MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lgsfval Structured version   Unicode version

Theorem lgsfval 23702
Description: Value of the function  F which defines the Legendre symbol at the primes. (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
lgsfval  |-  ( M  e.  NN  ->  ( F `  M )  =  if ( M  e. 
Prime ,  ( if ( M  =  2 ,  if ( 2  ||  A ,  0 ,  if ( ( A  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( A ^ ( ( M  -  1 )  /  2 ) )  +  1 )  mod 
M )  -  1 ) ) ^ ( M  pCnt  N ) ) ,  1 ) )
Distinct variable groups:    A, n    n, M    n, N
Allowed substitution hint:    F( n)

Proof of Theorem lgsfval
StepHypRef Expression
1 eleq1 2529 . . 3  |-  ( n  =  M  ->  (
n  e.  Prime  <->  M  e.  Prime ) )
2 eqeq1 2461 . . . . 5  |-  ( n  =  M  ->  (
n  =  2  <->  M  =  2 ) )
3 oveq1 6303 . . . . . . . . . 10  |-  ( n  =  M  ->  (
n  -  1 )  =  ( M  - 
1 ) )
43oveq1d 6311 . . . . . . . . 9  |-  ( n  =  M  ->  (
( n  -  1 )  /  2 )  =  ( ( M  -  1 )  / 
2 ) )
54oveq2d 6312 . . . . . . . 8  |-  ( n  =  M  ->  ( A ^ ( ( n  -  1 )  / 
2 ) )  =  ( A ^ (
( M  -  1 )  /  2 ) ) )
65oveq1d 6311 . . . . . . 7  |-  ( n  =  M  ->  (
( A ^ (
( n  -  1 )  /  2 ) )  +  1 )  =  ( ( A ^ ( ( M  -  1 )  / 
2 ) )  +  1 ) )
7 id 22 . . . . . . 7  |-  ( n  =  M  ->  n  =  M )
86, 7oveq12d 6314 . . . . . 6  |-  ( n  =  M  ->  (
( ( A ^
( ( n  - 
1 )  /  2
) )  +  1 )  mod  n )  =  ( ( ( A ^ ( ( M  -  1 )  /  2 ) )  +  1 )  mod 
M ) )
98oveq1d 6311 . . . . 5  |-  ( n  =  M  ->  (
( ( ( A ^ ( ( n  -  1 )  / 
2 ) )  +  1 )  mod  n
)  -  1 )  =  ( ( ( ( A ^ (
( M  -  1 )  /  2 ) )  +  1 )  mod  M )  - 
1 ) )
102, 9ifbieq2d 3969 . . . 4  |-  ( n  =  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 ) )  =  if ( M  =  2 ,  if ( 2 
||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
( M  -  1 )  /  2 ) )  +  1 )  mod  M )  - 
1 ) ) )
11 oveq1 6303 . . . 4  |-  ( n  =  M  ->  (
n  pCnt  N )  =  ( M  pCnt  N ) )
1210, 11oveq12d 6314 . . 3  |-  ( n  =  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
) )  =  ( if ( M  =  2 ,  if ( 2  ||  A , 
0 ,  if ( ( A  mod  8
)  e.  { 1 ,  7 } , 
1 ,  -u 1
) ) ,  ( ( ( ( A ^ ( ( M  -  1 )  / 
2 ) )  +  1 )  mod  M
)  -  1 ) ) ^ ( M 
pCnt  N ) ) )
131, 12ifbieq1d 3967 . 2  |-  ( n  =  M  ->  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 )  =  if ( M  e.  Prime ,  ( if ( M  =  2 ,  if ( 2 
||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
( M  -  1 )  /  2 ) )  +  1 )  mod  M )  - 
1 ) ) ^
( M  pCnt  N
) ) ,  1 ) )
14 lgsval.1 . 2  |-  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 ) )
15 ovex 6324 . . 3  |-  ( if ( M  =  2 ,  if ( 2 
||  A ,  0 ,  if ( ( A  mod  8 )  e.  { 1 ,  7 } ,  1 ,  -u 1 ) ) ,  ( ( ( ( A ^ (
( M  -  1 )  /  2 ) )  +  1 )  mod  M )  - 
1 ) ) ^
( M  pCnt  N
) )  e.  _V
16 1ex 9608 . . 3  |-  1  e.  _V
1715, 16ifex 4013 . 2  |-  if ( M  e.  Prime ,  ( if ( M  =  2 ,  if ( 2  ||  A , 
0 ,  if ( ( A  mod  8
)  e.  { 1 ,  7 } , 
1 ,  -u 1
) ) ,  ( ( ( ( A ^ ( ( M  -  1 )  / 
2 ) )  +  1 )  mod  M
)  -  1 ) ) ^ ( M 
pCnt  N ) ) ,  1 )  e.  _V
1813, 14, 17fvmpt 5956 1  |-  ( M  e.  NN  ->  ( F `  M )  =  if ( M  e. 
Prime ,  ( if ( M  =  2 ,  if ( 2  ||  A ,  0 ,  if ( ( A  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) ) ,  ( ( ( ( A ^ ( ( M  -  1 )  /  2 ) )  +  1 )  mod 
M )  -  1 ) ) ^ ( M  pCnt  N ) ) ,  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   ifcif 3944   {cpr 4034   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512    - cmin 9824   -ucneg 9825    / cdiv 10227   NNcn 10556   2c2 10606   7c7 10611   8c8 10612    mod cmo 11999   ^cexp 12169    || cdvds 13998   Primecprime 14229    pCnt cpc 14372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-1cn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299
This theorem is referenced by:  lgsval2lem  23707
  Copyright terms: Public domain W3C validator