Theorem sn-redivcld

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

	Ref	Expression
Hypotheses	redivvald.a	$$⊢\phi \to A\in ℝ$$
	redivvald.b	$$⊢\phi \to B\in ℝ$$
	redivvald.z	$$⊢\phi \to B\ne 0$$
Assertion	sn-redivcld	$$⊢\phi \to A{/}_{ℝ}B\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		redivvald.a	$$⊢\phi \to A\in ℝ$$
2		redivvald.b	$$⊢\phi \to B\in ℝ$$
3		redivvald.z	$$⊢\phi \to B\ne 0$$
4	1 2 3	redivvald	$$⊢\phi \to A{/}_{ℝ}B=\left(\iota x\in ℝ|Bx=A\right)$$
5	1 2 3	rediveud	$$⊢\phi \to \exists !x\in ℝBx=A$$
6		riotacl	$$⊢\exists !x\in ℝBx=A\to \left(\iota x\in ℝ|Bx=A\right)\in ℝ$$
7	5 6	syl	$$⊢\phi \to \left(\iota x\in ℝ|Bx=A\right)\in ℝ$$
8	4 7	eqeltrd	$$⊢\phi \to A{/}_{ℝ}B\in ℝ$$