Theorem rediveq1d

Description: Equality in terms of unit ratio. (Contributed by SN, 2-Apr-2026)

	Ref	Expression
Hypotheses	redivcan2d.a	|- ( ph -> A e. RR )
	redivcan2d.b	|- ( ph -> B e. RR )
	redivcan2d.z	|- ( ph -> B =/= 0 )
Assertion	rediveq1d	|- ( ph -> ( ( A /R B ) = 1 <-> A = B ) )

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		1red	|- ( ph -> 1 e. RR )
5	1 4 2 3	redivmul2d	|- ( ph -> ( ( A /R B ) = 1 <-> A = ( B x. 1 ) ) )
6		ax-1rid	|- ( B e. RR -> ( B x. 1 ) = B )
7	2 6	syl	|- ( ph -> ( B x. 1 ) = B )
8	7	eqeq2d	|- ( ph -> ( A = ( B x. 1 ) <-> A = B ) )
9	5 8	bitrd	|- ( ph -> ( ( A /R B ) = 1 <-> A = B ) )