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 ∈ States → ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		isst	⊢ ( 𝑆 ∈ States ↔ ( 𝑆 : Cℋ ⟶ ( 0 [,] 1 ) ∧ ( 𝑆 ‘ ℋ ) = 1 ∧ ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ) ) )
2	1	simp1bi	⊢ ( 𝑆 ∈ States → 𝑆 : Cℋ ⟶ ( 0 [,] 1 ) )
3		ffvelcdm	⊢ ( ( 𝑆 : Cℋ ⟶ ( 0 [,] 1 ) ∧ 𝐴 ∈ Cℋ ) → ( 𝑆 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) )
4	3	ex	⊢ ( 𝑆 : Cℋ ⟶ ( 0 [,] 1 ) → ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) ) )
5	2 4	syl	⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) ) )