Metamath Proof Explorer


Theorem hst1a

Description: Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion hst1a ( 𝑆 ∈ CHStates → ( norm ‘ ( 𝑆 ‘ ℋ ) ) = 1 )

Proof

Step Hyp Ref Expression
1 ishst ( 𝑆 ∈ CHStates ↔ ( 𝑆 : C ⟶ ℋ ∧ ( norm ‘ ( 𝑆 ‘ ℋ ) ) = 1 ∧ ∀ 𝑥C𝑦C ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( ( ( 𝑆𝑥 ) ·ih ( 𝑆𝑦 ) ) = 0 ∧ ( 𝑆 ‘ ( 𝑥 𝑦 ) ) = ( ( 𝑆𝑥 ) + ( 𝑆𝑦 ) ) ) ) ) )
2 1 simp2bi ( 𝑆 ∈ CHStates → ( norm ‘ ( 𝑆 ‘ ℋ ) ) = 1 )