Theorem ipcnd

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

	Ref	Expression
Hypotheses	recld.1	$$⊢\phi \to A\in ℂ$$
	readdd.2	$$⊢\phi \to B\in ℂ$$
Assertion	ipcnd	$$⊢\phi \to \Re \left(A\bar{B}\right)=\Re \left(A\right)\Re \left(B\right)+\Im \left(A\right)\Im \left(B\right)$$

Proof

Step	Hyp	Ref	Expression
1		recld.1	$$⊢\phi \to A\in ℂ$$
2		readdd.2	$$⊢\phi \to B\in ℂ$$
3		ipcnval	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \Re \left(A\bar{B}\right)=\Re \left(A\right)\Re \left(B\right)+\Im \left(A\right)\Im \left(B\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \Re \left(A\bar{B}\right)=\Re \left(A\right)\Re \left(B\right)+\Im \left(A\right)\Im \left(B\right)$$