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Metamath Proof Explorer

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 ) )