Theorem ipcnd

Description: Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	readdd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	ipcnd	⊢ ( 𝜑 → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		readdd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		ipcnval	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) )