Metamath Proof Explorer


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 : C ⟶ ( 0 [,] 1 ) ∧ ( 𝑆 ‘ ℋ ) = 1 ∧ ∀ 𝑥C𝑦C ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 𝑦 ) ) = ( ( 𝑆𝑥 ) + ( 𝑆𝑦 ) ) ) ) )
2 1 simp2bi ( 𝑆 ∈ States → ( 𝑆 ‘ ℋ ) = 1 )