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Theorem lgslem4 22638
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.)
Hypothesis
Ref Expression
lgslem2.z  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
Assertion
Ref Expression
lgslem4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  -  1 )  e.  Z )
Distinct variable group:    x, A
Allowed substitution hints:    P( x)    Z( x)

Proof of Theorem lgslem4
StepHypRef Expression
1 simpll 753 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  A  e.  ZZ )
2 oddprm 13882 . . . . . . . . . . 11  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
32ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN )
43nnnn0d 10636 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN0 )
5 zexpcl 11880 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( ( P  - 
1 )  /  2
)  e.  NN0 )  ->  ( A ^ (
( P  -  1 )  /  2 ) )  e.  ZZ )
61, 4, 5syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  ZZ )
76zred 10747 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  RR )
8 0red 9387 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  0  e.  RR )
9 1red 9401 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  1  e.  RR )
10 eldifi 3478 . . . . . . . . . . . 12  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
1110ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  Prime )
12 prmuz2 13781 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
1311, 12syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  ( ZZ>= `  2 )
)
14 eluz2b2 10927 . . . . . . . . . 10  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
1513, 14sylib 196 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( P  e.  NN  /\  1  <  P ) )
1615simpld 459 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  NN )
1716nnrpd 11026 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  RR+ )
18 0zd 10658 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  0  e.  ZZ )
19 simpr 461 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  ||  A )
20 dvdsval3 13539 . . . . . . . . . . . 12  |-  ( ( P  e.  NN  /\  A  e.  ZZ )  ->  ( P  ||  A  <->  ( A  mod  P )  =  0 ) )
2116, 1, 20syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( P  ||  A  <->  ( A  mod  P )  =  0 ) )
2219, 21mpbid 210 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A  mod  P )  =  0 )
23 0mod 11739 . . . . . . . . . . 11  |-  ( P  e.  RR+  ->  ( 0  mod  P )  =  0 )
2417, 23syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
0  mod  P )  =  0 )
2522, 24eqtr4d 2478 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A  mod  P )  =  ( 0  mod  P
) )
26 modexp 11999 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  0  e.  ZZ )  /\  ( ( ( P  -  1 )  /  2 )  e. 
NN0  /\  P  e.  RR+ )  /\  ( A  mod  P )  =  ( 0  mod  P
) )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( ( 0 ^ ( ( P  -  1 )  / 
2 ) )  mod 
P ) )
271, 18, 4, 17, 25, 26syl221anc 1229 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( ( 0 ^ ( ( P  -  1 )  / 
2 ) )  mod 
P ) )
2830expd 12024 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
0 ^ ( ( P  -  1 )  /  2 ) )  =  0 )
2928oveq1d 6106 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( 0 ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( 0  mod 
P ) )
3027, 29eqtrd 2475 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( 0  mod 
P ) )
31 modadd1 11745 . . . . . . 7  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  e.  RR  /\  0  e.  RR )  /\  ( 1  e.  RR  /\  P  e.  RR+ )  /\  (
( A ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( 0  mod 
P ) )  -> 
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  ( ( 0  +  1 )  mod  P ) )
327, 8, 9, 17, 30, 31syl221anc 1229 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  ( ( 0  +  1 )  mod 
P ) )
33 0p1e1 10433 . . . . . . 7  |-  ( 0  +  1 )  =  1
3433oveq1i 6101 . . . . . 6  |-  ( ( 0  +  1 )  mod  P )  =  ( 1  mod  P
)
3532, 34syl6eq 2491 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  ( 1  mod 
P ) )
3616nnred 10337 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  RR )
3715simprd 463 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  1  <  P )
38 1mod 11740 . . . . . 6  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
3936, 37, 38syl2anc 661 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
1  mod  P )  =  1 )
4035, 39eqtrd 2475 . . . 4  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  1 )
4140oveq1d 6106 . . 3  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  =  ( 1  -  1 ) )
42 1m1e0 10390 . . . 4  |-  ( 1  -  1 )  =  0
43 lgslem2.z . . . . . 6  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
4443lgslem2 22636 . . . . 5  |-  ( -u
1  e.  Z  /\  0  e.  Z  /\  1  e.  Z )
4544simp2i 998 . . . 4  |-  0  e.  Z
4642, 45eqeltri 2513 . . 3  |-  ( 1  -  1 )  e.  Z
4741, 46syl6eqel 2531 . 2  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
48 lgslem1 22635 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  e.  { 0 ,  2 } )
49 elpri 3897 . . . 4  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  e.  { 0 ,  2 }  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  0  \/  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  2 ) )
50 oveq1 6098 . . . . . 6  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  0  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  =  ( 0  -  1 ) )
51 df-neg 9598 . . . . . . 7  |-  -u 1  =  ( 0  -  1 )
5244simp1i 997 . . . . . . 7  |-  -u 1  e.  Z
5351, 52eqeltrri 2514 . . . . . 6  |-  ( 0  -  1 )  e.  Z
5450, 53syl6eqel 2531 . . . . 5  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  0  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
55 oveq1 6098 . . . . . 6  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  2  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  =  ( 2  -  1 ) )
56 2m1e1 10436 . . . . . . 7  |-  ( 2  -  1 )  =  1
5744simp3i 999 . . . . . . 7  |-  1  e.  Z
5856, 57eqeltri 2513 . . . . . 6  |-  ( 2  -  1 )  e.  Z
5955, 58syl6eqel 2531 . . . . 5  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  2  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
6054, 59jaoi 379 . . . 4  |-  ( ( ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  0  \/  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  2 )  ->  ( ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  - 
1 )  e.  Z
)
6148, 49, 603syl 20 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
62613expa 1187 . 2  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
6347, 62pm2.61dan 789 1  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  -  1 )  e.  Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2719    \ cdif 3325   {csn 3877   {cpr 3879   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    < clt 9418    <_ cle 9419    - cmin 9595   -ucneg 9596    / cdiv 9993   NNcn 10322   2c2 10371   NN0cn0 10579   ZZcz 10646   ZZ>=cuz 10861   RR+crp 10991    mod cmo 11708   ^cexp 11865   abscabs 12723    || cdivides 13535   Primecprime 13763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-prm 13764  df-phi 13841
This theorem is referenced by:  lgsfcl2  22641
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