Theorem divscan2wd

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	divscan2wd	⊢ ( 𝜑 → ( 𝐵 ·s ( 𝐴 /su 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		eqid	⊢ ( 𝐴 /su 𝐵 ) = ( 𝐴 /su 𝐵 )
6	1 2 3 4	divsclwd	⊢ ( 𝜑 → ( 𝐴 /su 𝐵 ) ∈ No )
7	1 6 2 3 4	divmulswd	⊢ ( 𝜑 → ( ( 𝐴 /su 𝐵 ) = ( 𝐴 /su 𝐵 ) ↔ ( 𝐵 ·s ( 𝐴 /su 𝐵 ) ) = 𝐴 ) )
8	5 7	mpbii	⊢ ( 𝜑 → ( 𝐵 ·s ( 𝐴 /su 𝐵 ) ) = 𝐴 )