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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