Theorem sticl

Description: [ 0 , 1 ] closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	sticl	|- ( S e. States -> ( A e. CH -> ( S ` A ) e. ( 0 [,] 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		isst	|- ( S e. States <-> ( S : CH --> ( 0 [,] 1 ) /\ ( S ` ~H ) = 1 /\ A. x e. CH A. y e. CH ( x C_ ( _|_ ` y ) -> ( S ` ( x vH y ) ) = ( ( S ` x ) + ( S ` y ) ) ) ) )
2	1	simp1bi	|- ( S e. States -> S : CH --> ( 0 [,] 1 ) )
3		ffvelrn	|- ( ( S : CH --> ( 0 [,] 1 ) /\ A e. CH ) -> ( S ` A ) e. ( 0 [,] 1 ) )
4	3	ex	|- ( S : CH --> ( 0 [,] 1 ) -> ( A e. CH -> ( S ` A ) e. ( 0 [,] 1 ) ) )
5	2 4	syl	|- ( S e. States -> ( A e. CH -> ( S ` A ) e. ( 0 [,] 1 ) ) )