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	|- ( ph -> A = 0 )
Assertion	abs00bd	|- ( ph -> ( abs ` A ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		abs00bd.1	|- ( ph -> A = 0 )
2		0cn	|- 0 e. CC
3	1 2	eqeltrdi	|- ( ph -> A e. CC )
4	3	abs00ad	|- ( ph -> ( ( abs ` A ) = 0 <-> A = 0 ) )
5	1 4	mpbird	|- ( ph -> ( abs ` A ) = 0 )