Theorem npcans

Description: Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025)

	Ref	Expression
Assertion	npcans	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 -s 𝐵 ) +s 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		subscl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 -s 𝐵 ) ∈ No )
2		simpr	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
3	1 2	addscomd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 -s 𝐵 ) +s 𝐵 ) = ( 𝐵 +s ( 𝐴 -s 𝐵 ) ) )
4		pncan3s	⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ) → ( 𝐵 +s ( 𝐴 -s 𝐵 ) ) = 𝐴 )
5	4	ancoms	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐵 +s ( 𝐴 -s 𝐵 ) ) = 𝐴 )
6	3 5	eqtrd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 -s 𝐵 ) +s 𝐵 ) = 𝐴 )