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 ) ) ) |
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 | ffvelrn | ⊢ ( ( 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 ) ) ) |