Theorem divdiv3d

Description: Division into a fraction. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypotheses	divdiv3d.1	|- ( ph -> A e. CC )
	divdiv3d.2	|- ( ph -> B e. CC )
	divdiv3d.3	|- ( ph -> C e. CC )
	divdiv3d.4	|- ( ph -> B =/= 0 )
	divdiv3d.5	|- ( ph -> C =/= 0 )
Assertion	divdiv3d	|- ( ph -> ( ( A / B ) / C ) = ( A / ( C x. B ) ) )

Proof

Step	Hyp	Ref	Expression
1		divdiv3d.1	|- ( ph -> A e. CC )
2		divdiv3d.2	|- ( ph -> B e. CC )
3		divdiv3d.3	|- ( ph -> C e. CC )
4		divdiv3d.4	|- ( ph -> B =/= 0 )
5		divdiv3d.5	|- ( ph -> C =/= 0 )
6	1 2 3 4 5	divdiv1d	|- ( ph -> ( ( A / B ) / C ) = ( A / ( B x. C ) ) )
7	2 3	mulcomd	|- ( ph -> ( B x. C ) = ( C x. B ) )
8	7	oveq2d	|- ( ph -> ( A / ( B x. C ) ) = ( A / ( C x. B ) ) )
9	6 8	eqtrd	|- ( ph -> ( ( A / B ) / C ) = ( A / ( C x. B ) ) )