Theorem cjsub

Description: Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005)

	Ref	Expression
Assertion	cjsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( 𝐴 − 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) − ( ∗ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		negcl	⊢ ( 𝐵 ∈ ℂ → - 𝐵 ∈ ℂ )
2		cjadd	⊢ ( ( 𝐴 ∈ ℂ ∧ - 𝐵 ∈ ℂ ) → ( ∗ ‘ ( 𝐴 + - 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) + ( ∗ ‘ - 𝐵 ) ) )
3	1 2	sylan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( 𝐴 + - 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) + ( ∗ ‘ - 𝐵 ) ) )
4		negsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
5	4	fveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( 𝐴 + - 𝐵 ) ) = ( ∗ ‘ ( 𝐴 − 𝐵 ) ) )
6		cjneg	⊢ ( 𝐵 ∈ ℂ → ( ∗ ‘ - 𝐵 ) = - ( ∗ ‘ 𝐵 ) )
7	6	adantl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ - 𝐵 ) = - ( ∗ ‘ 𝐵 ) )
8	7	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ 𝐴 ) + ( ∗ ‘ - 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) + - ( ∗ ‘ 𝐵 ) ) )
9		cjcl	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) ∈ ℂ )
10		cjcl	⊢ ( 𝐵 ∈ ℂ → ( ∗ ‘ 𝐵 ) ∈ ℂ )
11		negsub	⊢ ( ( ( ∗ ‘ 𝐴 ) ∈ ℂ ∧ ( ∗ ‘ 𝐵 ) ∈ ℂ ) → ( ( ∗ ‘ 𝐴 ) + - ( ∗ ‘ 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) − ( ∗ ‘ 𝐵 ) ) )
12	9 10 11	syl2an	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ 𝐴 ) + - ( ∗ ‘ 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) − ( ∗ ‘ 𝐵 ) ) )
13	8 12	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ∗ ‘ 𝐴 ) + ( ∗ ‘ - 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) − ( ∗ ‘ 𝐵 ) ) )
14	3 5 13	3eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∗ ‘ ( 𝐴 − 𝐵 ) ) = ( ( ∗ ‘ 𝐴 ) − ( ∗ ‘ 𝐵 ) ) )