Theorem abs00

Description: The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of Gleason p. 133. (Contributed by NM, 26-Sep-2005) (Proof shortened by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	abs00	$$⊢A\in ℂ\to \left(\left|A\right|=0\leftrightarrow A=0\right)$$

Proof

Step	Hyp	Ref	Expression
1		absrpcl	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \left|A\right|\in {ℝ}^{+}$$
2	1	rpne0d	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \left|A\right|\ne 0$$
3	2	ex	$$⊢A\in ℂ\to \left(A\ne 0\to \left|A\right|\ne 0\right)$$
4	3	necon4d	$$⊢A\in ℂ\to \left(\left|A\right|=0\to A=0\right)$$
5		fveq2	$$⊢A=0\to \left|A\right|=\left|0\right|$$
6		abs0	$$⊢\left|0\right|=0$$
7	5 6	eqtrdi	$$⊢A=0\to \left|A\right|=0$$
8	4 7	impbid1	$$⊢A\in ℂ\to \left(\left|A\right|=0\leftrightarrow A=0\right)$$