Theorem pncans

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

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

Proof

Step	Hyp	Ref	Expression
1		addscom	|- ( ( A e. No /\ B e. No ) -> ( A +s B ) = ( B +s A ) )
2	1	eqcomd	|- ( ( A e. No /\ B e. No ) -> ( B +s A ) = ( A +s B ) )
3		addscl	|- ( ( A e. No /\ B e. No ) -> ( A +s B ) e. No )
4		simpr	|- ( ( A e. No /\ B e. No ) -> B e. No )
5		simpl	|- ( ( A e. No /\ B e. No ) -> A e. No )
6	3 4 5	subaddsd	|- ( ( A e. No /\ B e. No ) -> ( ( ( A +s B ) -s B ) = A <-> ( B +s A ) = ( A +s B ) ) )
7	2 6	mpbird	|- ( ( A e. No /\ B e. No ) -> ( ( A +s B ) -s B ) = A )