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Theorem lgsneg 20574
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 3584 . . . . . . . . 9  |-  ( A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
21adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
32oveq1d 5889 . . . . . . 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 5882 . . . . . . . . . 10  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  ( -u 1  x.  -u 1 ) )
5 neg1cn 9829 . . . . . . . . . . . 12  |-  -u 1  e.  CC
65mulm1i 9240 . . . . . . . . . . 11  |-  ( -u
1  x.  -u 1
)  =  -u -u 1
7 ax-1cn 8811 . . . . . . . . . . . 12  |-  1  e.  CC
87negnegi 9132 . . . . . . . . . . 11  |-  -u -u 1  =  1
96, 8eqtri 2316 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
104, 9syl6eq 2344 . . . . . . . . 9  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  1 )
11 oveq2 5882 . . . . . . . . . 10  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  (
-u 1  x.  1 ) )
127mulm1i 9240 . . . . . . . . . 10  |-  ( -u
1  x.  1 )  =  -u 1
1311, 12syl6eq 2344 . . . . . . . . 9  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  -u
1 )
1410, 13ifsb 3587 . . . . . . . 8  |-  ( -u
1  x.  if ( N  <  0 , 
-u 1 ,  1 ) )  =  if ( N  <  0 ,  1 ,  -u
1 )
15 simpr 447 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  A  <  0 )
1615biantrud 493 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  ( N  <  0  /\  A  <  0 ) ) )
1716ifbid 3596 . . . . . . . . 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 ) )
1817oveq2d 5890 . . . . . . . 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 ) ) )
19 simpl2 959 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  e.  ZZ )
2019zred 10133 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  e.  RR )
21 0re 8854 . . . . . . . . . . . . 13  |-  0  e.  RR
22 ltlen 8938 . . . . . . . . . . . . 13  |-  ( ( N  e.  RR  /\  0  e.  RR )  ->  ( N  <  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
2320, 21, 22sylancl 643 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
24 simpl3 960 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  =/=  0 )
2524necomd 2542 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
0  =/=  N )
2625biantrud 493 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <_  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
2723, 26bitr4d 247 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  N  <_  0 ) )
2820le0neg1d 9360 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <_  0  <->  0  <_  -u N ) )
2920renegcld 9226 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  -u N  e.  RR )
30 lenlt 8917 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  -u N  e.  RR )  ->  ( 0  <_  -u N  <->  -.  -u N  <  0 ) )
3121, 29, 30sylancr 644 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( 0  <_  -u N  <->  -.  -u N  <  0
) )
3227, 28, 313bitrd 270 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  -.  -u N  <  0
) )
3332ifbid 3596 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( N  <  0 ,  1 ,  -u
1 )  =  if ( -.  -u N  <  0 ,  1 , 
-u 1 ) )
34 ifnot 3616 . . . . . . . . 9  |-  if ( -.  -u N  <  0 ,  1 ,  -u
1 )  =  if ( -u N  <  0 ,  -u 1 ,  1 )
3533, 34syl6eq 2344 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( N  <  0 ,  1 ,  -u
1 )  =  if ( -u N  <  0 ,  -u 1 ,  1 ) )
3614, 18, 353eqtr3a 2352 . . . . . . 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 ) )
3715biantrud 493 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( -u N  <  0  <->  (
-u N  <  0  /\  A  <  0
) ) )
3837ifbid 3596 . . . . . . 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 ) )
393, 36, 383eqtrd 2332 . . . . . 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 ) )
40 1t1e1 9886 . . . . . . 7  |-  ( 1  x.  1 )  =  1
41 iffalse 3585 . . . . . . . . 9  |-  ( -.  A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  1 )
4241adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if ( A  <  0 ,  -u
1 ,  1 )  =  1 )
43 simpr 447 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  A  <  0 )
4443intnand 882 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  ( N  <  0  /\  A  <  0 ) )
45 iffalse 3585 . . . . . . . . 9  |-  ( -.  ( N  <  0  /\  A  <  0
)  ->  if (
( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
4644, 45syl 15 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if (
( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
4742, 46oveq12d 5892 . . . . . . 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 ) )
4843intnand 882 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  ( -u N  <  0  /\  A  <  0 ) )
49 iffalse 3585 . . . . . . . 8  |-  ( -.  ( -u N  <  0  /\  A  <  0 )  ->  if ( ( -u N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  =  1 )
