Metamath Proof Explorer


Theorem cjcjd

Description: The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypothesis recld.1 ( 𝜑𝐴 ∈ ℂ )
Assertion cjcjd ( 𝜑 → ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) = 𝐴 )

Proof

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