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Theorem lgsfcl2 21078
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 10285 . . . . . . . 8  |-  0  e.  ZZ
2 0le1 9543 . . . . . . . 8  |-  0  <_  1
3 fveq2 5720 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( abs `  x )  =  ( abs `  0
) )
4 abs0 12082 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
53, 4syl6eq 2483 . . . . . . . . . 10  |-  ( x  =  0  ->  ( abs `  x )  =  0 )
65breq1d 4214 . . . . . . . . 9  |-  ( x  =  0  ->  (
( abs `  x
)  <_  1  <->  0  <_  1 ) )
7 lgsfcl2.z . . . . . . . . 9  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
86, 7elrab2 3086 . . . . . . . 8  |-  ( 0  e.  Z  <->  ( 0  e.  ZZ  /\  0  <_  1 ) )
91, 2, 8mpbir2an 887 . . . . . . 7  |-  0  e.  Z
10 1z 10303 . . . . . . . . 9  |-  1  e.  ZZ
11 1le1 9642 . . . . . . . . 9  |-  1  <_  1
12 fveq2 5720 . . . . . . . . . . . 12  |-  ( x  =  1  ->  ( abs `  x )  =  ( abs `  1
) )
13 abs1 12094 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
1412, 13syl6eq 2483 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( abs `  x )  =  1 )
1514breq1d 4214 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( abs `  x
)  <_  1  <->  1  <_  1 ) )
1615, 7elrab2 3086 . . . . . . . . 9  |-  ( 1  e.  Z  <->  ( 1  e.  ZZ  /\  1  <_  1 ) )
1710, 11, 16mpbir2an 887 . . . . . . . 8  |-  1  e.  Z
18 znegcl 10305 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
1910, 18ax-mp 8 . . . . . . . . 9  |-  -u 1  e.  ZZ
20 fveq2 5720 . . . . . . . . . . . 12  |-  ( x  =  -u 1  ->  ( abs `  x )  =  ( abs `  -u 1
) )
21 ax-1cn 9040 . . . . . . . . . . . . . 14  |-  1  e.  CC
2221absnegi 12195 . . . . . . . . . . . . 13  |-  ( abs `  -u 1 )  =  ( abs `  1
)
2322, 13eqtri 2455 . . . . . . . . . . . 12  |-  ( abs `  -u 1 )  =  1
2420, 23syl6eq 2483 . . . . . . . . . . 11  |-  ( x  =  -u 1  ->  ( abs `  x )  =  1 )
2524breq1d 4214 . . . . . . . . . 10  |-  ( x  =  -u 1  ->  (
( abs `  x
)  <_  1  <->  1  <_  1 ) )
2625, 7elrab2 3086 . . . . . . . . 9  |-  ( -u
1  e.  Z  <->  ( -u 1  e.  ZZ  /\  1  <_ 
1 ) )
2719, 11, 26mpbir2an 887 . . . . . . . 8  |-  -u 1  e.  Z
2817, 27keepel 3788 . . . . . . 7  |-  if ( ( A  mod  8
)  e.  { 1 ,  7 } , 
1 ,  -u 1
)  e.  Z
299, 28keepel 3788 . . . . . 6  |-  if ( 2  ||  A , 
0 ,  if ( ( A  mod  8
)  e.  { 1 ,  7 } , 
1 ,  -u 1
) )  e.  Z
3029a1i 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 )
31 simpl1 960 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  ->  A  e.  ZZ )
3231ad2antrr 707 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  A  e.  ZZ )
33 simplr 732 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  n  e.  Prime )
34 simpr 448 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  -.  n  = 
2 )
3534neneqad 2668 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  n  =/=  2
)
36 eldifsn 3919 . . . . . . 7  |-  ( n  e.  ( Prime  \  {
2 } )  <->  ( n  e.  Prime  /\  n  =/=  2 ) )
3733, 35, 36sylanbrc 646 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  /\  -.  n  =  2 )  ->  n  e.  ( Prime  \  { 2 } ) )
387lgslem4 21075 . . . . . 6  |-  ( ( A  e.  ZZ  /\  n  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( A ^
( ( n  - 
1 )  /  2
) )  +  1 )  mod  n )  -  1 )  e.  Z )
3932, 37, 38syl2anc 643 . . . . 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
)
4030, 39ifclda 3758 . . . 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
)
41 simpr 448 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  ->  n  e.  Prime )
42 simpll2 997 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  ->  N  e.  ZZ )
43 simpll3 998 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  ->  N  =/=  0 )
44 pczcl 13214 . . . . 5  |-  ( ( n  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( n  pCnt  N
)  e.  NN0 )
4541, 42, 43, 44syl12anc 1182 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  n  e.  Prime )  ->  (
n  pCnt  N )  e.  NN0 )
46 ssrab2 3420 . . . . . . 7  |-  { x  e.  ZZ  |  ( abs `  x )  <_  1 }  C_  ZZ
477, 46eqsstri 3370 . . . . . 6  |-  Z  C_  ZZ
48 zsscn 10282 . . . . . 6  |-  ZZ  C_  CC
4947, 48sstri 3349 . . . . 5  |-  Z  C_  CC
507lgslem3 21074 . . . . 5  |-  ( ( a  e.  Z  /\  b  e.  Z )  ->  ( a  x.  b
)  e.  Z )
5149, 50, 17expcllem 11384 . . . 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 )
5240, 45, 51syl2anc 643 . . 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
)
5317a1i 11 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  /\  n  e.  NN )  /\  -.  n  e.  Prime )  -> 
1  e.  Z )
5452, 53ifclda 3758 . 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 )
55 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 ) )
5654, 55fmptd 5885 1  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  F : NN --> Z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701    \ cdif 3309   ifcif 3731   {csn 3806   {cpr 3807   class class class wbr 4204    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    <_ cle 9113    - cmin 9283   -ucneg 9284    / cdiv 9669   NNcn 9992   2c2 10041   7c7 10046   8c8 10047   NN0cn0 10213   ZZcz 10274    mod cmo 11242   ^cexp 11374   abscabs 12031    || cdivides 12844   Primecprime 13071    pCnt cpc 13202
This theorem is referenced by:  lgscllem  21079  lgsfcl  21080  lgsfle1  21081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-phi 13147  df-pc 13203
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