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Theorem lgsfcl2 24309
Description: The function  F is closed in integers with absolute value less than  1 (namely  { -u
1 ,  0 ,  1 } although this representation is less useful to us). (Contributed by Mario Carneiro, 4-Feb-2015.)
Hypotheses
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 ) )
lgsfcl2.z  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
Assertion
Ref Expression
lgsfcl2  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  F : NN --> Z )
Distinct variable groups:    x, n, A    x, F    n, N, x    n, Z
Allowed substitution hints:    F( n)    Z( x)

Proof of Theorem lgsfcl2
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0z 10972 . . . . . . . 8  |-  0  e.  ZZ
2 0le1 10158 . . . . . . . 8  |-  0  <_  1
3 fveq2 5879 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( abs `  x )  =  ( abs `  0
) )
4 abs0 13425 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
53, 4syl6eq 2521 . . . . . . . . . 10  |-  ( x  =  0  ->  ( abs `  x )  =  0 )
65breq1d 4405 . . . . . . . . 9  |-  ( x  =  0  ->  (
( abs `  x
)  <_  1  <->  0  <_  1 ) )
7 lgsfcl2.z . . . . . . . . 9  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
86, 7elrab2 3186 . . . . . . . 8  |-  ( 0  e.  Z  <->  ( 0  e.  ZZ  /\  0  <_  1 ) )
91, 2, 8mpbir2an 934 . . . . . . 7  |-  0  e.  Z
10 1z 10991 . . . . . . . . 9  |-  1  e.  ZZ
11 1le1 10262 . . . . . . . . 9  |-  1  <_  1
12 fveq2 5879 . . . . . . . . . . . 12  |-  ( x  =  1  ->  ( abs `  x )  =  ( abs `  1
) )
13 abs1 13437 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
1412, 13syl6eq 2521 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( abs `  x )  =  1 )
1514breq1d 4405 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( abs `  x
)  <_  1  <->  1  <_  1 ) )
1615, 7elrab2 3186 . . . . . . . . 9  |-  ( 1  e.  Z  <->  ( 1  e.  ZZ  /\  1  <_  1 ) )
1710, 11, 16mpbir2an 934 . . . . . . . 8  |-  1  e.  Z
18 neg1z 10997 . . . . . . . . 9  |-  -u 1  e.  ZZ
19 fveq2 5879 . . . . . . . . . . . 12  |-  ( x  =  -u 1  ->  ( abs `  x )  =  ( abs `  -u 1
) )
20 ax-1cn 9615 . . . . . . . . . . . . . 14  |-  1  e.  CC
2120absnegi 13539 . . . . . . . . . . . . 13  |-  ( abs `  -u 1 )  =  ( abs `  1
)
2221, 13eqtri 2493 . . . . . . . . . . . 12  |-  ( abs `  -u 1 )  =  1
2319, 22syl6eq 2521 . . . . . . . . . . 11  |-  ( x  =  -u 1  ->  ( abs `  x )  =  1 )
2423breq1d 4405 . . . . . . . . . 10  |-  ( x  =  -u 1  ->  (
( abs `  x
)  <_  1  <->  1  <_  1 ) )
2524, 7elrab2 3186 . . . . . . . . 9  |-  ( -u
1  e.  Z  <->  ( -u 1  e.  ZZ  /\  1  <_ 
1 ) )
2618, 11, 25mpbir2an 934 . . . . . . . 8  |-  -u 1  e.  Z
2717, 26keepel 3939 . . . . . . 7  |-  if ( ( A  mod  8
)  e.  { 1 ,  7 } , 
1 ,  -u 1
)  e.  Z
289, 27keepel 3939 . . . . . 6  |-  if ( 2  ||  A , 
0 ,  if ( ( A  mod  8
)  e.  { 1 ,  7 } , 
1 ,  -u 1
) )  e.  Z
2928a1i 11 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  n  =  2 )  ->  if ( 2  ||  A ,  0 ,  if ( ( A  mod  8 )  e.  {
1 ,  7 } ,  1 ,  -u
1 ) )  e.  Z )
30 simpl1 1033 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  ->  A  e.  ZZ )
3130ad2antrr 740 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  A  e.  ZZ )
32 simplr 770 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  n  e.  Prime )
33 simpr 468 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  -.  n  = 
2 )
3433neqned 2650 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  n  =/=  2
)
35 eldifsn 4088 . . . . . . 7  |-  ( n  e.  ( Prime  \  {
2 } )  <->  ( n  e.  Prime  /\  n  =/=  2 ) )
3632, 34, 35sylanbrc 677 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  n  e.  ( Prime  \  { 2 } ) )
377lgslem4 24306 . . . . . 6  |-  ( ( A  e.  ZZ  /\  n  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( A ^
( ( n  - 
1 )  /  2
) )  +  1 )  mod  n )  -  1 )  e.  Z )
3831, 36, 37syl2anc 673 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  ( ( ( ( A ^ (
( n  -  1 )  /  2 ) )  +  1 )  mod  n )  - 
1 )  e.  Z
)
3929, 38ifclda 3904 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  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 ) )  e.  Z
)
40 simpr 468 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  ->  n  e.  Prime )
41 simpll2 1070 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  ->  N  e.  ZZ )
42 simpll3 1071 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  ->  N  =/=  0 )
43 pczcl 14877 . . . . 5  |-  ( ( n  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( n  pCnt  N
)  e.  NN0 )
4440, 41, 42, 43syl12anc 1290 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  ->  (
n  pCnt  N )  e.  NN0 )
45 ssrab2 3500 . . . . . . 7  |-  { x  e.  ZZ  |  ( abs `  x )  <_  1 }  C_  ZZ
467, 45eqsstri 3448 . . . . . 6  |-  Z  C_  ZZ
47 zsscn 10969 . . . . . 6  |-  ZZ  C_  CC
4846, 47sstri 3427 . . . . 5  |-  Z  C_  CC
497lgslem3 24305 . . . . 5  |-  ( ( a  e.  Z  /\  b  e.  Z )  ->  ( a  x.  b
)  e.  Z )
5048, 49, 17expcllem 12321 . . . 4  |-  ( ( 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 ) )  e.  Z  /\  ( n  pCnt  N )  e.  NN0 )  -> 
( 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 )
)  e.  Z )
5139, 44, 50syl2anc 673 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  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
) )  e.  Z
)
5217a1i 11 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  -.  n  e.  Prime )  -> 
1  e.  Z )
5351, 52ifclda 3904 . 2  |-  ( ( ( 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 )  e.  Z )
54 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 ) )
5553, 54fmptd 6061 1  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  F : NN --> Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760    \ cdif 3387   ifcif 3872   {csn 3959   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558    + caddc 9560    <_ cle 9694    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   2c2 10681   7c7 10686   8c8 10687   NN0cn0 10893   ZZcz 10961    mod cmo 12129   ^cexp 12310   abscabs 13374    || cdvds 14382   Primecprime 14701    pCnt cpc 14865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793  df-pc 14866
This theorem is referenced by:  lgscllem  24310  lgsfcl  24311  lgsfle1  24312
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