Theorem subsid1

Description: Identity law for subtraction. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	subsid1	|- ( A e. No -> ( A -s 0s ) = A )

Proof

Step	Hyp	Ref	Expression
1		0sno	|- 0s e. No
2		subsval	|- ( ( A e. No /\ 0s e. No ) -> ( A -s 0s ) = ( A +s ( -us ` 0s ) ) )
3	1 2	mpan2	|- ( A e. No -> ( A -s 0s ) = ( A +s ( -us ` 0s ) ) )
4		negs0s	|- ( -us ` 0s ) = 0s
5	4	oveq2i	|- ( A +s ( -us ` 0s ) ) = ( A +s 0s )
6		addsrid	|- ( A e. No -> ( A +s 0s ) = A )
7	5 6	eqtrid	|- ( A e. No -> ( A +s ( -us ` 0s ) ) = A )
8	3 7	eqtrd	|- ( A e. No -> ( A -s 0s ) = A )