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Metamath Proof Explorer

Theorem addcjd

Description: A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

		Ref	Expression
	Hypothesis	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	Assertion	addcjd	⊢ ( 𝜑 → ( 𝐴 + ( ∗ ‘ 𝐴 ) ) = ( 2 · ( ℜ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		addcj	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + ( ∗ ‘ 𝐴 ) ) = ( 2 · ( ℜ ‘ 𝐴 ) ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 + ( ∗ ‘ 𝐴 ) ) = ( 2 · ( ℜ ‘ 𝐴 ) ) )