Theorem abscjd

Description: The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	abscjd	⊢ ( 𝜑 → ( abs ‘ ( ∗ ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		abscj	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ∗ ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) )
3	1 2	syl	⊢ ( 𝜑 → ( abs ‘ ( ∗ ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) )