Theorem cjdivd

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

	Ref	Expression
Hypotheses	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	readdd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	cjdivd.2	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	cjdivd	⊢ ( 𝜑 → ( ∗ ‘ ( 𝐴 / 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) / ( ∗ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		readdd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		cjdivd.2	⊢ ( 𝜑 → 𝐵 ≠ 0 )
4		cjdiv	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ∗ ‘ ( 𝐴 / 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) / ( ∗ ‘ 𝐵 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ∗ ‘ ( 𝐴 / 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) / ( ∗ ‘ 𝐵 ) ) )