Theorem redivcan3d

Description: A cancellation law for division. (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	redivcan3d	⊢ ( 𝜑 → ( ( 𝐵 · 𝐴 ) /ℝ 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		redivcan2d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		redivcan2d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		redivcan2d.z	⊢ ( 𝜑 → 𝐵 ≠ 0 )
4		eqidd	⊢ ( 𝜑 → ( 𝐵 · 𝐴 ) = ( 𝐵 · 𝐴 ) )
5	2 1	remulcld	⊢ ( 𝜑 → ( 𝐵 · 𝐴 ) ∈ ℝ )
6	5 1 2 3	redivmuld	⊢ ( 𝜑 → ( ( ( 𝐵 · 𝐴 ) /ℝ 𝐵 ) = 𝐴 ↔ ( 𝐵 · 𝐴 ) = ( 𝐵 · 𝐴 ) ) )
7	4 6	mpbird	⊢ ( 𝜑 → ( ( 𝐵 · 𝐴 ) /ℝ 𝐵 ) = 𝐴 )