Theorem znadd

Description: The additive structure of Z/nZ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015) (Revised by AV, 13-Jun-2019) (Revised by AV, 3-Nov-2024)

	Ref	Expression
Hypotheses	znval2.s	⊢ 𝑆 = ( RSpan ‘ ℤring )
	znval2.u	⊢ 𝑈 = ( ℤring /s ( ℤring ~QG ( 𝑆 ‘ { 𝑁 } ) ) )
	znval2.y	⊢ 𝑌 = ( ℤ/nℤ ‘ 𝑁 )
Assertion	znadd	⊢ ( 𝑁 ∈ ℕ0 → ( +g ‘ 𝑈 ) = ( +g ‘ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		znval2.s	⊢ 𝑆 = ( RSpan ‘ ℤring )
2		znval2.u	⊢ 𝑈 = ( ℤring /s ( ℤring ~QG ( 𝑆 ‘ { 𝑁 } ) ) )
3		znval2.y	⊢ 𝑌 = ( ℤ/nℤ ‘ 𝑁 )
4		plusgid	⊢ +g = Slot ( +g ‘ ndx )
5		plendxnplusgndx	⊢ ( le ‘ ndx ) ≠ ( +g ‘ ndx )
6	5	necomi	⊢ ( +g ‘ ndx ) ≠ ( le ‘ ndx )
7	1 2 3 4 6	znbaslem	⊢ ( 𝑁 ∈ ℕ0 → ( +g ‘ 𝑈 ) = ( +g ‘ 𝑌 ) )