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Metamath Proof Explorer

Theorem reim0bd

Description: A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses recld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
reim0bd.2 ⊢ ( 𝜑 → ( ℑ ‘ 𝐴 ) = 0 )
Assertion reim0bd ( 𝜑 → 𝐴 ∈ ℝ )

Proof

Step Hyp Ref Expression
1 recld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 reim0bd.2 ⊢ ( 𝜑 → ( ℑ ‘ 𝐴 ) = 0 )
3 reim0b ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ ↔ ( ℑ ‘ 𝐴 ) = 0 ) )
4 1 3 syl ⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ↔ ( ℑ ‘ 𝐴 ) = 0 ) )
5 2 4 mpbird ⊢ ( 𝜑 → 𝐴 ∈ ℝ )