Theorem rediv11d

Description: One-to-one relationship for division. (Contributed by SN, 9-Apr-2026)

	Ref	Expression
Hypotheses	rediv23d.a	|- ( ph -> A e. RR )
	rediv23d.b	|- ( ph -> B e. RR )
	rediv23d.c	|- ( ph -> C e. RR )
	rediv23d.z	|- ( ph -> C =/= 0 )
Assertion	rediv11d	|- ( ph -> ( ( A /R C ) = ( B /R C ) <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		rediv23d.a	|- ( ph -> A e. RR )
2		rediv23d.b	|- ( ph -> B e. RR )
3		rediv23d.c	|- ( ph -> C e. RR )
4		rediv23d.z	|- ( ph -> C =/= 0 )
5	2 3 4	sn-redivcld	|- ( ph -> ( B /R C ) e. RR )
6	1 5 3 4	redivmul2d	|- ( ph -> ( ( A /R C ) = ( B /R C ) <-> A = ( C x. ( B /R C ) ) ) )
7	2 3 4	redivcan2d	|- ( ph -> ( C x. ( B /R C ) ) = B )
8	7	eqeq2d	|- ( ph -> ( A = ( C x. ( B /R C ) ) <-> A = B ) )
9	6 8	bitrd	|- ( ph -> ( ( A /R C ) = ( B /R C ) <-> A = B ) )