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Metamath Proof Explorer

Theorem ldilval

Description: Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012)

Ref Expression
Hypotheses ldilval.b 
|- B = ( Base ` K )
ldilval.l 
|- .<_ = ( le ` K )
ldilval.h 
|- H = ( LHyp ` K )
ldilval.d 
|- D = ( ( LDil ` K ) ` W )
Assertion ldilval 
|- ( ( ( K e. V /\ W e. H ) /\ F e. D /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )

Proof

Step Hyp Ref Expression
1 ldilval.b 
 |-  B = ( Base ` K )
2 ldilval.l 
 |-  .<_ = ( le ` K )
3 ldilval.h 
 |-  H = ( LHyp ` K )
4 ldilval.d 
 |-  D = ( ( LDil ` K ) ` W )
5 eqid 
 |-  ( LAut ` K ) = ( LAut ` K )
6 1 2 3 5 4 isldil 
 |-  ( ( K e. V /\ W e. H ) -> ( F e. D <-> ( F e. ( LAut ` K ) /\ A. x e. B ( x .<_ W -> ( F ` x ) = x ) ) ) )
7 simpr 
 |-  ( ( F e. ( LAut ` K ) /\ A. x e. B ( x .<_ W -> ( F ` x ) = x ) ) -> A. x e. B ( x .<_ W -> ( F ` x ) = x ) )
8 6 7 biimtrdi 
 |-  ( ( K e. V /\ W e. H ) -> ( F e. D -> A. x e. B ( x .<_ W -> ( F ` x ) = x ) ) )
9 breq1 
 |-  ( x = X -> ( x .<_ W <-> X .<_ W ) )
10 fveq2 
 |-  ( x = X -> ( F ` x ) = ( F ` X ) )
11 id 
 |-  ( x = X -> x = X )
12 10 11 eqeq12d 
 |-  ( x = X -> ( ( F ` x ) = x <-> ( F ` X ) = X ) )
13 9 12 imbi12d 
 |-  ( x = X -> ( ( x .<_ W -> ( F ` x ) = x ) <-> ( X .<_ W -> ( F ` X ) = X ) ) )
14 13 rspccv 
 |-  ( A. x e. B ( x .<_ W -> ( F ` x ) = x ) -> ( X e. B -> ( X .<_ W -> ( F ` X ) = X ) ) )
15 14 impd 
 |-  ( A. x e. B ( x .<_ W -> ( F ` x ) = x ) -> ( ( X e. B /\ X .<_ W ) -> ( F ` X ) = X ) )
16 8 15 syl6 
 |-  ( ( K e. V /\ W e. H ) -> ( F e. D -> ( ( X e. B /\ X .<_ W ) -> ( F ` X ) = X ) ) )
17 16 3imp 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. D /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )