Metamath Proof Explorer


Theorem lgscl2

Description: The Legendre symbol is an integer with absolute value less than or equal to 1. (Contributed by Mario Carneiro, 4-Feb-2015)

Ref Expression
Hypothesis lgscl2.z
|- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
Assertion lgscl2
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. Z )

Proof

Step Hyp Ref Expression
1 lgscl2.z
 |-  Z = { x e. ZZ | ( abs ` x ) <_ 1 }
2 eqid
 |-  ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) ) = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )
3 2 1 lgscllem
 |-  ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. Z )