Theorem rdiv

Description: Right-division. (Contributed by BJ, 6-Jun-2019)

	Ref	Expression
Hypotheses	ldiv.a	|- ( ph -> A e. CC )
	ldiv.b	|- ( ph -> B e. CC )
	ldiv.c	|- ( ph -> C e. CC )
	rdiv.an0	|- ( ph -> A =/= 0 )
Assertion	rdiv	|- ( ph -> ( ( A x. B ) = C <-> B = ( C / A ) ) )

Proof

Step	Hyp	Ref	Expression
1		ldiv.a	|- ( ph -> A e. CC )
2		ldiv.b	|- ( ph -> B e. CC )
3		ldiv.c	|- ( ph -> C e. CC )
4		rdiv.an0	|- ( ph -> A =/= 0 )
5	1 2	mulcomd	|- ( ph -> ( A x. B ) = ( B x. A ) )
6	5	eqeq1d	|- ( ph -> ( ( A x. B ) = C <-> ( B x. A ) = C ) )
7	2 1 3 4	ldiv	|- ( ph -> ( ( B x. A ) = C <-> B = ( C / A ) ) )
8	6 7	bitrd	|- ( ph -> ( ( A x. B ) = C <-> B = ( C / A ) ) )