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Metamath Proof Explorer

Theorem rerebd

Description: A real number equals its real part. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses recld.1 ⊢ φ → A ∈ ℂ
rerebd.2 ⊢ φ → ℜ ⁡ A = A
Assertion rerebd ⊢ φ → A ∈ ℝ

Proof

Step Hyp Ref Expression
1 recld.1 ⊢ φ → A ∈ ℂ
2 rerebd.2 ⊢ φ → ℜ ⁡ A = A
3 rereb ⊢ A ∈ ℂ → A ∈ ℝ ↔ ℜ ⁡ A = A
4 1 3 syl ⊢ φ → A ∈ ℝ ↔ ℜ ⁡ A = A
5 2 4 mpbird ⊢ φ → A ∈ ℝ