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 )