Theorem rediveq0d

Description: A ratio is zero iff the numerator is zero. (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	rediveq0d	$$⊢\phi \to \left(A{/}_{ℝ}B=0\leftrightarrow A=0\right)$$

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		0red	$$⊢\phi \to 0\in ℝ$$
5	1 4 2 3	redivmul2d	$$⊢\phi \to \left(A{/}_{ℝ}B=0\leftrightarrow A=B\cdot 0\right)$$
6		remul01	$$⊢B\in ℝ\to B\cdot 0=0$$
7	2 6	syl	$$⊢\phi \to B\cdot 0=0$$
8	7	eqeq2d	$$⊢\phi \to \left(A=B\cdot 0\leftrightarrow A=0\right)$$
9	5 8	bitrd	$$⊢\phi \to \left(A{/}_{ℝ}B=0\leftrightarrow A=0\right)$$