Theorem sthil

Description: The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	sthil	$$⊢S\in \mathrm{States}\to S\left(\mathscr{H}\right)=1$$

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	simp2bi	$$⊢S\in \mathrm{States}\to S\left(\mathscr{H}\right)=1$$