Theorem ldiv

Description: Left-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 )
	ldiv.bn0	|- ( ph -> B =/= 0 )
Assertion	ldiv	|- ( ph -> ( ( A x. B ) = C <-> A = ( C / B ) ) )

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		ldiv.bn0	|- ( ph -> B =/= 0 )
5		oveq1	|- ( ( A x. B ) = C -> ( ( A x. B ) / B ) = ( C / B ) )
6	1 2 4	divcan4d	|- ( ph -> ( ( A x. B ) / B ) = A )
7	6	eqeq1d	|- ( ph -> ( ( ( A x. B ) / B ) = ( C / B ) <-> A = ( C / B ) ) )
8	5 7	syl5ib	|- ( ph -> ( ( A x. B ) = C -> A = ( C / B ) ) )
9		oveq1	|- ( A = ( C / B ) -> ( A x. B ) = ( ( C / B ) x. B ) )
10	3 2 4	divcan1d	|- ( ph -> ( ( C / B ) x. B ) = C )
11	10	eqeq2d	|- ( ph -> ( ( A x. B ) = ( ( C / B ) x. B ) <-> ( A x. B ) = C ) )
12	9 11	syl5ib	|- ( ph -> ( A = ( C / B ) -> ( A x. B ) = C ) )
13	8 12	impbid	|- ( ph -> ( ( A x. B ) = C <-> A = ( C / B ) ) )