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Theorem lgsneg 14092
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 3539 . . . . . . . . 9  |-  ( A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
21adantl 277 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
32oveq1d 5884 . . . . . . 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 simpl2 1001 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  e.  ZZ )
5 0z 9253 . . . . . . . . . . 11  |-  0  e.  ZZ
6 zdclt 9319 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  <  0 )
75, 6mpan2 425 . . . . . . . . . 10  |-  ( N  e.  ZZ  -> DECID  N  <  0
)
8 oveq2 5877 . . . . . . . . . . . 12  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  ( -u 1  x.  -u 1 ) )
9 neg1mulneg1e1 9120 . . . . . . . . . . . 12  |-  ( -u
1  x.  -u 1
)  =  1
108, 9eqtrdi 2226 . . . . . . . . . . 11  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  1 )
11 oveq2 5877 . . . . . . . . . . . 12  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  (
-u 1  x.  1 ) )
12 ax-1cn 7895 . . . . . . . . . . . . 13  |-  1  e.  CC
1312mulm1i 8350 . . . . . . . . . . . 12  |-  ( -u
1  x.  1 )  =  -u 1
1411, 13eqtrdi 2226 . . . . . . . . . . 11  |-  ( if ( N  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  -u
1 )
1510, 14ifsbdc 3546 . . . . . . . . . 10  |-  (DECID  N  <  0  ->  ( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  if ( N  <  0 ,  1 ,  -u 1 ) )
167, 15syl 14 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  ( -u 1  x.  if ( N  <  0 , 
-u 1 ,  1 ) )  =  if ( N  <  0 ,  1 ,  -u
1 ) )
174, 16syl 14 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( -u 1  x.  if ( N  <  0 ,  -u 1 ,  1 ) )  =  if ( N  <  0 ,  1 ,  -u
1 ) )
18 simpr 110 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  A  <  0 )
1918biantrud 304 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  ( N  <  0  /\  A  <  0 ) ) )
2019ifbid 3555 . . . . . . . . 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 ) )
2120oveq2d 5885 . . . . . . . 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 ) ) )
22 simpl3 1002 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  =/=  0 )
2322necomd 2433 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
0  =/=  N )
24 zltlen 9320 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N  <  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
254, 5, 24sylancl 413 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
2623, 25mpbiran2d 442 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  N  <_  0 ) )
274zred 9364 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  N  e.  RR )
2827le0neg1d 8464 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <_  0  <->  0  <_  -u N ) )
29 0re 7948 . . . . . . . . . . . 12  |-  0  e.  RR
3027renegcld 8327 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  -u N  e.  RR )
31 lenlt 8023 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  -u N  e.  RR )  ->  ( 0  <_  -u N  <->  -.  -u N  <  0 ) )
3229, 30, 31sylancr 414 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( 0  <_  -u N  <->  -.  -u N  <  0
) )
3326, 28, 323bitrd 214 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( N  <  0  <->  -.  -u N  <  0
) )
3433ifbid 3555 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( N  <  0 ,  1 ,  -u
1 )  =  if ( -.  -u N  <  0 ,  1 , 
-u 1 ) )
35 znegcl 9273 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
36 zdclt 9319 . . . . . . . . . . . 12  |-  ( (
-u N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  -u N  <  0
)
3735, 5, 36sylancl 413 . . . . . . . . . . 11  |-  ( N  e.  ZZ  -> DECID  -u N  <  0
)
38 ifnotdc 3570 . . . . . . . . . . 11  |-  (DECID  -u N  <  0  ->  if ( -.  -u N  <  0 ,  1 ,  -u
1 )  =  if ( -u N  <  0 ,  -u 1 ,  1 ) )
3937, 38syl 14 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  if ( -.  -u N  <  0 ,  1 , 
-u 1 )  =  if ( -u N  <  0 ,  -u 1 ,  1 ) )
404, 39syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( -.  -u N  <  0 ,  1 , 
-u 1 )  =  if ( -u N  <  0 ,  -u 1 ,  1 ) )
4134, 40eqtrd 2210 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  ->  if ( N  <  0 ,  1 ,  -u
1 )  =  if ( -u N  <  0 ,  -u 1 ,  1 ) )
4217, 21, 413eqtr3d 2218 . . . . . . 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 ) )
4318biantrud 304 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  A  <  0 )  -> 
( -u N  <  0  <->  (
-u N  <  0  /\  A  <  0
) ) )
4443ifbid 3555 . . . . . . 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 ) )
453, 42, 443eqtrd 2214 . . . . . 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 ) )
46 1t1e1 9060 . . . . . . 7  |-  ( 1  x.  1 )  =  1
47 iffalse 3542 . . . . . . . . 9  |-  ( -.  A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  1 )
4847adantl 277 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if ( A  <  0 ,  -u
1 ,  1 )  =  1 )
49 simpr 110 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  A  <  0 )
5049intnand 931 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  ( N  <  0  /\  A  <  0 ) )
5150iffalsed 3544 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if (
( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
5248, 51oveq12d 5887 . . . . . . 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 ) )
5349intnand 931 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  -.  ( -u N  <  0  /\  A  <  0 ) )
5453iffalsed 3544 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  -.  A  <  0
)  ->  if (
( -u N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
5546, 52, 543eqtr4a 2236 . . . . . 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 ) )
56 simp1 997 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  A  e.  ZZ )
57 zdclt 9319 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  0  e.  ZZ )  -> DECID  A  <  0 )
