Theorem hst0h

Description: The norm of a Hilbert-space-valued state equals zero iff the state value equals zero. (Contributed by NM, 30-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	hst0h	⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 0 ↔ ( 𝑆 ‘ 𝐴 ) = 0ℎ ) )

Proof

Step	Hyp	Ref	Expression
1		hstcl	⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ( 𝑆 ‘ 𝐴 ) ∈ ℋ )
2		norm-i	⊢ ( ( 𝑆 ‘ 𝐴 ) ∈ ℋ → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 0 ↔ ( 𝑆 ‘ 𝐴 ) = 0ℎ ) )
3	1 2	syl	⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 0 ↔ ( 𝑆 ‘ 𝐴 ) = 0ℎ ) )