E ) /\ ( R ` ( F o. `' ( S ` G ) ) ) .<_ X ) ) )