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Theorem lgsneg 22617
Description: The Legendre symbol is either even or odd under negation with respect to the second parameter according to the sign of the first. (Contributed by Mario Carneiro, 4-Feb-2015.)
Assertion
Ref Expression
lgsneg  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( A  /L -u N
)  =  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  ( A  /L N ) ) )

Proof of Theorem lgsneg
Dummy variables  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iftrue 3794 . . . . . . . . 9  |-  ( A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
21adantl 463 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
32oveq1d 6105 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( if ( A  <  0 ,  -u
1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 ) )  =  (
-u 1  x.  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 ) ) )
4 oveq2 6098 . . . . . . . . . 10  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  ( -u 1  x.  -u 1 ) )
5 neg1mulneg1e1 10535 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
64, 5syl6eq 2489 . . . . . . . . 9  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  1 )
7 oveq2 6098 . . . . . . . . . 10  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  (
-u 1  x.  1 ) )
8 ax-1cn 9336 . . . . . . . . . . 11  |-  1  e.  CC
98mulm1i 9785 . . . . . . . . . 10  |-  ( -u
1  x.  1 )  =  -u 1
107, 9syl6eq 2489 . . . . . . . . 9  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  -u
1 )
116, 10ifsb 3799 . . . . . . . 8  |-  ( -u
1  x.  if ( N  <  0 , 
-u 1 ,  1 ) )  =  if ( N  <  0 ,  1 ,  -u
1 )
12 simpr 458 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  A  <  0 )
1312biantrud 504 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  ( N  <  0  /\  A  <  0 ) ) )
1413ifbid 3808 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( N  <  0 ,  -u 1 ,  1 )  =  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 ) )
1514oveq2d 6106 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  (
-u 1  x.  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 ) ) )
16 simpl2 987 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  e.  ZZ )
1716zred 10743 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  e.  RR )
18 0re 9382 . . . . . . . . . . . . 13  |-  0  e.  RR
19 ltlen 9472 . . . . . . . . . . . . 13  |-  ( ( N  e.  RR  /\  0  e.  RR )  ->  ( N  <  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
2017, 18, 19sylancl 657 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
21 simpl3 988 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  =/=  0 )
2221necomd 2693 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
0  =/=  N )
2322biantrud 504 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <_  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
2420, 23bitr4d 256 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  N  <_  0 ) )
2517le0neg1d 9907 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <_  0  <->  0  <_  -u N ) )
2617renegcld 9771 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  -u N  e.  RR )
27 lenlt 9449 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  -u N  e.  RR )  ->  ( 0  <_  -u N  <->  -.  -u N  <  0 ) )
2818, 26, 27sylancr 658 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( 0  <_  -u N  <->  -.  -u N  <  0
) )
2924, 25, 283bitrd 279 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  -.  -u N  <  0
) )
3029ifbid 3808 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( N  <  0 ,  1 ,  -u
1 )  =  if ( -.  -u N  <  0 ,  1 , 
-u 1 ) )
31 ifnot 3831 . . . . . . . . 9  |-  if ( -.  -u N  <  0 ,  1 ,  -u
1 )  =  if ( -u N  <  0 ,  -u 1 ,  1 )
3230, 31syl6eq 2489 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( N  <  0 ,  1 ,  -u
1 )  =  if ( -u N  <  0 ,  -u 1 ,  1 ) )
3311, 15, 323eqtr3a 2497 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( -u 1  x.  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 ) )  =  if (
-u N  <  0 ,  -u 1 ,  1 ) )
3412biantrud 504 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( -u N  <  0  <->  (
-u N  <  0  /\  A  <  0
) ) )
3534ifbid 3808 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( -u N  <  0 ,  -u 1 ,  1 )  =  if ( ( -u N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 ) )
363, 33, 353eqtrd 2477 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( if ( A  <  0 ,  -u
1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 ) )  =  if ( ( -u N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 ) )
37 1t1e1 10465 . . . . . . 7  |-  ( 1  x.  1 )  =  1
38 iffalse 3796 . . . . . . . . 9  |-  ( -.  A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  1 )
3938adantl 463 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if ( A  <  0 ,  -u
1 ,  1 )  =  1 )
40 simpr 458 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  A  <  0 )
4140intnand 902 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  ( N  <  0  /\  A  <  0 ) )
42 iffalse 3796 . . . . . . . . 9  |-  ( -.  ( N  <  0  /\  A  <  0
)  ->  if (
( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
4341, 42syl 16 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if (
( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
4439, 43oveq12d 6108 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 ) )  =  ( 1  x.  1 ) )
4540intnand 902 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  ( -u N  <  0  /\  A  <  0 ) )
46 iffalse 3796 . . . . . . . 8  |-  ( -.  ( -u N  <  0  /\  A  <  0 )  ->  if ( ( -u N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  =  1 )
4745, 46syl 16 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if (
( -u N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
4837, 44, 473eqtr4a 2499 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 ) )  =  if ( (
-u N  <  0  /\  A  <  0
) ,  -u 1 ,  1 ) )
4936, 48pm2.61dan 784 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 ) )  =  if ( (
-u N  <  0  /\  A  <  0
) ,  -u 1 ,  1 ) )
5049eqcomd 2446 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( ( -u N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  =  ( if ( A  <  0 , 
-u 1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 ) ) )
51 simpr 458 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  n  e.  Prime )
