Theorem pncans

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

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

Proof

Step	Hyp	Ref	Expression
1		addscom	Could not format ( ( A e. No /\ B e. No ) -> ( A +s B ) = ( B +s A ) ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( A +s B ) = ( B +s A ) ) with typecode |-
2	1	eqcomd	Could not format ( ( A e. No /\ B e. No ) -> ( B +s A ) = ( A +s B ) ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( B +s A ) = ( A +s B ) ) with typecode |-
3		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 |-
4		simpr	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to B\in \mathrm{No}$$
5		simpl	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to A\in \mathrm{No}$$
6	3 4 5	subaddsd	Could not format ( ( A e. No /\ B e. No ) -> ( ( ( A +s B ) -s B ) = A <-> ( B +s A ) = ( A +s B ) ) ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( ( ( A +s B ) -s B ) = A <-> ( B +s A ) = ( A +s B ) ) ) with typecode |-
7	2 6	mpbird	Could not format ( ( A e. No /\ B e. No ) -> ( ( A +s B ) -s B ) = A ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( ( A +s B ) -s B ) = A ) with typecode |-