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Theorem lgslem4 20554
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 730 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  A  e.  ZZ )
2 oddprm 12884 . . . . . . . . . . 11  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
32ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN )
43nnnn0d 10034 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN0 )
5 zexpcl 11134 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( ( P  - 
1 )  /  2
)  e.  NN0 )  ->  ( A ^ (
( P  -  1 )  /  2 ) )  e.  ZZ )
61, 4, 5syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  ZZ )
76zred 10133 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  RR )
8 0re 8854 . . . . . . . 8  |-  0  e.  RR
98a1i 10 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  0  e.  RR )
10 1re 8853 . . . . . . . 8  |-  1  e.  RR
1110a1i 10 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  1  e.  RR )
12 eldifi 3311 . . . . . . . . . . . 12  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
1312ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  Prime )
14 prmuz2 12792 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
1513, 14syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  ( ZZ>= `  2 )
)
16 eluz2b2 10306 . . . . . . . . . 10  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
1715, 16sylib 188 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( P  e.  NN  /\  1  <  P ) )
1817simpld 445 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  NN )
1918nnrpd 10405 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  RR+ )
20 0z 10051 . . . . . . . . . 10  |-  0  e.  ZZ
2120a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  0  e.  ZZ )
22 simpr 447 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  ||  A )
23 dvdsval3 12551 . . . . . . . . . . . 12  |-  ( ( P  e.  NN  /\  A  e.  ZZ )  ->  ( P  ||  A  <->  ( A  mod  P )  =  0 ) )
2418, 1, 23syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( P  ||  A  <->  ( A  mod  P )  =  0 ) )
2522, 24mpbid 201 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A  mod  P )  =  0 )
26 0mod 11011 . . . . . . . . . . 11  |-  ( P  e.  RR+  ->  ( 0  mod  P )  =  0 )
2719, 26syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
0  mod  P )  =  0 )
2825, 27eqtr4d 2331 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  ( A  mod  P )  =  ( 0  mod  P
) )
29 modexp 11252 . . . . . . . . 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 ) )
301, 21, 4, 19, 28, 29syl221anc 1193 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( ( 0 ^ ( ( P  -  1 )  / 
2 ) )  mod 
P ) )
3130expd 11277 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
0 ^ ( ( P  -  1 )  /  2 ) )  =  0 )
3231oveq1d 5889 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( 0 ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( 0  mod 
P ) )
3330, 32eqtrd 2328 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( 0  mod 
P ) )
34 modadd1 11017 . . . . . . 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 ) )
357, 9, 11, 19, 33, 34syl221anc 1193 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  ( ( 0  +  1 )  mod 
P ) )
36 0p1e1 9855 . . . . . . 7  |-  ( 0  +  1 )  =  1
3736oveq1i 5884 . . . . . 6  |-  ( ( 0  +  1 )  mod  P )  =  ( 1  mod  P
)
3835, 37syl6eq 2344 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  ( 1  mod 
P ) )
3918nnred 9777 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  P  e.  RR )
4017simprd 449 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  1  <  P )
41 1mod 11012 . . . . . 6  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
4239, 40, 41syl2anc 642 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
1  mod  P )  =  1 )
4338, 42eqtrd 2328 . . . 4  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  1 )
4443oveq1d 5889 . . 3  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  =  ( 1  -  1 ) )
45 1m1e0 9830 . . . 4  |-  ( 1  -  1 )  =  0
46 lgslem2.z . . . . . 6  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
4746lgslem2 20552 . . . . 5  |-  ( -u
1  e.  Z  /\  0  e.  Z  /\  1  e.  Z )
4847simp2i 965 . . . 4  |-  0  e.  Z
4945, 48eqeltri 2366 . . 3  |-  ( 1  -  1 )  e.  Z
5044, 49syl6eqel 2384 . 2  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
51 lgslem1 20551 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  e.  { 0 ,  2 } )
52 elpri 3673 . . . 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 ) )
53 oveq1 5881 . . . . . 6  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  0  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  =  ( 0  -  1 ) )
54 df-neg 9056 . . . . . . 7  |-  -u 1  =  ( 0  -  1 )
5547simp1i 964 . . . . . . 7  |-  -u 1  e.  Z
5654, 55eqeltrri 2367 . . . . . 6  |-  ( 0  -  1 )  e.  Z
5753, 56syl6eqel 2384 . . . . 5  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  0  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
58 oveq1 5881 . . . . . 6  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  2  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  =  ( 2  -  1 ) )
59 2cn 9832 . . . . . . . 8  |-  2  e.  CC
60 ax-1cn 8811 . . . . . . . 8  |-  1  e.  CC
61 1p1e2 9856 . . . . . . . 8  |-  ( 1  +  1 )  =  2
6259, 60, 60, 61subaddrii 9151 . . . . . . 7  |-  ( 2  -  1 )  =  1
6347simp3i 966 . . . . . . 7  |-  1  e.  Z
6462, 63eqeltri 2366 . . . . . 6  |-  ( 2  -  1 )  e.  Z
6558, 64syl6eqel 2384 . . . . 5  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  =  2  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
6657, 65jaoi 368 . . . 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
)
6751, 52, 663syl 18 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
68673expa 1151 . 2  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  e.  Z )
6950, 68pm2.61dan 766 1  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  -  1 )  e.  Z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560    \ cdif 3162   {csn 3653   {cpr 3654   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370    mod cmo 10989   ^cexp 11120   abscabs 11735    || cdivides 12547   Primecprime 12774
This theorem is referenced by:  lgsfcl2  20557
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-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
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