Theorem ipcni

Description: Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999)

	Ref	Expression
Hypotheses	recl.1	⊢ 𝐴 ∈ ℂ
	readdi.2	⊢ 𝐵 ∈ ℂ
Assertion	ipcni	⊢ ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recl.1	⊢ 𝐴 ∈ ℂ
2		readdi.2	⊢ 𝐵 ∈ ℂ
3		ipcnval	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) ) )
4	1 2 3	mp2an	⊢ ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) )