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Theorem lgsval 23776
Description: Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
Hypothesis
Ref Expression
lgsval.1  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2  ||  A ,  0 ,  if ( ( A  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( A ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 ) ) ^ (
n  pCnt  N )
) ,  1 ) )
Assertion
Ref Expression
lgsval  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L
N )  =  if ( N  =  0 ,  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 )  x.  (  seq 1 (  x.  ,  F ) `  ( abs `  N ) ) ) ) )
Distinct variable groups:    A, n    n, N
Allowed substitution hint:    F( n)

Proof of Theorem lgsval
Dummy variables  a  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 459 . . . 4  |-  ( ( a  =  A  /\  m  =  N )  ->  m  =  N )
21eqeq1d 2456 . . 3  |-  ( ( a  =  A  /\  m  =  N )  ->  ( m  =  0  <-> 
N  =  0 ) )
3 simpl 455 . . . . . 6  |-  ( ( a  =  A  /\  m  =  N )  ->  a  =  A )
43oveq1d 6285 . . . . 5  |-  ( ( a  =  A  /\  m  =  N )  ->  ( a ^ 2 )  =  ( A ^ 2 ) )
54eqeq1d 2456 . . . 4  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( a ^
2 )  =  1  <-> 
( A ^ 2 )  =  1 ) )
65ifbid 3951 . . 3  |-  ( ( a  =  A  /\  m  =  N )  ->  if ( ( a ^ 2 )  =  1 ,  1 ,  0 )  =  if ( ( A ^
2 )  =  1 ,  1 ,  0 ) )
71breq1d 4449 . . . . . 6  |-  ( ( a  =  A  /\  m  =  N )  ->  ( m  <  0  <->  N  <  0 ) )
83breq1d 4449 . . . . . 6  |-  ( ( a  =  A  /\  m  =  N )  ->  ( a  <  0  <->  A  <  0 ) )
97, 8anbi12d 708 . . . . 5  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( m  <  0  /\  a  <  0 )  <->  ( N  <  0  /\  A  <  0 ) ) )
109ifbid 3951 . . . 4  |-  ( ( a  =  A  /\  m  =  N )  ->  if ( ( m  <  0  /\  a  <  0 ) ,  -u
1 ,  1 )  =  if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 ) )
113breq2d 4451 . . . . . . . . . . . 12  |-  ( ( a  =  A  /\  m  =  N )  ->  ( 2  ||  a  <->  2 
||  A ) )
123oveq1d 6285 . . . . . . . . . . . . . 14  |-  ( ( a  =  A  /\  m  =  N )  ->  ( a  mod  8
)  =  ( A  mod  8 ) )
1312eleq1d 2523 . . . . . . . . . . . . 13  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( a  mod  8 )  e.  {
1 ,  7 }  <-> 
( A  mod  8
)  e.  { 1 ,  7 } ) )
1413ifbid 3951 . . . . . . . . . . . 12  |-  ( ( a  =  A  /\  m  =  N )  ->  if ( ( a  mod  8 )  e. 
{ 1 ,  7 } ,  1 , 
-u 1 )  =  if ( ( A  mod  8 )  e. 
