Theorem cjre

Description: A real number equals its complex conjugate. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by NM, 8-Oct-1999)

	Ref	Expression
Assertion	cjre	⊢ ( 𝐴 ∈ ℝ → ( ∗ ‘ 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2		cjreb	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ ↔ ( ∗ ‘ 𝐴 ) = 𝐴 ) )
3	2	biimpd	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ → ( ∗ ‘ 𝐴 ) = 𝐴 ) )
4	1 3	mpcom	⊢ ( 𝐴 ∈ ℝ → ( ∗ ‘ 𝐴 ) = 𝐴 )