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	$$⊢S\in \mathrm{States}\to \left(A\in {\mathbf{C}}_{\mathscr{H}}\to S\left(A\right)\in ℝ\right)$$

Proof

Step	Hyp	Ref	Expression
1		sticl	$$⊢S\in \mathrm{States}\to \left(A\in {\mathbf{C}}_{\mathscr{H}}\to S\left(A\right)\in \left[0,1\right]\right)$$
2		unitssre	$$⊢\left[0,1\right]\subseteq ℝ$$
3	2	sseli	$$⊢S\left(A\right)\in \left[0,1\right]\to S\left(A\right)\in ℝ$$
4	1 3	syl6	$$⊢S\in \mathrm{States}\to \left(A\in {\mathbf{C}}_{\mathscr{H}}\to S\left(A\right)\in ℝ\right)$$