Theorem cjdiv

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

	Ref	Expression
Assertion	cjdiv	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ∗ ‘ ( 𝐴 / 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) / ( ∗ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		divcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) ∈ ℂ )
2		cjcl	⊢ ( ( 𝐴 / 𝐵 ) ∈ ℂ → ( ∗ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ∗ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℂ )
4		simp2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℂ )
5		cjcl	⊢ ( 𝐵 ∈ ℂ → ( ∗ ‘ 𝐵 ) ∈ ℂ )
6	4 5	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ∗ ‘ 𝐵 ) ∈ ℂ )
7		simp3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐵 ≠ 0 )
8		cjne0	⊢ ( 𝐵 ∈ ℂ → ( 𝐵 ≠ 0 ↔ ( ∗ ‘ 𝐵 ) ≠ 0 ) )
9	4 8	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 ≠ 0 ↔ ( ∗ ‘ 𝐵 ) ≠ 0 ) )
10	7 9	mpbid	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ∗ ‘ 𝐵 ) ≠ 0 )
11	3 6 10	divcan4d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( ( ∗ ‘ ( 𝐴 / 𝐵 ) ) · ( ∗ ‘ 𝐵 ) ) / ( ∗ ‘ 𝐵 ) ) = ( ∗ ‘ ( 𝐴 / 𝐵 ) ) )
12		cjmul	⊢ ( ( ( 𝐴 / 𝐵 ) ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( ( 𝐴 / 𝐵 ) · 𝐵 ) ) = ( ( ∗ ‘ ( 𝐴 / 𝐵 ) ) · ( ∗ ‘ 𝐵 ) ) )
13	1 4 12	syl2anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ∗ ‘ ( ( 𝐴 / 𝐵 ) · 𝐵 ) ) = ( ( ∗ ‘ ( 𝐴 / 𝐵 ) ) · ( ∗ ‘ 𝐵 ) ) )
14		divcan1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) · 𝐵 ) = 𝐴 )
15	14	fveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ∗ ‘ ( ( 𝐴 / 𝐵 ) · 𝐵 ) ) = ( ∗ ‘ 𝐴 ) )
16	13 15	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( ∗ ‘ ( 𝐴 / 𝐵 ) ) · ( ∗ ‘ 𝐵 ) ) = ( ∗ ‘ 𝐴 ) )
17	16	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( ( ∗ ‘ ( 𝐴 / 𝐵 ) ) · ( ∗ ‘ 𝐵 ) ) / ( ∗ ‘ 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) / ( ∗ ‘ 𝐵 ) ) )
18	11 17	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ∗ ‘ ( 𝐴 / 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) / ( ∗ ‘ 𝐵 ) ) )