Theorem redivvald

Description: Value of real division, which is the (unique) real x such that ( B x. x ) = A . (Contributed by SN, 25-Nov-2025)

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

Proof

Step	Hyp	Ref	Expression
1		redivvald.a	|- ( ph -> A e. RR )
2		redivvald.b	|- ( ph -> B e. RR )
3		redivvald.z	|- ( ph -> B =/= 0 )
4	2 3	eldifsnd	|- ( ph -> B e. ( RR \ { 0 } ) )
5		eqeq2	|- ( z = A -> ( ( y x. x ) = z <-> ( y x. x ) = A ) )
6	5	riotabidv	|- ( z = A -> ( iota_ x e. RR ( y x. x ) = z ) = ( iota_ x e. RR ( y x. x ) = A ) )
7		oveq1	|- ( y = B -> ( y x. x ) = ( B x. x ) )
8	7	eqeq1d	|- ( y = B -> ( ( y x. x ) = A <-> ( B x. x ) = A ) )
9	8	riotabidv	|- ( y = B -> ( iota_ x e. RR ( y x. x ) = A ) = ( iota_ x e. RR ( B x. x ) = A ) )
10		df-rediv	|- /R = ( z e. RR , y e. ( RR \ { 0 } ) |-> ( iota_ x e. RR ( y x. x ) = z ) )
11		riotaex	|- ( iota_ x e. RR ( B x. x ) = A ) e. _V
12	6 9 10 11	ovmpo	|- ( ( A e. RR /\ B e. ( RR \ { 0 } ) ) -> ( A /R B ) = ( iota_ x e. RR ( B x. x ) = A ) )
13	1 4 12	syl2anc	|- ( ph -> ( A /R B ) = ( iota_ x e. RR ( B x. x ) = A ) )