Theorem divscan1wd

Description: A weak cancellation law for surreal division. (Contributed by Scott Fenton, 13-Mar-2025)

	Ref	Expression
Hypotheses	divscan2wd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	divscan2wd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	divscan2wd.3	⊢ ( 𝜑 → 𝐵 ≠ 0s )
	divscan2wd.4	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s )
Assertion	divscan1wd	⊢ ( 𝜑 → ( ( 𝐴 /su 𝐵 ) ·s 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		divscan2wd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		divscan2wd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		divscan2wd.3	⊢ ( 𝜑 → 𝐵 ≠ 0s )
4		divscan2wd.4	⊢ ( 𝜑 → ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s )
5	1 2 3 4	divsclwd	⊢ ( 𝜑 → ( 𝐴 /su 𝐵 ) ∈ No )
6	5 2	mulscomd	⊢ ( 𝜑 → ( ( 𝐴 /su 𝐵 ) ·s 𝐵 ) = ( 𝐵 ·s ( 𝐴 /su 𝐵 ) ) )
7	1 2 3 4	divscan2wd	⊢ ( 𝜑 → ( 𝐵 ·s ( 𝐴 /su 𝐵 ) ) = 𝐴 )
8	6 7	eqtrd	⊢ ( 𝜑 → ( ( 𝐴 /su 𝐵 ) ·s 𝐵 ) = 𝐴 )