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Metamath Proof Explorer

Theorem ldilset

Description: The set of lattice dilations for a fiducial co-atom W . (Contributed by NM, 11-May-2012)

Ref Expression
Hypotheses ldilset.b 
|- B = ( Base ` K )
ldilset.l 
|- .<_ = ( le ` K )
ldilset.h 
|- H = ( LHyp ` K )
ldilset.i 
|- I = ( LAut ` K )
ldilset.d 
|- D = ( ( LDil ` K ) ` W )
Assertion ldilset 
|- ( ( K e. C /\ W e. H ) -> D = { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } )

Proof

Step Hyp Ref Expression
1 ldilset.b 
 |-  B = ( Base ` K )
2 ldilset.l 
 |-  .<_ = ( le ` K )
3 ldilset.h 
 |-  H = ( LHyp ` K )
4 ldilset.i 
 |-  I = ( LAut ` K )
5 ldilset.d 
 |-  D = ( ( LDil ` K ) ` W )
6 1 2 3 4 ldilfset 
 |-  ( K e. C -> ( LDil ` K ) = ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) )
7 6 fveq1d 
 |-  ( K e. C -> ( ( LDil ` K ) ` W ) = ( ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) ` W ) )
8 breq2 
 |-  ( w = W -> ( x .<_ w <-> x .<_ W ) )
9 8 imbi1d 
 |-  ( w = W -> ( ( x .<_ w -> ( f ` x ) = x ) <-> ( x .<_ W -> ( f ` x ) = x ) ) )
10 9 ralbidv 
 |-  ( w = W -> ( A. x e. B ( x .<_ w -> ( f ` x ) = x ) <-> A. x e. B ( x .<_ W -> ( f ` x ) = x ) ) )
11 10 rabbidv 
 |-  ( w = W -> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } = { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } )
12 eqid 
 |-  ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) = ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } )
13 4 fvexi 
 |-  I e. _V
14 13 rabex 
 |-  { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } e. _V
15 11 12 14 fvmpt 
 |-  ( W e. H -> ( ( w e. H |-> { f e. I | A. x e. B ( x .<_ w -> ( f ` x ) = x ) } ) ` W ) = { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } )
16 7 15 sylan9eq 
 |-  ( ( K e. C /\ W e. H ) -> ( ( LDil ` K ) ` W ) = { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } )
17 5 16 eqtrid 
 |-  ( ( K e. C /\ W e. H ) -> D = { f e. I | A. x e. B ( x .<_ W -> ( f ` x ) = x ) } )