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 ) ) ) )