Theorem cjmulvali

Description: A complex number times its conjugate. (Contributed by NM, 2-Oct-1999)

	Ref	Expression
Hypothesis	recl.1	⊢ 𝐴 ∈ ℂ
Assertion	cjmulvali	⊢ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		recl.1	⊢ 𝐴 ∈ ℂ
2		cjmulval	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) )
3	1 2	ax-mp	⊢ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) )