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Theorem lgslem4 24306
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 768 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  A  e.  ZZ )
2 oddprm 14844 . . . . . . . . . . 11  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
32ad2antlr 741 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN )
43nnnn0d 10949 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN0 )
5 zexpcl 12325 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( ( P  - 
1 )  /  2
)  e.  NN0 )  ->  ( A ^ (
( P  -  1 )  /  2 ) )  e.  ZZ )
61, 4, 5syl2anc 673 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  ZZ )
76zred 11063 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  RR )
8 0red 9662 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  0  e.  RR )
9 1red 9676 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  1  e.  RR )
10 eldifi 3544 . . . . . . . . . . . 12  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
1110ad2antlr 741 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  Prime )
12 prmuz2 14721 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
1311, 12syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  ( ZZ>= `  2 )
)
14 eluz2b2 11254 . . . . . . . . . 10  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
1513, 14sylib 201 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( P  e.  NN  /\  1  <  P ) )
1615simpld 466 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  NN )
1716nnrpd 11362 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  RR+ )
18 0zd 10973 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  0  e.  ZZ )
19 simpr 468 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  ||  A )
20 dvdsval3 14386 . . . . . . . . . . . 12  |-  ( ( P  e.  NN  /\  A  e.  ZZ )  ->  ( P  ||  A  <->  ( A  mod  P )  =  0 ) )
2116, 1, 20syl2anc 673 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( P  ||  A  <->  ( A  mod  P )  =  0 ) )
2219, 21mpbid 215 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A  mod  P )  =  0 )
23 0mod 12161 . . . . . . . . . . 11  |-  ( P  e.  RR+  ->  ( 0  mod  P )  =  0 )
2417, 23syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
0  mod  P )  =  0 )
2522, 24eqtr4d 2508 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A  mod  P )  =  ( 0  mod  P
) )
26 modexp 12445 . . . . . . . . 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 1303 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( ( 0 ^ ( ( P  -  1 )  / 
2 ) )  mod 
P ) )
2830expd 12470 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
0 ^ ( ( P  -  1 )  /  2 ) )  =  0 )
2928oveq1d 6323 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( 0 ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( 0  mod 
P ) )
3027, 29eqtrd 2505 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( 0  mod 
P ) )
31 modadd1 12167 . . . . . . 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 1303 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  ( ( 0  +  1 )  mod 
P ) )
33 0p1e1 10743 . . . . . . 7  |-  ( 0  +  1 )  =  1
3433oveq1i 6318 . . . . . 6  |-  ( ( 0  +  1 )  mod  P )  =  ( 1  mod  P
)
3532, 34syl6eq 2521 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  ( 1  mod 
P ) )
3616nnred 10646 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  RR )
3715simprd 470 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  1  <  P )
38 1mod 12162 . . . . . 6  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
3936, 37, 38syl2anc 673 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
1  mod  P )  =  1 )
4035, 39eqtrd 2505 . . . 4  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  1 )
4140oveq1d 6323 . . 3  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  =  ( 1  -  1 ) )
42 1m1e0 10700 . . . 4  |-  ( 1  -  1 )  =  0
43 lgslem2.z . . . . . 6  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
4443lgslem2 24304 . . . . 5  |-  ( -u
1  e.  Z  /\  0  e.  Z  /\  1  e.  Z )
4544simp2i 1040 . . . 4  |-  0  e.  Z
4642, 45eqeltri 2545 . . 3  |-  ( 1  -  1 )  e.  Z
4741, 46syl6eqel 2557 . 2  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
48 lgslem1 24303 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  e.  { 0 ,  2 } )
49 elpri 3976 . . . 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 6315 . . . . . 6  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  0  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  =  ( 0  -  1 ) )
51 df-neg 9883 . . . . . . 7  |-  -u 1  =  ( 0  -  1 )
5244simp1i 1039 . . . . . . 7  |-  -u 1  e.  Z
5351, 52eqeltrri 2546 . . . . . 6  |-  ( 0  -  1 )  e.  Z
5450, 53syl6eqel 2557 . . . . 5  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  0  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
55 oveq1 6315 . . . . . 6  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  2  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  =  ( 2  -  1 ) )
56 2m1e1 10746 . . . . . . 7  |-  ( 2  -  1 )  =  1
5744simp3i 1041 . . . . . . 7  |-  1  e.  Z
5856, 57eqeltri 2545 . . . . . 6  |-  ( 2  -  1 )  e.  Z
5955, 58syl6eqel 2557 . . . . 5  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  2  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
6054, 59jaoi 386 . . . 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 18 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
62613expa 1231 . 2  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
6347, 62pm2.61dan 808 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 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {crab 2760    \ cdif 3387   {csn 3959   {cpr 3961   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325    mod cmo 12129   ^cexp 12310   abscabs 13374    || cdvds 14382   Primecprime 14701
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-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
This theorem is referenced by:  lgsfcl2  24309
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