Theorem redivcan2d

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	redivcan2d	⊢ ( 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 → ( 𝐴 /ℝ 𝐵 ) = ( 𝐴 /ℝ 𝐵 ) )
5	1 2 3	sn-redivcld	⊢ ( 𝜑 → ( 𝐴 /ℝ 𝐵 ) ∈ ℝ )
6	1 5 2 3	redivmuld	⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐵 ) = ( 𝐴 /ℝ 𝐵 ) ↔ ( 𝐵 · ( 𝐴 /ℝ 𝐵 ) ) = 𝐴 ) )
7	4 6	mpbid	⊢ ( 𝜑 → ( 𝐵 · ( 𝐴 /ℝ 𝐵 ) ) = 𝐴 )