Theorem redivcan2d

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

	Ref	Expression
Hypotheses	redivcan2d.a	$$⊢\phi \to A\in ℝ$$
	redivcan2d.b	$$⊢\phi \to B\in ℝ$$
	redivcan2d.z	$$⊢\phi \to B\ne 0$$
Assertion	redivcan2d	$$⊢\phi \to B\left(A{/}_{ℝ}B\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		redivcan2d.a	$$⊢\phi \to A\in ℝ$$
2		redivcan2d.b	$$⊢\phi \to B\in ℝ$$
3		redivcan2d.z	$$⊢\phi \to B\ne 0$$
4		eqidd	$$⊢\phi \to A{/}_{ℝ}B=A{/}_{ℝ}B$$
5	1 2 3	sn-redivcld	$$⊢\phi \to A{/}_{ℝ}B\in ℝ$$
6	1 5 2 3	redivmuld	$$⊢\phi \to \left(A{/}_{ℝ}B=A{/}_{ℝ}B\leftrightarrow B\left(A{/}_{ℝ}B\right)=A\right)$$
7	4 6	mpbid	$$⊢\phi \to B\left(A{/}_{ℝ}B\right)=A$$