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Metamath Proof Explorer

Theorem rere

Description: A real number equals its real part. One direction of Proposition 10-3.4(f) of Gleason p. 133. (Contributed by Paul Chapman, 7-Sep-2007)

Ref Expression
Assertion rere ⊢ A ∈ ℝ → ℜ ⁡ A = A

Proof

Step Hyp Ref Expression
1 recn ⊢ A ∈ ℝ → A ∈ ℂ
2 rereb ⊢ A ∈ ℂ → A ∈ ℝ ↔ ℜ ⁡ A = A
3 1 2 syl ⊢ A ∈ ℝ → A ∈ ℝ ↔ ℜ ⁡ A = A
4 3 ibi ⊢ A ∈ ℝ → ℜ ⁡ A = A