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Metamath Proof Explorer

Theorem isdilN

Description: The predicate "is a dilation". (Contributed by NM, 4-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses dilset.a 
|- A = ( Atoms ` K )
dilset.s 
|- S = ( PSubSp ` K )
dilset.w 
|- W = ( WAtoms ` K )
dilset.m 
|- M = ( PAut ` K )
dilset.l 
|- L = ( Dil ` K )
Assertion isdilN 
|- ( ( K e. B /\ D e. A ) -> ( F e. ( L ` D ) <-> ( F e. M /\ A. x e. S ( x C_ ( W ` D ) -> ( F ` x ) = x ) ) ) )

Proof

Step Hyp Ref Expression
1 dilset.a 
 |-  A = ( Atoms ` K )
2 dilset.s 
 |-  S = ( PSubSp ` K )
3 dilset.w 
 |-  W = ( WAtoms ` K )
4 dilset.m 
 |-  M = ( PAut ` K )
5 dilset.l 
 |-  L = ( Dil ` K )
6 1 2 3 4 5 dilsetN 
 |-  ( ( K e. B /\ D e. A ) -> ( L ` D ) = { f e. M | A. x e. S ( x C_ ( W ` D ) -> ( f ` x ) = x ) } )
7 6 eleq2d 
 |-  ( ( K e. B /\ D e. A ) -> ( F e. ( L ` D ) <-> F e. { f e. M | A. x e. S ( x C_ ( W ` D ) -> ( f ` x ) = x ) } ) )
8 fveq1 
 |-  ( f = F -> ( f ` x ) = ( F ` x ) )
9 8 eqeq1d 
 |-  ( f = F -> ( ( f ` x ) = x <-> ( F ` x ) = x ) )
10 9 imbi2d 
 |-  ( f = F -> ( ( x C_ ( W ` D ) -> ( f ` x ) = x ) <-> ( x C_ ( W ` D ) -> ( F ` x ) = x ) ) )
11 10 ralbidv 
 |-  ( f = F -> ( A. x e. S ( x C_ ( W ` D ) -> ( f ` x ) = x ) <-> A. x e. S ( x C_ ( W ` D ) -> ( F ` x ) = x ) ) )
12 11 elrab 
 |-  ( F e. { f e. M | A. x e. S ( x C_ ( W ` D ) -> ( f ` x ) = x ) } <-> ( F e. M /\ A. x e. S ( x C_ ( W ` D ) -> ( F ` x ) = x ) ) )
13 7 12 bitrdi 
 |-  ( ( K e. B /\ D e. A ) -> ( F e. ( L ` D ) <-> ( F e. M /\ A. x e. S ( x C_ ( W ` D ) -> ( F ` x ) = x ) ) ) )