Theorem addcji

Description: A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of Gleason p. 133. (Contributed by NM, 2-Oct-1999)

	Ref	Expression
Hypothesis	recl.1	⊢ 𝐴 ∈ ℂ
Assertion	addcji	⊢ ( 𝐴 + ( ∗ ‘ 𝐴 ) ) = ( 2 · ( ℜ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		recl.1	⊢ 𝐴 ∈ ℂ
2		addcj	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + ( ∗ ‘ 𝐴 ) ) = ( 2 · ( ℜ ‘ 𝐴 ) ) )
3	1 2	ax-mp	⊢ ( 𝐴 + ( ∗ ‘ 𝐴 ) ) = ( 2 · ( ℜ ‘ 𝐴 ) )