Theorem redivcan2d

Description: A cancellation law for division. (Contributed by SN, 25-Nov-2025)

	Ref	Expression
Hypotheses	redivcan2d.a	|- ( ph -> A e. RR )
	redivcan2d.b	|- ( ph -> B e. RR )
	redivcan2d.z	|- ( ph -> B =/= 0 )
Assertion	redivcan2d	|- ( ph -> ( B x. ( A /R B ) ) = A )

Proof

Step	Hyp	Ref	Expression
1		redivcan2d.a	|- ( ph -> A e. RR )
2		redivcan2d.b	|- ( ph -> B e. RR )
3		redivcan2d.z	|- ( ph -> B =/= 0 )
4		eqidd	|- ( ph -> ( A /R B ) = ( A /R B ) )
5	1 2 3	sn-redivcld	|- ( ph -> ( A /R B ) e. RR )
6	1 5 2 3	redivmuld	|- ( ph -> ( ( A /R B ) = ( A /R B ) <-> ( B x. ( A /R B ) ) = A ) )
7	4 6	mpbid	|- ( ph -> ( B x. ( A /R B ) ) = A )