Theorem pncans

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

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

Proof

Step	Hyp	Ref	Expression
1		addscom	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 +s 𝐵 ) = ( 𝐵 +s 𝐴 ) )
2	1	eqcomd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐵 +s 𝐴 ) = ( 𝐴 +s 𝐵 ) )
3		addscl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 +s 𝐵 ) ∈ No )
4		simpr	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
5		simpl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
6	3 4 5	subaddsd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( ( 𝐴 +s 𝐵 ) -s 𝐵 ) = 𝐴 ↔ ( 𝐵 +s 𝐴 ) = ( 𝐴 +s 𝐵 ) ) )
7	2 6	mpbird	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 +s 𝐵 ) -s 𝐵 ) = 𝐴 )