MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lgsval4 Unicode version

Theorem lgsval4 20571
Description: Restate lgsval 20555 for nonzero  N, where the function  F has been abbreviated into a self-referential expression taking the value of  / L on the primes as given. (Contributed by Mario Carneiro, 4-Feb-2015.)
Hypothesis
Ref Expression
lgsval4.1  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) )
Assertion
Ref Expression
lgsval4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( A  / L N )  =  ( 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 lgsval4
StepHypRef Expression
1 eqid 2296 . . . 4  |-  ( 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 ) )  =  ( 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 ) )
21lgsval 20555 . . 3  |-  ( ( 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.  , 
( 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 ) ) ) `  ( abs `  N ) ) ) ) )
323adant3 975 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( A  / L N )  =  if ( N  =  0 ,  if ( ( A ^
2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( N  <  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  N )
) ,  1 ) ) ) `  ( abs `  N ) ) ) ) )
4 simp3 957 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  N  =/=  0 )
54neneqd 2475 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  -.  N  =  0 )
6 iffalse 3585 . . 3  |-  ( -.  N  =  0  ->  if ( N  =  0 ,  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( N  <  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  N )
) ,  1 ) ) ) `  ( abs `  N ) ) ) )  =  ( if ( ( N  <  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  N )
) ,  1 ) ) ) `  ( abs `  N ) ) ) )
75, 6syl 15 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  if ( N  =  0 ,  if ( ( A ^ 2 )  =  1 ,  1 ,  0 ) ,  ( if ( ( N  <  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  N )
) ,  1 ) ) ) `  ( abs `  N ) ) ) )  =  ( if ( ( N  <  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  N )
) ,  1 ) ) ) `  ( abs `  N ) ) ) )
81lgsval4lem 20562 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
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 ) )  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L
n ) ^ (
n  pCnt  N )
) ,  1 ) ) )
9 lgsval4.1 . . . . . 6  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( ( A  / L n ) ^
( n  pCnt  N
) ) ,  1 ) )
108, 9syl6eqr 2346 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
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 ) )  =  F )
1110seqeq3d 11070 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  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  N )
) ,  1 ) ) )  =  seq  1 (  x.  ,  F ) )
1211fveq1d 5543 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (  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  N )
) ,  1 ) ) ) `  ( abs `  N ) )  =  (  seq  1
(  x.  ,  F
) `  ( abs `  N ) ) )
1312oveq2d 5890 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( if ( ( N  <  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  N )
) ,  1 ) ) ) `  ( abs `  N ) ) )  =  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  (  seq  1
(  x.  ,  F
) `  ( abs `  N ) ) ) )
143, 7, 133eqtrd 2332 1  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( A  / L N )  =  ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  (  seq  1 (  x.  ,  F ) `
 ( abs `  N
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   ifcif 3578   {cpr 3654   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   7c7 9816   8c8 9817   ZZcz 10040    mod cmo 10989    seq cseq 11062   ^cexp 11120   abscabs 11735    || cdivides 12547   Primecprime 12774    pCnt cpc 12905    / Lclgs 20549
This theorem is referenced by:  lgsval4a  20573  lgsneg  20574  lgsdir  20585  lgsdi  20587  lgsne0  20588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-phi 12850  df-pc 12906  df-lgs 20550
  Copyright terms: Public domain W3C validator