Theorem divscan2d

Description: A cancellation law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Hypotheses	divscan2d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	divscan2d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	divscan2d.3	⊢ ( 𝜑 → 𝐵 ≠ 0s )
Assertion	divscan2d	⊢ ( 𝜑 → ( 𝐵 ·s ( 𝐴 /su 𝐵 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		divscan2d.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		divscan2d.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		divscan2d.3	⊢ ( 𝜑 → 𝐵 ≠ 0s )
4	2 3	recsexd	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s )
5	1 2 3 4	divscan2wd	⊢ ( 𝜑 → ( 𝐵 ·s ( 𝐴 /su 𝐵 ) ) = 𝐴 )