Theorem rered

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

	Ref	Expression
Hypothesis	crred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion	rered	⊢ ( 𝜑 → ( ℜ ‘ 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		crred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		rere	⊢ ( 𝐴 ∈ ℝ → ( ℜ ‘ 𝐴 ) = 𝐴 )
3	1 2	syl	⊢ ( 𝜑 → ( ℜ ‘ 𝐴 ) = 𝐴 )