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Metamath Proof Explorer

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+ )