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Theorem lgsneg 22663
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 3802 . . . . . . . . 9  |-  ( A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
21adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
32oveq1d 6111 . . . . . . 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 6104 . . . . . . . . . 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 10544 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
64, 5syl6eq 2491 . . . . . . . . 9  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  1 )
7 oveq2 6104 . . . . . . . . . 10  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  (
-u 1  x.  1 ) )
8 ax-1cn 9345 . . . . . . . . . . 11  |-  1  e.  CC
98mulm1i 9794 . . . . . . . . . 10  |-  ( -u
1  x.  1 )  =  -u 1
107, 9syl6eq 2491 . . . . . . . . 9  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  -u
1 )
116, 10ifsb 3807 . . . . . . . 8  |-  ( -u
1  x.  if ( N  <  0 , 
-u 1 ,  1 ) )  =  if ( N  <  0 ,  1 ,  -u
1 )
12 simpr 461 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  A  <  0 )
1312biantrud 507 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  ( N  <  0  /\  A  <  0 ) ) )
1413ifbid 3816 . . . . . . . . 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 6112 . . . . . . . 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 992 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  e.  ZZ )
1716zred 10752 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  e.  RR )
18 0re 9391 . . . . . . . . . . . . 13  |-  0  e.  RR
19 ltlen 9481 . . . . . . . . . . . . 13  |-  ( ( N  e.  RR  /\  0  e.  RR )  ->  ( N  <  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
2017, 18, 19sylancl 662 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
21 simpl3 993 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  =/=  0 )
2221necomd 2700 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
0  =/=  N )
2322biantrud 507 . . . . . . . . . . . 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 9916 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <_  0  <->  0  <_  -u N ) )
2617renegcld 9780 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  -u N  e.  RR )
27 lenlt 9458 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  -u N  e.  RR )  ->  ( 0  <_  -u N  <->  -.  -u N  <  0 ) )
2818, 26, 27sylancr 663 . . . . . . . . . . 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 3816 . . . . . . . . 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 3839 . . . . . . . . 9  |-  if ( -.  -u N  <  0 ,  1 ,  -u
1 )  =  if ( -u N  <  0 ,  -u 1 ,  1 )
3230, 31syl6eq 2491 . . . . . . . 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 2499 . . . . . . 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 507 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( -u N  <  0  <->  (
-u N  <  0  /\  A  <  0
) ) )
3534ifbid 3816 . . . . . . 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 2479 . . . . . 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 10474 . . . . . . 7  |-  ( 1  x.  1 )  =  1
38 iffalse 3804 . . . . . . . . 9  |-  ( -.  A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  1 )
3938adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if ( A  <  0 ,  -u
1 ,  1 )  =  1 )
40 simpr 461 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  A  <  0 )
4140intnand 907 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  ( N  <  0  /\  A  <  0 ) )
42 iffalse 3804 . . . . . . . . 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 6114 . . . . . . 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 907 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  ( -u N  <  0  /\  A  <  0 ) )
46 iffalse 3804 . . . . . . . 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 2501 . . . . . 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 789 . . . . 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 2448 . . . 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 461 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  n  e.  Prime )
52 simpl2 992 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  N  e.  ZZ )
53 zq 10964 . . . . . . . . . . 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 13945 . . . . . . . . . 10  |-  ( ( n  e.  Prime  /\  N  e.  QQ )  ->  (
n  pCnt  -u N )  =  ( n  pCnt  N ) )
5651, 54, 55syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  -> 
( n  pCnt  -u N
)  =  ( n 
pCnt  N ) )
5756oveq2d 6112 . . . . . . . 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 3824 . . . . . . 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 4384 . . . . . 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 11819 . . . . 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 10656 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
62613ad2ant2 1010 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  CC )
6362absnegd 12940 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  -u N )  =  ( abs `  N
) )
6460, 63fveq12d 5702 . . . 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 6114 . . 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 10430 . . . . . 6  |-  -u 1  e.  CC
6766, 8keepel 3862 . . . . 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 3862 . . . . 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 12818 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
72713adant1 1006 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  N )  e.  NN )
73 nnuz 10901 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
7472, 73syl6eleq 2533 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  N )  e.  ( ZZ>= `  1 )
)
75 eqid 2443 . . . . . . . 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 22661 . . . . . . 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 11483 . . . . . . 7  |-  ( x  e.  ( 1 ... ( abs `  N
) )  ->  x  e.  NN )
78 ffvelrn 5846 . . . . . . 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 477 . . . . . 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 10698 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
8180adantl 466 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  x.  y )  e.  ZZ )
8274, 79, 81seqcl 11831 . . . . 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 10753 . . . 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 9414 . . 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 2475 . 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 988 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  A  e.  ZZ )
87 znegcl 10685 . . . 4  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
88873ad2ant2 1010 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u N  e.  ZZ )
89 simp3 990 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  =/=  0 )
9062, 89negne0d 9722 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u N  =/=  0 )
91 eqid 2443 . . . 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 22660 . . 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 1218 . 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 22660 . . 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 6112 . 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 2485 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   ifcif 3796   class class class wbr 4297    e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    x. cmul 9292    < clt 9423    <_ cle 9424   -ucneg 9601   NNcn 10327   ZZcz 10651   ZZ>=cuz 10866   QQcq 10958   ...cfz 11442    seqcseq 11811   ^cexp 11870   abscabs 12728   Primecprime 13768    pCnt cpc 13908    /Lclgs 22638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-dvds 13541  df-gcd 13696  df-prm 13769  df-phi 13846  df-pc 13909  df-lgs 22639
This theorem is referenced by:  lgsneg1  22664
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