Theorem redivcan3d

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	redivcan3d	|- ( 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 -> ( B x. A ) = ( B x. A ) )
5	2 1	remulcld	|- ( ph -> ( B x. A ) e. RR )
6	5 1 2 3	redivmuld	|- ( ph -> ( ( ( B x. A ) /R B ) = A <-> ( B x. A ) = ( B x. A ) ) )
7	4 6	mpbird	|- ( ph -> ( ( B x. A ) /R B ) = A )