Theorem stcl

Description: Real closure of the value of a state. (Contributed by NM, 24-Oct-1999) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	stcl	⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) ∈ ℝ ) )

Proof

Step	Hyp	Ref	Expression
1		sticl	⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) ) )
2		unitssre	⊢ ( 0 [,] 1 ) ⊆ ℝ
3	2	sseli	⊢ ( ( 𝑆 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) → ( 𝑆 ‘ 𝐴 ) ∈ ℝ )
4	1 3	syl6	⊢ ( 𝑆 ∈ States → ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) ∈ ℝ ) )