Theorem imdivd

Description: Imaginary part of a division. Related to remul2 . (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	crred.1	|- ( ph -> A e. RR )
	remul2d.2	|- ( ph -> B e. CC )
	redivd.2	|- ( ph -> A =/= 0 )
Assertion	imdivd	|- ( ph -> ( Im ` ( B / A ) ) = ( ( Im ` B ) / A ) )

Proof

Step	Hyp	Ref	Expression
1		crred.1	|- ( ph -> A e. RR )
2		remul2d.2	|- ( ph -> B e. CC )
3		redivd.2	|- ( ph -> A =/= 0 )
4		imdiv	|- ( ( B e. CC /\ A e. RR /\ A =/= 0 ) -> ( Im ` ( B / A ) ) = ( ( Im ` B ) / A ) )
5	2 1 3 4	syl3anc	|- ( ph -> ( Im ` ( B / A ) ) = ( ( Im ` B ) / A ) )