Theorem lgscl

Description: The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015)

	Ref	Expression
Assertion	lgscl	|- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )

Proof

Step	Hyp	Ref	Expression
1		ssrab2	|- { x e. ZZ | ( abs ` x ) <_ 1 } C_ ZZ
2		eqid	|- { x e. ZZ | ( abs ` x ) <_ 1 } = { x e. ZZ | ( abs ` x ) <_ 1 }
3	2	lgscl2	|- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. { x e. ZZ | ( abs ` x ) <_ 1 } )
4	1 3	sseldi	|- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )