Theorem sn-redivcld

Description: Closure law for real division. (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	sn-redivcld	|- ( ph -> ( A /R B ) e. RR )

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	1 2 3	redivvald	|- ( ph -> ( A /R B ) = ( iota_ x e. RR ( B x. x ) = A ) )
5	1 2 3	rediveud	|- ( ph -> E! x e. RR ( B x. x ) = A )
6		riotacl	|- ( E! x e. RR ( B x. x ) = A -> ( iota_ x e. RR ( B x. x ) = A ) e. RR )
7	5 6	syl	|- ( ph -> ( iota_ x e. RR ( B x. x ) = A ) e. RR )
8	4 7	eqeltrd	|- ( ph -> ( A /R B ) e. RR )