Theorem stge0

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

	Ref	Expression
Assertion	stge0	$$⊢S\in \mathrm{States}\to \left(A\in {\mathbf{C}}_{\mathscr{H}}\to 0\le S\left(A\right)\right)$$

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		elicc01	$$⊢S\left(A\right)\in \left[0,1\right]\leftrightarrow \left(S\left(A\right)\in ℝ\wedge 0\le S\left(A\right)\wedge S\left(A\right)\le 1\right)$$
3	2	simp2bi	$$⊢S\left(A\right)\in \left[0,1\right]\to 0\le S\left(A\right)$$
4	1 3	syl6	$$⊢S\in \mathrm{States}\to \left(A\in {\mathbf{C}}_{\mathscr{H}}\to 0\le S\left(A\right)\right)$$