Theorem npcans

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

	Ref	Expression
Assertion	npcans	|- ( ( A e. No /\ B e. No ) -> ( ( A -s B ) +s B ) = A )

Proof

Step	Hyp	Ref	Expression
1		subscl	|- ( ( A e. No /\ B e. No ) -> ( A -s B ) e. No )
2		simpr	|- ( ( A e. No /\ B e. No ) -> B e. No )
3	1 2	addscomd	|- ( ( A e. No /\ B e. No ) -> ( ( A -s B ) +s B ) = ( B +s ( A -s B ) ) )
4		pncan3s	|- ( ( B e. No /\ A e. No ) -> ( B +s ( A -s B ) ) = A )
5	4	ancoms	|- ( ( A e. No /\ B e. No ) -> ( B +s ( A -s B ) ) = A )
6	3 5	eqtrd	|- ( ( A e. No /\ B e. No ) -> ( ( A -s B ) +s B ) = A )