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Theorem lgsneg 23977
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 3893 . . . . . . . . 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 6295 . . . . . . 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 6288 . . . . . . . . . 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 10796 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
64, 5syl6eq 2461 . . . . . . . . 9  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  1 )
7 oveq2 6288 . . . . . . . . . 10  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  (
-u 1  x.  1 ) )
8 ax-1cn 9582 . . . . . . . . . . 11  |-  1  e.  CC
98mulm1i 10044 . . . . . . . . . 10  |-  ( -u
1  x.  1 )  =  -u 1
107, 9syl6eq 2461 . . . . . . . . 9  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  -u
1 )
116, 10ifsb 3900 . . . . . . . 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 3909 . . . . . . . . 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 6296 . . . . . . . 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 1003 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  e.  ZZ )
1716zred 11010 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  e.  RR )
18 0re 9628 . . . . . . . . . . . . 13  |-  0  e.  RR
19 ltlen 9719 . . . . . . . . . . . . 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 1004 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  =/=  0 )
2221necomd 2676 . . . . . . . . . . . . 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 258 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  N  <_  0 ) )
2517le0neg1d 10166 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <_  0  <->  0  <_  -u N ) )
2617renegcld 10029 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  -u N  e.  RR )
27 lenlt 9696 . . . . . . . . . . . 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 281 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  -.  -u N  <  0
) )
3029ifbid 3909 . . . . . . . . 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 3932 . . . . . . . . 9  |-  if ( -.  -u N  <  0 ,  1 ,  -u
1 )  =  if ( -u N  <  0 ,  -u 1 ,  1 )
3230, 31syl6eq 2461 . . . . . . . 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 2469 . . . . . . 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 3909 . . . . . . 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 2449 . . . . . 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 10726 . . . . . . 7  |-  ( 1  x.  1 )  =  1
38 iffalse 3896 . . . . . . . . 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 919 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  ( N  <  0  /\  A  <  0 ) )
4241iffalsed 3898 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if (
( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
4339, 42oveq12d 6298 . . . . . . 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 ) )
4440intnand 919 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  ( -u N  <  0  /\  A  <  0 ) )
4544iffalsed 3898 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if (
( -u N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
4637, 43, 453eqtr4a 2471 . . . . . 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 ) )
4736, 46pm2.61dan 794 . . . . 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 ) )
4847eqcomd 2412 . . . 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 ) ) )
49 simpr 461 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  n  e.  Prime )
50 simpl2 1003 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  N  e.  ZZ )
51 zq 11235 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  QQ )
5250, 51syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  N  e.  QQ )
53 pcneg 14608 . . . . . . . . . 10  |-  ( ( n  e.  Prime  /\  N  e.  QQ )  ->  (
n  pCnt  -u N )  =  ( n  pCnt  N ) )
5449, 52, 53syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  -> 
( n  pCnt  -u N
)  =  ( n 
pCnt  N ) )
5554oveq2d 6296 . . . . . . . 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 )
) )
5655ifeq1da 3917 . . . . . . 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 ) )
5756mpteq2dv 4484 . . . . . 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 ) ) )
5857seqeq3d 12161 . . . . 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 ) ) ) )
59 zcn 10912 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
60593ad2ant2 1021 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  CC )
6160absnegd 13431 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  -u N )  =  ( abs `  N
) )
6258, 61fveq12d 5857 . . . 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
) ) )
6348, 62oveq12d 6298 . . 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 ) ) ) )
64 neg1cn 10682 . . . . . 6  |-  -u 1  e.  CC
6564, 8keepel 3954 . . . . 5  |-  if ( A  <  0 , 
-u 1 ,  1 )  e.  CC
6665a1i 11 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  e.  CC )
6764, 8keepel 3954 . . . . 5  |-  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  e.  CC
6867a1i 11 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  e.  CC )
69 nnabscl 13309 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
70693adant1 1017 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  N )  e.  NN )
71 nnuz 11164 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
7270, 71syl6eleq 2502 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  N )  e.  ( ZZ>= `  1 )
)
73 eqid 2404 . . . . . . . 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 ) )
7473lgsfcl3 23975 . . . . . . 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 )
75 elfznn 11770 . . . . . . 7  |-  ( x  e.  ( 1 ... ( abs `  N
) )  ->  x  e.  NN )
76 ffvelrn 6009 . . . . . . 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 )
7774, 75, 76syl2an 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 )
78 zmulcl 10955 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
7978adantl 466 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  x.  y )  e.  ZZ )
8072, 77, 79seqcl 12173 . . . . 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 )
8180zcnd 11011 . . . 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 )
8266, 68, 81mulassd 9651 . . 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 ) ) ) ) )
8363, 82eqtrd 2445 . 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 ) ) ) ) )
84 simp1 999 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  A  e.  ZZ )
85 znegcl 10942 . . . 4  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
86853ad2ant2 1021 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u N  e.  ZZ )
87 simp3 1001 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  =/=  0 )
8860, 87negne0d 9967 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u N  =/=  0 )
89 eqid 2404 . . . 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 ) )
9089lgsval4 23974 . . 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 ) ) ) )
9184, 86, 88, 90syl3anc 1232 . 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
) ) ) )
9273lgsval4 23974 . . 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 ) ) ) )
9392oveq2d 6296 . 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 ) ) ) ) )
9483, 91, 933eqtr4d 2455 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600   ifcif 3887   class class class wbr 4397    |-> cmpt 4455   -->wf 5567   ` cfv 5571  (class class class)co 6280   CCcc 9522   RRcr 9523   0cc0 9524   1c1 9525    x. cmul 9529    < clt 9660    <_ cle 9661   -ucneg 9844   NNcn 10578   ZZcz 10907   ZZ>=cuz 11129   QQcq 11229   ...cfz 11728    seqcseq 12153   ^cexp 12212   abscabs 13218   Primecprime 14428    pCnt cpc 14571    /Lclgs 23952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-q 11230  df-rp 11268  df-fz 11729  df-fzo 11857  df-fl 11968  df-mod 12037  df-seq 12154  df-exp 12213  df-hash 12455  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-dvds 14198  df-gcd 14356  df-prm 14429  df-phi 14507  df-pc 14572  df-lgs 23953
This theorem is referenced by:  lgsneg1  23978
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