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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 → 𝐴 = 0 )
	Assertion	abs00bd	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		abs00bd.1	⊢ ( 𝜑 → 𝐴 = 0 )
2		0cn	⊢ 0 ∈ ℂ
3	1 2	eqeltrdi	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
4	3	abs00ad	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) )
5	1 4	mpbird	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) = 0 )