Metamath Proof Explorer

Theorem lgsval4lem

Description: Lemma for lgsval4 . (Contributed by Mario Carneiro, 4-Feb-2015)

Ref Expression
Hypothesis lgsval.1 
|- F = ( 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 ) )
Assertion lgsval4lem 
|- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) )

Proof

Step Hyp Ref Expression
1 lgsval.1 
 |-  F = ( 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 ) )
2 eqid 
 |-  ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) ) = ( m e. NN |-> if ( m e. Prime , ( if ( m = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( m - 1 ) / 2 ) ) + 1 ) mod m ) - 1 ) ) ^ ( m pCnt n ) ) , 1 ) )
3 2 lgsval2lem 
 |-  ( ( A e. ZZ /\ n e. Prime ) -> ( A /L n ) = 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 ) ) )
4 3 3ad2antl1 
 |-  ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> ( A /L n ) = 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 ) ) )
5 4 oveq1d 
 |-  ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> ( ( A /L n ) ^ ( n pCnt N ) ) = ( 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 ) ) )
6 5 ifeq1da 
 |-  ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) = 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 ) )
7 6 mpteq2dv 
 |-  ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( 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 ) ) )
8 1 7 eqtr4id 
 |-  ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> F = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) )