Theorem rediveq0d

Description: A ratio is zero iff the numerator is zero. (Contributed by SN, 25-Nov-2025)

	Ref	Expression
Hypotheses	redivcan2d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	redivcan2d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	redivcan2d.z	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	rediveq0d	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐵 ) = 0 ↔ 𝐴 = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		redivcan2d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		redivcan2d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		redivcan2d.z	⊢ ( 𝜑 → 𝐵 ≠ 0 )
4		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
5	1 4 2 3	redivmul2d	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐵 ) = 0 ↔ 𝐴 = ( 𝐵 · 0 ) ) )
6		remul01	⊢ ( 𝐵 ∈ ℝ → ( 𝐵 · 0 ) = 0 )
7	2 6	syl	⊢ ( 𝜑 → ( 𝐵 · 0 ) = 0 )
8	7	eqeq2d	⊢ ( 𝜑 → ( 𝐴 = ( 𝐵 · 0 ) ↔ 𝐴 = 0 ) )
9	5 8	bitrd	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐵 ) = 0 ↔ 𝐴 = 0 ) )