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Metamath Proof Explorer

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	$⊢ φ \to A = 0$
	Assertion	abs00bd	$⊢ φ \to \|A\| = 0$

Proof

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