Theorem imsub

Description: Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005)

	Ref	Expression
Assertion	imsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 − 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) − ( ℑ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		negcl	⊢ ( 𝐵 ∈ ℂ → - 𝐵 ∈ ℂ )
2		imadd	⊢ ( ( 𝐴 ∈ ℂ ∧ - 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 + - 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ - 𝐵 ) ) )
3	1 2	sylan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 + - 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ - 𝐵 ) ) )
4		imneg	⊢ ( 𝐵 ∈ ℂ → ( ℑ ‘ - 𝐵 ) = - ( ℑ ‘ 𝐵 ) )
5	4	adantl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ - 𝐵 ) = - ( ℑ ‘ 𝐵 ) )
6	5	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ℑ ‘ 𝐴 ) + ( ℑ ‘ - 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) + - ( ℑ ‘ 𝐵 ) ) )
7	3 6	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 + - 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) + - ( ℑ ‘ 𝐵 ) ) )
8		negsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
9	8	fveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 + - 𝐵 ) ) = ( ℑ ‘ ( 𝐴 − 𝐵 ) ) )
10		imcl	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ )
11	10	recnd	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℂ )
12		imcl	⊢ ( 𝐵 ∈ ℂ → ( ℑ ‘ 𝐵 ) ∈ ℝ )
13	12	recnd	⊢ ( 𝐵 ∈ ℂ → ( ℑ ‘ 𝐵 ) ∈ ℂ )
14		negsub	⊢ ( ( ( ℑ ‘ 𝐴 ) ∈ ℂ ∧ ( ℑ ‘ 𝐵 ) ∈ ℂ ) → ( ( ℑ ‘ 𝐴 ) + - ( ℑ ‘ 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) − ( ℑ ‘ 𝐵 ) ) )
15	11 13 14	syl2an	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ℑ ‘ 𝐴 ) + - ( ℑ ‘ 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) − ( ℑ ‘ 𝐵 ) ) )
16	7 9 15	3eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 − 𝐵 ) ) = ( ( ℑ ‘ 𝐴 ) − ( ℑ ‘ 𝐵 ) ) )