Theorem abs00bd

Description: If a complex number is zero, its absolute value is zero. Converse of abs00d . One-way deduction form of abs00 . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypothesis	abs00bd.1	$$⊢\phi \to A=0$$
Assertion	abs00bd	$$⊢\phi \to \left|A\right|=0$$

Proof

Step	Hyp	Ref	Expression
1		abs00bd.1	$$⊢\phi \to A=0$$
2		0cn	$$⊢0\in ℂ$$
3	1 2	eqeltrdi	$$⊢\phi \to A\in ℂ$$
4	3	abs00ad	$$⊢\phi \to \left(\left|A\right|=0\leftrightarrow A=0\right)$$
5	1 4	mpbird	$$⊢\phi \to \left|A\right|=0$$