Theorem pncan2s

Description: Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 16-Apr-2025)

	Ref	Expression
Assertion	pncan2s	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 +s 𝐵 ) -s 𝐴 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 𝐴 +s 𝐵 ) = ( 𝐴 +s 𝐵 )
2		addscl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 +s 𝐵 ) ∈ No )
3		simpl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
4		simpr	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
5	2 3 4	subaddsd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( ( 𝐴 +s 𝐵 ) -s 𝐴 ) = 𝐵 ↔ ( 𝐴 +s 𝐵 ) = ( 𝐴 +s 𝐵 ) ) )
6	1 5	mpbiri	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 +s 𝐵 ) -s 𝐴 ) = 𝐵 )