Theorem nvgt0

Description: A nonzero norm is positive. (Contributed by NM, 20-Nov-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvgt0.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	nvgt0.5	$$⊢Z=0_{\mathrm{vec}}\left(U\right)$$
	nvgt0.6	$$⊢N={norm}_{CV}\left(U\right)$$
Assertion	nvgt0	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to \left(A\ne Z\leftrightarrow 0<N\left(A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nvgt0.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		nvgt0.5	$$⊢Z=0_{\mathrm{vec}}\left(U\right)$$
3		nvgt0.6	$$⊢N={norm}_{CV}\left(U\right)$$
4	1 2 3	nvz	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to \left(N\left(A\right)=0\leftrightarrow A=Z\right)$$
5	4	necon3bid	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to \left(N\left(A\right)\ne 0\leftrightarrow A\ne Z\right)$$
6	1 3	nvcl	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to N\left(A\right)\in ℝ$$
7	1 3	nvge0	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to 0\le N\left(A\right)$$
8		ne0gt0	$$⊢\left(N\left(A\right)\in ℝ\wedge 0\le N\left(A\right)\right)\to \left(N\left(A\right)\ne 0\leftrightarrow 0<N\left(A\right)\right)$$
9	6 7 8	syl2anc	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to \left(N\left(A\right)\ne 0\leftrightarrow 0<N\left(A\right)\right)$$
10	5 9	bitr3d	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to \left(A\ne Z\leftrightarrow 0<N\left(A\right)\right)$$