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Theorem lgsval 23331
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 461 . . . 4  |-  ( ( a  =  A  /\  m  =  N )  ->  m  =  N )
21eqeq1d 2469 . . 3  |-  ( ( a  =  A  /\  m  =  N )  ->  ( m  =  0  <-> 
N  =  0 ) )
3 simpl 457 . . . . . 6  |-  ( ( a  =  A  /\  m  =  N )  ->  a  =  A )
43oveq1d 6299 . . . . 5  |-  ( ( a  =  A  /\  m  =  N )  ->  ( a ^ 2 )  =  ( A ^ 2 ) )
54eqeq1d 2469 . . . 4  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( a ^
2 )  =  1  <-> 
( A ^ 2 )  =  1 ) )
65ifbid 3961 . . 3  |-  ( ( a  =  A  /\  m  =  N )  ->  if ( ( a ^ 2 )  =  1 ,  1 ,  0 )  =  if ( ( A ^
2 )  =  1 ,  1 ,  0 ) )
71breq1d 4457 . . . . . 6  |-  ( ( a  =  A  /\  m  =  N )  ->  ( m  <  0  <->  N  <  0 ) )
83breq1d 4457 . . . . . 6  |-  ( ( a  =  A  /\  m  =  N )  ->  ( a  <  0  <->  A  <  0 ) )
97, 8anbi12d 710 . . . . 5  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( m  <  0  /\  a  <  0 )  <->  ( N  <  0  /\  A  <  0 ) ) )
109ifbid 3961 . . . 4  |-  ( ( a  =  A  /\  m  =  N )  ->  if ( ( m  <  0  /\  a  <  0 ) ,  -u
1 ,  1 )  =  if ( ( N  <  0  /\  A  <  0 ) ,  -u 1 ,  1 ) )
113breq2d 4459 . . . . . . . . . . . 12  |-  ( ( a  =  A  /\  m  =  N )  ->  ( 2  ||  a  <->  2 
||  A ) )
123oveq1d 6299 . . . . . . . . . . . . . 14  |-  ( ( a  =  A  /\  m  =  N )  ->  ( a  mod  8
)  =  ( A  mod  8 ) )
1312eleq1d 2536 . . . . . . . . . . . . 13  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( a  mod  8 )  e.  {
1 ,  7 }  <-> 
( A  mod  8
)  e.  { 1 ,  7 } ) )
1413ifbid 3961 . . . . . . . . . . . 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 3964 . . . . . . . . . . 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 6299 . . . . . . . . . . . . . 14  |-  ( ( a  =  A  /\  m  =  N )  ->  ( a ^ (
( n  -  1 )  /  2 ) )  =  ( A ^ ( ( n  -  1 )  / 
2 ) ) )
1716oveq1d 6299 . . . . . . . . . . . . 13  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( a ^
( ( n  - 
1 )  /  2
) )  +  1 )  =  ( ( A ^ ( ( n  -  1 )  /  2 ) )  +  1 ) )
1817oveq1d 6299 . . . . . . . . . . . 12  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( ( a ^ ( ( n  -  1 )  / 
2 ) )  +  1 )  mod  n
)  =  ( ( ( A ^ (
( n  -  1 )  /  2 ) )  +  1 )  mod  n ) )
1918oveq1d 6299 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  m  =  N )  ->  ( ( ( ( a ^ ( ( n  -  1 )  /  2 ) )  +  1 )  mod  n )  -  1 )  =  ( ( ( ( A ^
( ( n  - 
1 )  /  2
) )  +  1 )  mod  n )  -  1 ) )
2015, 19ifeq12d 3959 . . . . . . . . . 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 6300 . . . . . . . . . 10  |-  ( ( a  =  A  /\  m  =  N )  ->  ( n  pCnt  m
)  =  ( n 
pCnt  N ) )
2220, 21oveq12d 6302 . . . . . . . . 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 3957 . . . . . . . 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 4534 . . . . . . 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 2526 . . . . . 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 12083 . . . . 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 5870 . . . . 5  |-  ( ( a  =  A  /\  m  =  N )  ->  ( abs `  m
)  =  ( abs `  N ) )
2927, 28fveq12d 5872 . . . 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 6302 . . 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 3966 . 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 23326 . 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 10811 . . . . 5  |-  1  e.  NN0
34 0nn0 10810 . . . . 5  |-  0  e.  NN0
3533, 34keepel 4007 . . . 4  |-  if ( ( A ^ 2 )  =  1 ,  1 ,  0 )  e.  NN0
3635elexi 3123 . . 3  |-  if ( ( A ^ 2 )  =  1 ,  1 ,  0 )  e.  _V
37 ovex 6309 . . 3  |-  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq 1
(  x.  ,  F
) `  ( abs `  N ) ) )  e.  _V
3836, 37ifex 4008 . 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 6417 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 369    = wceq 1379    e. wcel 1767   ifcif 3939   {cpr 4029   class class class wbr 4447    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    - cmin 9805   -ucneg 9806    / cdiv 10206   NNcn 10536   2c2 10585   7c7 10590   8c8 10591   NN0cn0 10795   ZZcz 10864    mod cmo 11964    seqcseq 12075   ^cexp 12134   abscabs 13030    || cdivides 13847   Primecprime 14076    pCnt cpc 14219    /Lclgs 23325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-mulcl 9554  ax-i2m1 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-nn 10537  df-n0 10796  df-seq 12076  df-lgs 23326
This theorem is referenced by:  lgscllem  23334  lgsval2lem  23337  lgs0  23340  lgsval4  23347
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