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 )