{ 1 ,  7 } ,  1 , 
-u 1 ) )
1511, 14ifbieq2d 3954 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  m  =  N )  ->  if ( 2  ||  a ,  0 ,  if ( ( a  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) )  =  if ( 2  ||  A ,  0 ,  if ( ( A  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) )
163oveq1d 6285 . . . . . . . . . . . . . 14  |-  ( ( a  =  A  /\  m  =  N )  ->  ( a ^ (
( n  -  1 )  /  2 ) )  =  ( A ^ ( ( n  -  1 )  / 
2 ) ) )
1716oveq1d 6285 . . . . . . . . . . . . 13  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( a ^
( ( n  - 
1 )  /  2
) )  +  1 )  =  ( ( A ^ ( ( n  -  1 )  /  2 ) )  +  1 ) )
1817oveq1d 6285 . . . . . . . . . . . 12  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( ( a ^ ( ( n  -  1 )  / 
2 ) )  +  1 )  mod  n
)  =  ( ( ( A ^ (
( n  -  1 )  /  2 ) )  +  1 )  mod  n ) )
1918oveq1d 6285 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( ( ( a ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 )  =  ( ( ( ( A ^
( ( n  - 
1 )  /  2
) )  +  1 )  mod  n )  -  1 ) )
2015, 19ifeq12d 3949 . . . . . . . . . 10  |-  ( ( a  =  A  /\  m  =  N )  ->  if ( n  =  2 ,  if ( 2  ||  a ,  0 ,  if ( ( a  mod  8
)  e.  { 1 ,  7 } , 
1 ,  -u 1
) ) ,  ( ( ( ( a ^ ( ( n  -  1 )  / 
2 ) )  +  1 )  mod  n
)  -  1 ) )  =  if ( n  =  2 ,  if ( 2  ||  A ,  0 ,  if ( ( A  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( A ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 ) ) )
211oveq2d 6286 . . . . . . . . . 10  |-  ( ( a  =  A  /\  m  =  N )  ->  ( n  pCnt  m
)  =  ( n 
pCnt  N ) )
2220, 21oveq12d 6288 . . . . . . . . 9  |-  ( ( a  =  A  /\  m  =  N )  ->  ( if ( n  =  2 ,  if ( 2  ||  a ,  0 ,  if ( ( a  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( a ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 ) ) ^ (
n  pCnt  m )
)  =  ( if ( n  =  2 ,  if ( 2 
||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
( n  -  1 )  /  2 ) )  +  1 )  mod  n )  - 
1 ) ) ^
( n  pCnt  N
) ) )
2322ifeq1d 3947 . . . . . . . 8  |-  ( ( a  =  A  /\  m  =  N )  ->  if ( n  e. 
Prime ,  ( if ( n  =  2 ,  if ( 2  ||  a ,  0 ,  if ( ( a  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( a ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 ) ) ^ (
n  pCnt  m )
) ,  1 )  =  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2 
||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
( n  -  1 )  /  2 ) )  +  1 )  mod  n )  - 
1 ) ) ^
( n  pCnt  N
) ) ,  1 ) )
2423mpteq2dv 4526 . . . . . . 7  |-  ( ( a  =  A  /\  m  =  N )  ->  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2  ||  a ,  0 ,  if ( ( a  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( a ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 ) ) ^ (
n  pCnt  m )
) ,  1 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2  ||  A , 
0 ,  if ( ( A  mod  8
)  e.  { 1 ,  7 } , 
1 ,  -u 1
) ) ,  ( ( ( ( A ^ ( ( n  -  1 )  / 
2 ) )  +  1 )  mod  n
)  -  1 ) ) ^ ( n 
pCnt  N ) ) ,  1 ) ) )
25 lgsval.1 . . . . . . 7  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2  ||  A ,  0 ,  if ( ( A  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( A ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 ) ) ^ (
n  pCnt  N )
) ,  1 ) )
2624, 25syl6eqr 2513 . . . . . 6  |-  ( ( a  =  A  /\  m  =  N )  ->  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2  ||  a ,  0 ,  if ( ( a  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( a ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 ) ) ^ (
n  pCnt  m )
) ,  1 ) )  =  F )
