Theorem redivmuld

Description: Relationship between division and multiplication. (Contributed by SN, 25-Nov-2025)

	Ref	Expression
Hypotheses	redivmuld.a	$$⊢\phi \to A\in ℝ$$
	redivmuld.b	$$⊢\phi \to B\in ℝ$$
	redivmuld.c	$$⊢\phi \to C\in ℝ$$
	redivmuld.z	$$⊢\phi \to C\ne 0$$
Assertion	redivmuld	$$⊢\phi \to \left(A{/}_{ℝ}C=B\leftrightarrow CB=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		redivmuld.a	$$⊢\phi \to A\in ℝ$$
2		redivmuld.b	$$⊢\phi \to B\in ℝ$$
3		redivmuld.c	$$⊢\phi \to C\in ℝ$$
4		redivmuld.z	$$⊢\phi \to C\ne 0$$
5	1 3 4	redivvald	$$⊢\phi \to A{/}_{ℝ}C=\left(\iota x\in ℝ|Cx=A\right)$$
6	5	eqeq1d	$$⊢\phi \to \left(A{/}_{ℝ}C=B\leftrightarrow \left(\iota x\in ℝ|Cx=A\right)=B\right)$$
7	1 3 4	rediveud	$$⊢\phi \to \exists !x\in ℝCx=A$$
8		oveq2	$$⊢x=B\to Cx=CB$$
9	8	eqeq1d	$$⊢x=B\to \left(Cx=A\leftrightarrow CB=A\right)$$
10	9	riota2	$$⊢\left(B\in ℝ\wedge \exists !x\in ℝCx=A\right)\to \left(CB=A\leftrightarrow \left(\iota x\in ℝ|Cx=A\right)=B\right)$$
11	2 7 10	syl2anc	$$⊢\phi \to \left(CB=A\leftrightarrow \left(\iota x\in ℝ|Cx=A\right)=B\right)$$
12	6 11	bitr4d	$$⊢\phi \to \left(A{/}_{ℝ}C=B\leftrightarrow CB=A\right)$$