52 simpl2 987 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  N  e.  ZZ )
53 zq 10955 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  QQ )
5452, 53syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  N  e.  QQ )
55 pcneg 13936 . . . . . . . . . 10  |-  ( ( n  e.  Prime  /\  N  e.  QQ )  ->  (
n  pCnt  -u N )  =  ( n  pCnt  N ) )
5651, 54, 55syl2anc 656 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  -> 
( n  pCnt  -u N
)  =  ( n 
pCnt  N ) )
5756oveq2d 6106 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  -> 
( ( A  /L n ) ^
( n  pCnt  -u N
) )  =  ( ( A  /L
n ) ^ (
n  pCnt  N )
) )
5857ifeq1da 3816 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  -u N
) ) ,  1 )  =  if ( n  e.  Prime ,  ( ( A  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) )
5958mpteq2dv 4376 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  -u N
) ) ,  1 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) )
6059seqeq3d 11810 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  -u N
) ) ,  1 ) ) )  =  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) ) )
61 zcn 10647 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
62613ad2ant2 1005 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  CC )
6362absnegd 12931 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  -u N )  =  ( abs `  N
) )
6460, 63fveq12d 5694 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  -u N
) ) ,  1 ) ) ) `  ( abs `  -u N
) )  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) ) `
 ( abs `  N
) ) )
6550, 64oveq12d 6108 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( if ( ( -u N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  -u N
) ) ,  1 ) ) ) `  ( abs `  -u N
) ) )  =  ( ( if ( A  <  0 , 
-u 1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 ) )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) )
66 neg1cn 10421 . . . . . 6  |-  -u 1  e.  CC
6766, 8keepel 3854 . . . . 5  |-  if ( A  <  0 , 
-u 1 ,  1 )  e.  CC
6867a1i 11 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  e.  CC )
6966, 8keepel 3854 . . . . 5  |-  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  e.  CC
7069a1i 11 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  e.  CC )
71 nnabscl 12809 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
72713adant1 1001 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  N )  e.  NN )
73 nnuz 10892 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
7472, 73syl6eleq 2531 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  N )  e.  ( ZZ>= `  1 )
)
75 eqid 2441 . . . . . . . 8  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) )
7675lgsfcl3 22615 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) : NN --> ZZ )
77 elfznn 11474 . . . . . . 7  |-  ( x  e.  ( 1 ... ( abs `  N
) )  ->  x  e.  NN )
78 ffvelrn 5838 . . . . . . 7  |-  ( ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) : NN --> ZZ  /\  x  e.  NN )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) `  x )  e.  ZZ )
7976, 77, 78syl2an 474 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  x  e.  ( 1 ... ( abs `  N
) ) )  -> 
( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  x )  e.  ZZ )
80 zmulcl 10689 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
8180adantl 463 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  x.  y )  e.  ZZ )
8274, 79, 81seqcl 11822 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) )  e.  ZZ )
8382zcnd 10744 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) )  e.  CC )
8468, 70, 83mulassd 9405 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
( if ( A  <  0 ,  -u
1 ,  1 )  x.  if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 ) )  x.  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) )  =  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 )  x.  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
8565, 84eqtrd 2473 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( if ( ( -u N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  -u N
) ) ,  1 ) ) ) `  ( abs `  -u N
) ) )  =  ( if ( A  <  0 ,  -u
1 ,  1 )  x.  ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
86 simp1 983 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  A  e.  ZZ )
87 znegcl 10676 . . . 4  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
88873ad2ant2 1005 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u N  e.  ZZ )
89 simp3 985 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  =/=  0 )
9062, 89negne0d 9713 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u N  =/=  0 )
91 eqid 2441 . . . 4  |-  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L
n ) ^ (
n  pCnt  -u N ) ) ,  1 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L
n ) ^ (
n  pCnt  -u N ) ) ,  1 ) )
9291lgsval4 22614 . . 3  |-  ( ( A  e.  ZZ  /\  -u N  e.  ZZ  /\  -u N  =/=  0 )  ->  ( A  /L -u N )  =  ( if ( (
-u N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L
n ) ^ (
n  pCnt  -u N ) ) ,  1 ) ) ) `  ( abs `  -u N ) ) ) )
9386, 88, 90, 92syl3anc 1213 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( A  /L -u N
)  =  ( if ( ( -u N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  -u N
) ) ,  1 ) ) ) `  ( abs `  -u N
) ) ) )
9475lgsval4 22614 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( A  /L N )  =  ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) ) `  ( abs `  N ) ) ) )
9594oveq2d 6106 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  ( A  /L N ) )  =  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
9685, 93, 953eqtr4d 2483 1  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( A  /L -u N
)  =  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  ( A  /L N ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   ifcif 3788   class class class wbr 4289    e. cmpt 4347   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283    < clt 9414    <_ cle 9415   -ucneg 9592   NNcn 10318   ZZcz 10642   ZZ>=cuz 10857   QQcq 10949   ...cfz 11433    seqcseq 11802   ^cexp 11861   abscabs 12719   Primecprime 13759    pCnt cpc 13899    /Lclgs 22592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-prm 13760  df-phi 13837  df-pc 13900  df-lgs 22593
This theorem is referenced by:  lgsneg1  22618
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