Theorem subsid1

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

	Ref	Expression
Assertion	subsid1	⊢ ( 𝐴 ∈ No → ( 𝐴 -s 0s ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		0sno	⊢ 0s ∈ No
2		subsval	⊢ ( ( 𝐴 ∈ No ∧ 0s ∈ No ) → ( 𝐴 -s 0s ) = ( 𝐴 +s ( -us ‘ 0s ) ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ 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 ∈ No → ( 𝐴 +s 0s ) = 𝐴 )
7	5 6	eqtrid	⊢ ( 𝐴 ∈ No → ( 𝐴 +s ( -us ‘ 0s ) ) = 𝐴 )
8	3 7	eqtrd	⊢ ( 𝐴 ∈ No → ( 𝐴 -s 0s ) = 𝐴 )