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Metamath Proof Explorer

Theorem abs00i

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, 28-Jul-1999)

		Ref	Expression
	Hypothesis	absvalsqi.1	⊢ 𝐴 ∈ ℂ
	Assertion	abs00i	⊢ ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 )

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	⊢ 𝐴 ∈ ℂ
2		abs00	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) )
3	1 2	ax-mp	⊢ ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 )