Theorem cjdivi

Description: Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005) (Revised by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recl.1	⊢ 𝐴 ∈ ℂ
	readdi.2	⊢ 𝐵 ∈ ℂ
Assertion	cjdivi	⊢ ( 𝐵 ≠ 0 → ( ∗ ‘ ( 𝐴 / 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) / ( ∗ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recl.1	⊢ 𝐴 ∈ ℂ
2		readdi.2	⊢ 𝐵 ∈ ℂ
3		cjdiv	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ∗ ‘ ( 𝐴 / 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) / ( ∗ ‘ 𝐵 ) ) )
4	1 2 3	mp3an12	⊢ ( 𝐵 ≠ 0 → ( ∗ ‘ ( 𝐴 / 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) / ( ∗ ‘ 𝐵 ) ) )