5048, 49syl 15 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if (
( -u N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
5140, 47, 503eqtr4a 2354 . . . . . 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 ) )
5239, 51pm2.61dan 766 . . . . 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 ) )
5352eqcomd 2301 . . . 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 ) ) )
54 simpr 447 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  n  e.  Prime )
55 simpl2 959 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  N  e.  ZZ )
56 zq 10338 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  QQ )
5755, 56syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  N  e.  QQ )
58 pcneg 12942 . . . . . . . . . 10  |-  ( ( n  e.  Prime  /\  N  e.  QQ )  ->  (
n  pCnt  -u N )  =  ( n  pCnt  N ) )
5954, 57, 58syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  -> 
( n  pCnt  -u N
)  =  ( n 
pCnt  N ) )
6059oveq2d 5890 . . . . . . . 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 )
) )
6160ifeq1da 3603 . . . . . . 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 ) )
6261mpteq2dv 4123 . . . . . 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 ) ) )
6362seqeq3d 11070 . . . . 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 ) ) ) )
64 zcn 10045 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
65643ad2ant2 977 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  CC )
6665absnegd 11947 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  -u N )  =  ( abs `  N
) )
6763, 66fveq12d 5547 . . . 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
) ) )
6853, 67oveq12d 5892 . . 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 ) ) ) )
695, 7keepel 3635 . . . . 5  |-  if ( A  <  0 , 
-u 1 ,  1 )  e.  CC
7069a1i 10 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  e.  CC )
715, 7keepel 3635 . . . . 5  |-  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  e.  CC
7271a1i 10 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  e.  CC )
73 nnabscl 11825 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
74733adant1 973 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  N )  e.  NN )
75 nnuz 10279 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
7674, 75syl6eleq 2386 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  N )  e.  ( ZZ>= `  1 )
)
77 eqid 2296 . . . . . . . 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 ) )
7877lgsfcl3 20572 . . . . . . 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 )
79 elfznn 10835 . . . . . . 7  |-  ( x  e.  ( 1 ... ( abs `  N
) )  ->  x  e.  NN )
80 ffvelrn 5679 . . . . . . 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 )
8178, 79, 80syl2an 463 . . . . . 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 )
82 zmulcl 10082 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
8382adantl 452 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  x.  y )  e.  ZZ )
8476, 81, 83seqcl 11082 . . . . 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 )
8584zcnd 10134 . . . 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 )
8670, 72, 85mulassd 8874 . . 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 ) ) ) ) )
8768, 86eqtrd 2328 . 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 ) ) ) ) )
88 simp1 955 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  A  e.  ZZ )
89 znegcl 10071 . . . 4  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
90893ad2ant2 977 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u N  e.  ZZ )
91 simp3 957 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  =/=  0 )
9265, 91negne0d 9171 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u N  =/=  0 )
93 eqid 2296 . . . 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 ) )
9493lgsval4 20571 . . 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 ) ) ) )
9588, 90, 92, 94syl3anc 1182 . 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
) ) ) )
9677lgsval4 20571 . . 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 ) ) ) )
9796oveq2d 5890 . 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 ) ) ) ) )
9887, 95, 973eqtr4d 2338 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   ifcif 3578   class class class wbr 4039    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883    <_ cle 8884   -ucneg 9054   NNcn 9762   ZZcz 10040   ZZ>=cuz 10246   QQcq 10332   ...cfz 10798    seq cseq 11062   ^cexp 11120   abscabs 11735   Primecprime 12774    pCnt cpc 12905    / Lclgs 20549
This theorem is referenced by:  lgsneg1  20575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-phi 12850  df-pc 12906  df-lgs 20550
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