Theorem absrpcld

Description: The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	absne0d.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	absrpcld	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		absne0d.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		absrpcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( abs ‘ 𝐴 ) ∈ ℝ+ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) ∈ ℝ+ )