5856, 5, 57sylancl 413 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  -> DECID  A  <  0
)
59 exmiddc 836 . . . . . . 7  |-  (DECID  A  <  0  ->  ( A  <  0  \/  -.  A  <  0 ) )
6058, 59syl 14 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( A  <  0  \/  -.  A  <  0 ) )
6145, 55, 60mpjaodan 798 . . . . 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 ) )
6261eqcomd 2183 . . . 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 ) ) )
63 simpr 110 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  n  e.  Prime )
64 simpl2 1001 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  N  e.  ZZ )
65 zq 9615 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  N  e.  QQ )
6664, 65syl 14 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  ->  N  e.  QQ )
67 pcneg 12307 . . . . . . . . . . 11  |-  ( ( n  e.  Prime  /\  N  e.  QQ )  ->  (
n  pCnt  -u N )  =  ( n  pCnt  N ) )
6863, 66, 67syl2anc 411 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  -> 
( n  pCnt  -u N
)  =  ( n 
pCnt  N ) )
6968oveq2d 5885 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  Prime )  -> 
( ( A  /L n ) ^
( n  pCnt  -u N
) )  =  ( ( A  /L
n ) ^ (
n  pCnt  N )
) )
7069adantlr 477 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  ->  (
( A  /L
n ) ^ (
n  pCnt  -u N ) )  =  ( ( A  /L n ) ^ ( n 
pCnt  N ) ) )
71 prmdc 12113 . . . . . . . . 9  |-  ( n  e.  NN  -> DECID  n  e.  Prime )
7271adantl 277 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  -> DECID  n  e.  Prime )
7370, 72ifeq1dadc 3564 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  ->  if ( n  e. 
Prime ,  ( ( A  /L n ) ^ ( n  pCnt  -u N ) ) ,  1 )  =  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) )
7473mpteq2dva 4090 . . . . . 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 ) ) )
7574seqeq3d 10439 . . . . 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 ) ) ) )
76 zcn 9247 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
77763ad2ant2 1019 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  CC )
7877absnegd 11182 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  -u N )  =  ( abs `  N
) )
7975, 78fveq12d 5518 . . . 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
) ) )
8062, 79oveq12d 5887 . . 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 ) ) ) )
81 neg1cn 9013 . . . . . 6  |-  -u 1  e.  CC
8281a1i 9 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u 1  e.  CC )
8312a1i 9 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  1  e.  CC )
8482, 83, 58ifcldcd 3569 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  e.  CC )
8573ad2ant2 1019 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  -> DECID  N  <  0
)
86 dcan2 934 . . . . . 6  |-  (DECID  N  <  0  ->  (DECID  A  <  0  -> DECID 
( N  <  0  /\  A  <  0
) ) )
8785, 58, 86sylc 62 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  -> DECID  ( N  <  0  /\  A  <  0
) )
8882, 83, 87ifcldcd 3569 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  e.  CC )
89 nnuz 9552 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
90 1zzd 9269 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  1  e.  ZZ )
91 eqid 2177 . . . . . . . . 9  |-  ( 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 ) )
9291lgsfcl3 14089 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^
( n  pCnt  N
) ) ,  1 ) ) : NN --> ZZ )
9392ffvelcdmda 5647 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  x  e.  NN )  ->  ( ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  /L n ) ^ ( n 
pCnt  N ) ) ,  1 ) ) `  x )  e.  ZZ )
94 zmulcl 9295 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
9594adantl 277 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  x.  y )  e.  ZZ )
9689, 90, 93, 95seqf 10447 . . . . . 6  |-  ( ( 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 ) ) ) : NN --> ZZ )
97 nnabscl 11093 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
98973adant1 1015 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( abs `  N )  e.  NN )
9996, 98ffvelcdmd 5648 . . . . 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 )
10099zcnd 9365 . . . 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 )
10184, 88, 100mulassd 7971 . . 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 ) ) ) ) )
10280, 101eqtrd 2210 . 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 ) ) ) ) )
103353ad2ant2 1019 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u N  e.  ZZ )
104 simp3 999 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  =/=  0 )
10577, 104negne0d 8256 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -u N  =/=  0 )
106 eqid 2177 . . . 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 ) )
107106lgsval4 14088 . . 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 ) ) ) )
10856, 103, 105, 107syl3anc 1238 . 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
) ) ) )
10991lgsval4 14088 . . 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 ) ) ) )
110109oveq2d 5885 . 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 ) ) ) ) )
111102, 108, 1103eqtr4d 2220 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    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347   ifcif 3534   class class class wbr 4000    |-> cmpt 4061   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803    x. cmul 7807    < clt 7982    <_ cle 7983   -ucneg 8119   NNcn 8908   ZZcz 9242   QQcq 9608    seqcseq 10431   ^cexp 10505   abscabs 10990   Primecprime 12090    pCnt cpc 12267    /Lclgs 14065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-2o 6412  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543  df-dvds 11779  df-gcd 11927  df-prm 12091  df-phi 12194  df-pc 12268  df-lgs 14066
This theorem is referenced by:  lgsneg1  14093
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