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 ∈ States → ( 𝑆 ‘ ℋ ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		isst	⊢ ( 𝑆 ∈ States ↔ ( 𝑆 : Cℋ ⟶ ( 0 [,] 1 ) ∧ ( 𝑆 ‘ ℋ ) = 1 ∧ ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ) ) )
2	1	simp2bi	⊢ ( 𝑆 ∈ States → ( 𝑆 ‘ ℋ ) = 1 )