Theorem normne0

Description: A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	normne0	⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) ≠ 0 ↔ 𝐴 ≠ 0ℎ ) )

Proof

Step	Hyp	Ref	Expression
1		norm-i	⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) = 0 ↔ 𝐴 = 0ℎ ) )
2	1	necon3bid	⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) ≠ 0 ↔ 𝐴 ≠ 0ℎ ) )