Theorem sn-rediv0d

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

	Ref	Expression
Hypotheses	sn-rediv0d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	sn-rediv0d.z	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	sn-rediv0d	⊢ ( 𝜑 → ( 0 /ℝ 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		sn-rediv0d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		sn-rediv0d.z	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		eqidd	⊢ ( 𝜑 → 0 = 0 )
4		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
5	4 1 2	rediveq0d	⊢ ( 𝜑 → ( ( 0 /ℝ 𝐴 ) = 0 ↔ 0 = 0 ) )
6	3 5	mpbird	⊢ ( 𝜑 → ( 0 /ℝ 𝐴 ) = 0 )