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	$$⊢S\in \mathrm{States}\to \left(A\in {\mathbf{C}}_{\mathscr{H}}\to S\left(A\right)\in \left[0,1\right]\right)$$

Proof

Step	Hyp	Ref	Expression
1		isst	$$⊢S\in \mathrm{States}\leftrightarrow \left(S:{\mathbf{C}}_{\mathscr{H}}⟶\left[0,1\right]\wedge S\left(\mathscr{H}\right)=1\wedge \forall x\in {\mathbf{C}}_{\mathscr{H}}\forall y\in {\mathbf{C}}_{\mathscr{H}}\left(x\subseteq \perp \left(y\right)\to S\left(x{\vee }_{\mathscr{H}}y\right)=S\left(x\right)+S\left(y\right)\right)\right)$$
2	1	simp1bi	$$⊢S\in \mathrm{States}\to S:{\mathbf{C}}_{\mathscr{H}}⟶\left[0,1\right]$$
3		ffvelcdm	$$⊢\left(S:{\mathbf{C}}_{\mathscr{H}}⟶\left[0,1\right]\wedge A\in {\mathbf{C}}_{\mathscr{H}}\right)\to S\left(A\right)\in \left[0,1\right]$$
4	3	ex	$$⊢S:{\mathbf{C}}_{\mathscr{H}}⟶\left[0,1\right]\to \left(A\in {\mathbf{C}}_{\mathscr{H}}\to S\left(A\right)\in \left[0,1\right]\right)$$
5	2 4	syl	$$⊢S\in \mathrm{States}\to \left(A\in {\mathbf{C}}_{\mathscr{H}}\to S\left(A\right)\in \left[0,1\right]\right)$$