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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 )