Metamath Proof Explorer


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 C norm S A = 0 S A = 0

Proof

Step Hyp Ref Expression
1 hstcl S CHStates A C S A
2 norm-i S A norm S A = 0 S A = 0
3 1 2 syl S CHStates A C norm S A = 0 S A = 0