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	$$⊢A\in \mathscr{H}\to \left({norm}_{ℎ}\left(A\right)\ne 0\leftrightarrow A\ne 0_{ℎ}\right)$$

Proof

Step	Hyp	Ref	Expression
1		norm-i	$$⊢A\in \mathscr{H}\to \left({norm}_{ℎ}\left(A\right)=0\leftrightarrow A=0_{ℎ}\right)$$
2	1	necon3bid	$$⊢A\in \mathscr{H}\to \left({norm}_{ℎ}\left(A\right)\ne 0\leftrightarrow A\ne 0_{ℎ}\right)$$