Theorem cjmulval

Description: A complex number times its conjugate. (Contributed by NM, 1-Feb-2007) (Revised by Mario Carneiro, 14-Jul-2014)

	Ref	Expression
Assertion	cjmulval	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recl	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ )
2	1	recnd	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℂ )
3	2	sqvald	⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) ↑ 2 ) = ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) )
4		imcl	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ )
5	4	recnd	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℂ )
6	5	sqvald	⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ 𝐴 ) ↑ 2 ) = ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) )
7	3 6	oveq12d	⊢ ( 𝐴 ∈ ℂ → ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) )
8		ipcnval	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) )
9	8	anidms	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) )
10		cjmulrcl	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ )
11		rere	⊢ ( ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )
12	10 11	syl	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )
13	7 9 12	3eqtr2rd	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) )