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Metamath Proof Explorer

Theorem recjd

Description: Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypothesis recld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion recjd ( 𝜑 → ( ℜ ‘ ( ∗ ‘ 𝐴 ) ) = ( ℜ ‘ 𝐴 ) )

Proof

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