2726seqeq3d 12100 . . . . 5  |-  ( ( a  =  A  /\  m  =  N )  ->  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2 
||  a ,  0 ,  if ( ( a  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( a ^ (
( n  -  1 )  /  2 ) )  +  1 )  mod  n )  - 
1 ) ) ^
( n  pCnt  m
) ) ,  1 ) ) )  =  seq 1 (  x.  ,  F ) )
281fveq2d 5852 . . . . 5  |-  ( ( a  =  A  /\  m  =  N )  ->  ( abs `  m
)  =  ( abs `  N ) )
2927, 28fveq12d 5854 . . . 4  |-  ( ( a  =  A  /\  m  =  N )  ->  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2  ||  a ,  0 ,  if ( ( a  mod  8
)  e.  { 1 ,  7 } , 
1 ,  -u 1
) ) ,  ( ( ( ( a ^ ( ( n  -  1 )  / 
2 ) )  +  1 )  mod  n
)  -  1 ) ) ^ ( n 
pCnt  m ) ) ,  1 ) ) ) `
 ( abs `  m
) )  =  (  seq 1 (  x.  ,  F ) `  ( abs `  N ) ) )
3010, 29oveq12d 6288 . . 3  |-  ( ( a  =  A  /\  m  =  N )  ->  ( if ( ( m  <  0  /\  a  <  0 ) ,  -u 1 ,  1 )  x.  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2  ||  a ,  0 ,  if ( ( a  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( a ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 ) ) ^ (
n  pCnt  m )
) ,  1 ) ) ) `  ( abs `  m ) ) )  =  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  F
) `  ( abs `  N ) ) ) )
312, 6, 30ifbieq12d 3956 . 2  |-  ( ( a  =  A  /\  m  =  N )  ->  if ( m  =  0 ,  if ( ( a ^ 2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( m  <  0  /\  a  <  0
) ,  -u 1 ,  1 )  x.  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2  ||  a ,  0 ,  if ( ( a  mod  8
)  e.  { 1 ,  7 } , 
1 ,  -u 1
) ) ,  ( ( ( ( a ^ ( ( n  -  1 )  / 
2 ) )  +  1 )  mod  n
)  -  1 ) ) ^ ( n 
pCnt  m ) ) ,  1 ) ) ) `
 ( abs `  m
) ) ) )  =  if ( N  =  0 ,  if ( ( A ^
2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  F
) `  ( abs `  N ) ) ) ) )
32 df-lgs 23771 . 2  |-  /L 
=  ( a  e.  ZZ ,  m  e.  ZZ  |->  if ( m  =  0 ,  if ( ( a ^
2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( m  <  0  /\  a  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( if ( n  =  2 ,  if ( 2  ||  a ,  0 ,  if ( ( a  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( a ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 ) ) ^ (
n  pCnt  m )
) ,  1 ) ) ) `  ( abs `  m ) ) ) ) )
33 1nn0 10807 . . . . 5  |-  1  e.  NN0
34 0nn0 10806 . . . . 5  |-  0  e.  NN0
3533, 34keepel 3996 . . . 4  |-  if ( ( A ^ 2 )  =  1 ,  1 ,  0 )  e.  NN0
3635elexi 3116 . . 3  |-  if ( ( A ^ 2 )  =  1 ,  1 ,  0 )  e.  _V
37 ovex 6298 . . 3  |-  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  F
) `  ( abs `  N ) ) )  e.  _V
3836, 37ifex 3997 . 2  |-  if ( N  =  0 ,  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  F
) `  ( abs `  N ) ) ) )  e.  _V
3931, 32, 38ovmpt2a 6406 1  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  /L
N )  =  if ( N  =  0 ,  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 )  x.  (  seq 1 (  x.  ,  F ) `  ( abs `  N ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   ifcif 3929   {cpr 4018   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    - cmin 9796   -ucneg 9797    / cdiv 10202   NNcn 10531   2c2 10581   7c7 10586   8c8 10587   NN0cn0 10791   ZZcz 10860    mod cmo 11978    seqcseq 12092   ^cexp 12151   abscabs 13152    || cdvds 14073   Primecprime 14304    pCnt cpc 14447    /Lclgs 23770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-mulcl 9543  ax-i2m1 9549
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-nn 10532  df-n0 10792  df-seq 12093  df-lgs 23771
This theorem is referenced by:  lgscllem  23779  lgsval2lem  23782  lgs0  23785  lgsval4  23792
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