Metamath Proof Explorer


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 → ( 𝐴C → ( 𝑆𝐴 ) ∈ ( 0 [,] 1 ) ) )

Proof

Step Hyp Ref Expression
1 isst ( 𝑆 ∈ States ↔ ( 𝑆 : C ⟶ ( 0 [,] 1 ) ∧ ( 𝑆 ‘ ℋ ) = 1 ∧ ∀ 𝑥C𝑦C ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 𝑦 ) ) = ( ( 𝑆𝑥 ) + ( 𝑆𝑦 ) ) ) ) )
2 1 simp1bi ( 𝑆 ∈ States → 𝑆 : C ⟶ ( 0 [,] 1 ) )
3 ffvelrn ( ( 𝑆 : C ⟶ ( 0 [,] 1 ) ∧ 𝐴C ) → ( 𝑆𝐴 ) ∈ ( 0 [,] 1 ) )
4 3 ex ( 𝑆 : C ⟶ ( 0 [,] 1 ) → ( 𝐴C → ( 𝑆𝐴 ) ∈ ( 0 [,] 1 ) ) )
5 2 4 syl ( 𝑆 ∈ States → ( 𝐴C → ( 𝑆𝐴 ) ∈ ( 0 [,] 1 ) ) )