Theorem nmlnopne0

Description: A linear operator with a nonzero norm is nonzero. (Contributed by NM, 12-Aug-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	nmlnopne0	$$⊢T\in \mathrm{LinOp}\to \left({norm}_{\mathrm{op}}\left(T\right)\ne 0\leftrightarrow T\ne 0_{\mathrm{hop}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		nmlnop0	$$⊢T\in \mathrm{LinOp}\to \left({norm}_{\mathrm{op}}\left(T\right)=0\leftrightarrow T=0_{\mathrm{hop}}\right)$$
2	1	necon3bid	$$⊢T\in \mathrm{LinOp}\to \left({norm}_{\mathrm{op}}\left(T\right)\ne 0\leftrightarrow T\ne 0_{\mathrm{hop}}\right)$$