Theorem pncan2s

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

	Ref	Expression
Assertion	pncan2s	Could not format assertion : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( ( A +s B ) -s A ) = B ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		eqid	Could not format ( A +s B ) = ( A +s B ) : No typesetting found for |- ( A +s B ) = ( A +s B ) with typecode |-
2		addscl	Could not format ( ( A e. No /\ B e. No ) -> ( A +s B ) e. No ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( A +s B ) e. No ) with typecode |-
3		simpl	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to A\in \mathrm{No}$$
4		simpr	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to B\in \mathrm{No}$$
5	2 3 4	subaddsd	Could not format ( ( A e. No /\ B e. No ) -> ( ( ( A +s B ) -s A ) = B <-> ( A +s B ) = ( A +s B ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( ( ( A +s B ) -s A ) = B <-> ( A +s B ) = ( A +s B ) ) ) with typecode |-
6	1 5	mpbiri	Could not format ( ( A e. No /\ B e. No ) -> ( ( A +s B ) -s A ) = B ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( ( A +s B ) -s A ) = B ) with typecode |-