Theorem redivmuld

Description: Relationship between division and multiplication. (Contributed by SN, 25-Nov-2025)

	Ref	Expression
Hypotheses	redivmuld.a	|- ( ph -> A e. RR )
	redivmuld.b	|- ( ph -> B e. RR )
	redivmuld.c	|- ( ph -> C e. RR )
	redivmuld.z	|- ( ph -> C =/= 0 )
Assertion	redivmuld	|- ( ph -> ( ( A /R C ) = B <-> ( C x. B ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		redivmuld.a	|- ( ph -> A e. RR )
2		redivmuld.b	|- ( ph -> B e. RR )
3		redivmuld.c	|- ( ph -> C e. RR )
4		redivmuld.z	|- ( ph -> C =/= 0 )
5	1 3 4	redivvald	|- ( ph -> ( A /R C ) = ( iota_ x e. RR ( C x. x ) = A ) )
6	5	eqeq1d	|- ( ph -> ( ( A /R C ) = B <-> ( iota_ x e. RR ( C x. x ) = A ) = B ) )
7	1 3 4	rediveud	|- ( ph -> E! x e. RR ( C x. x ) = A )
8		oveq2	|- ( x = B -> ( C x. x ) = ( C x. B ) )
9	8	eqeq1d	|- ( x = B -> ( ( C x. x ) = A <-> ( C x. B ) = A ) )
10	9	riota2	|- ( ( B e. RR /\ E! x e. RR ( C x. x ) = A ) -> ( ( C x. B ) = A <-> ( iota_ x e. RR ( C x. x ) = A ) = B ) )
11	2 7 10	syl2anc	|- ( ph -> ( ( C x. B ) = A <-> ( iota_ x e. RR ( C x. x ) = A ) = B ) )
12	6 11	bitr4d	|- ( ph -> ( ( A /R C ) = B <-> ( C x. B ) = A ) )