Theorem nmgt0

Description: The norm of a nonzero element is a positive real. (Contributed by NM, 20-Nov-2007) (Revised by AV, 8-Oct-2021)

	Ref	Expression
Hypotheses	nmgt0.x	$$⊢X={\mathrm{Base}}_G$$
	nmgt0.n	$$⊢N=\mathrm{norm}\left(G\right)$$
	nmgt0.z	$$⊢\underset{\dot{}}{0}=0_G$$
Assertion	nmgt0	$$⊢\left(G\in \mathrm{NrmGrp}\wedge A\in X\right)\to \left(A\ne \underset{\dot{}}{0}\leftrightarrow 0<N\left(A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nmgt0.x	$$⊢X={\mathrm{Base}}_G$$
2		nmgt0.n	$$⊢N=\mathrm{norm}\left(G\right)$$
3		nmgt0.z	$$⊢\underset{\dot{}}{0}=0_G$$
4	1 2 3	nmeq0	$$⊢\left(G\in \mathrm{NrmGrp}\wedge A\in X\right)\to \left(N\left(A\right)=0\leftrightarrow A=\underset{\dot{}}{0}\right)$$
5	4	necon3bid	$$⊢\left(G\in \mathrm{NrmGrp}\wedge A\in X\right)\to \left(N\left(A\right)\ne 0\leftrightarrow A\ne \underset{\dot{}}{0}\right)$$
6	1 2	nmcl	$$⊢\left(G\in \mathrm{NrmGrp}\wedge A\in X\right)\to N\left(A\right)\in ℝ$$
7	1 2	nmge0	$$⊢\left(G\in \mathrm{NrmGrp}\wedge A\in X\right)\to 0\le N\left(A\right)$$
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	$$⊢\left(G\in \mathrm{NrmGrp}\wedge A\in X\right)\to \left(N\left(A\right)\ne 0\leftrightarrow 0<N\left(A\right)\right)$$
10	5 9	bitr3d	$$⊢\left(G\in \mathrm{NrmGrp}\wedge A\in X\right)\to \left(A\ne \underset{\dot{}}{0}\leftrightarrow 0<N\left(A\right)\right)$$