Theorem cjmulvald

Description: A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	cjmulvald	⊢ ( 𝜑 → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cjmulval	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) )