Theorem sn-rediv0d

Description: Division into zero is zero. (Contributed by SN, 2-Apr-2026)

	Ref	Expression
Hypotheses	sn-rediv0d.a	$$⊢\phi \to A\in ℝ$$
	sn-rediv0d.z	$$⊢\phi \to A\ne 0$$
Assertion	sn-rediv0d	$$⊢\phi \to 0{/}_{ℝ}A=0$$

Proof

Step	Hyp	Ref	Expression
1		sn-rediv0d.a	$$⊢\phi \to A\in ℝ$$
2		sn-rediv0d.z	$$⊢\phi \to A\ne 0$$
3		eqidd	$$⊢\phi \to 0=0$$
4		0red	$$⊢\phi \to 0\in ℝ$$
5	4 1 2	rediveq0d	$$⊢\phi \to \left(0{/}_{ℝ}A=0\leftrightarrow 0=0\right)$$
6	3 5	mpbird	$$⊢\phi \to 0{/}_{ℝ}A=0$$