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Theorem lgslem1 21033
Description: When  a is coprime to the prime  p,  a ^ ( ( p  -  1 )  / 
2 ) is equivalent  mod  p to  1 or  -u
1, and so adding  1 makes it equivalent to  0 or  2. (Contributed by Mario Carneiro, 4-Feb-2015.)
Assertion
Ref Expression
lgslem1  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  e.  { 0 ,  2 } )

Proof of Theorem lgslem1
StepHypRef Expression
1 eldifi 3429 . . . . . . . . 9  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
213ad2ant2 979 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  Prime )
3 prmnn 13037 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
42, 3syl 16 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  NN )
5 simp1 957 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  A  e.  ZZ )
6 prmz 13038 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
72, 6syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  ZZ )
8 gcdcom 12975 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( A  gcd  P
)  =  ( P  gcd  A ) )
95, 7, 8syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  ( P  gcd  A
) )
10 simp3 959 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  -.  P  ||  A )
11 coprm 13055 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
122, 5, 11syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
1310, 12mpbid 202 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  gcd  A )  =  1 )
149, 13eqtrd 2436 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  1 )
15 eulerth 13127 . . . . . . 7  |-  ( ( P  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
164, 5, 14, 15syl3anc 1184 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
17 phiprm 13121 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1 ) )
182, 17syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( P  -  1 ) )
19 nnm1nn0 10217 . . . . . . . . . 10  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
204, 19syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  -  1 )  e.  NN0 )
2118, 20eqeltrd 2478 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( phi `  P )  e. 
NN0 )
22 zexpcl 11351 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( phi `  P )  e.  NN0 )  -> 
( A ^ ( phi `  P ) )  e.  ZZ )
235, 21, 22syl2anc 643 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  e.  ZZ )
24 1z 10267 . . . . . . . 8  |-  1  e.  ZZ
2524a1i 11 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  1  e.  ZZ )
26 moddvds 12814 . . . . . . 7  |-  ( ( P  e.  NN  /\  ( A ^ ( phi `  P ) )  e.  ZZ  /\  1  e.  ZZ )  ->  (
( ( A ^
( phi `  P
) )  mod  P
)  =  ( 1  mod  P )  <->  P  ||  (
( A ^ ( phi `  P ) )  -  1 ) ) )
274, 23, 25, 26syl3anc 1184 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( phi `  P
) )  mod  P
)  =  ( 1  mod  P )  <->  P  ||  (
( A ^ ( phi `  P ) )  -  1 ) ) )
2816, 27mpbid 202 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  ||  ( ( A ^
( phi `  P
) )  -  1 ) )
2920nn0cnd 10232 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  -  1 )  e.  CC )
30 2cn 10026 . . . . . . . . . . . . 13  |-  2  e.  CC
3130a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  CC )
32 2ne0 10039 . . . . . . . . . . . . 13  |-  2  =/=  0
3332a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  =/=  0 )
3429, 31, 33divcan1d 9747 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( P  - 
1 )  /  2
)  x.  2 )  =  ( P  - 
1 ) )
3518, 34eqtr4d 2439 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( ( ( P  -  1 )  / 
2 )  x.  2 ) )
3635oveq2d 6056 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( A ^ (
( ( P  - 
1 )  /  2
)  x.  2 ) ) )
375zcnd 10332 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  A  e.  CC )
38 2nn0 10194 . . . . . . . . . . 11  |-  2  e.  NN0
3938a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  NN0 )
40 oddprm 13144 . . . . . . . . . . . 12  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
41403ad2ant2 979 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN )
4241nnnn0d 10230 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN0 )
4337, 39, 42expmuld 11481 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( ( P  -  1 )  /  2 )  x.  2 ) )  =  ( ( A ^
( ( P  - 
1 )  /  2
) ) ^ 2 ) )
4436, 43eqtrd 2436 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( ( A ^
( ( P  - 
1 )  /  2
) ) ^ 2 ) )
4544oveq1d 6055 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  1 ) )
46 sq1 11431 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
4746oveq2i 6051 . . . . . . 7  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) ) ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  1 )
4845, 47syl6eqr 2454 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  (
1 ^ 2 ) ) )
49 zexpcl 11351 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( ( P  - 
1 )  /  2
)  e.  NN0 )  ->  ( A ^ (
( P  -  1 )  /  2 ) )  e.  ZZ )
505, 42, 49syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  ZZ )
5150zcnd 10332 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  CC )
52 ax-1cn 9004 . . . . . . 7  |-  1  e.  CC
53 subsq 11443 . . . . . . 7  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  e.  CC  /\  1  e.  CC )  ->  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  (
1 ^ 2 ) )  =  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  x.  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 ) ) )
5451, 52, 53sylancl 644 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) ) ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( ( A ^ ( ( P  -  1 )  /  2 ) )  +  1 )  x.  ( ( A ^
( ( P  - 
1 )  /  2
) )  -  1 ) ) )
5548, 54eqtrd 2436 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  x.  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 ) ) )
5628, 55breqtrd 4196 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  ||  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  x.  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 ) ) )
5750peano2zd 10334 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  e.  ZZ )
58 peano2zm 10276 . . . . . 6  |-  ( ( A ^ ( ( P  -  1 )  /  2 ) )  e.  ZZ  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 )  e.  ZZ )
5950, 58syl 16 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 )  e.  ZZ )
60 euclemma 13063 . . . . 5  |-  ( ( P  e.  Prime  /\  (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  e.  ZZ  /\  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 )  e.  ZZ )  -> 
( P  ||  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  x.  ( ( A ^ ( ( P  -  1 )  /  2 ) )  -  1 ) )  <-> 
( P  ||  (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  \/  P  ||  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 ) ) ) )
612, 57, 59, 60syl3anc 1184 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  ||  ( ( ( A ^ ( ( P  -  1 )  /  2 ) )  +  1 )  x.  ( ( A ^
( ( P  - 
1 )  /  2
) )  -  1 ) )  <->  ( P  ||  ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  \/  P  ||  ( ( A ^
( ( P  - 
1 )  /  2
) )  -  1 ) ) ) )
6256, 61mpbid 202 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  \/  P  ||  ( ( A ^
( ( P  - 
1 )  /  2
) )  -  1 ) ) )
63 dvdsval3 12811 . . . . 5  |-  ( ( P  e.  NN  /\  ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  e.  ZZ )  ->  ( P  ||  ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  <->  ( ( ( A ^ ( ( P  -  1 )  /  2 ) )  +  1 )  mod 
P )  =  0 ) )
644, 57, 63syl2anc 643 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  <->  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  =  0 ) )
65 2z 10268 . . . . . . 7  |-  2  e.  ZZ
6665a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  ZZ )
67 moddvds 12814 . . . . . 6  |-  ( ( P  e.  NN  /\  ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( ( ( A ^ ( ( P  -  1 )  /  2 ) )  +  1 )  mod 
P )  =  ( 2  mod  P )  <-> 
P  ||  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  -  2 ) ) )
684, 57, 66, 67syl3anc 1184 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  ( 2  mod  P )  <->  P  ||  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  -  2 ) ) )
69 2re 10025 . . . . . . . 8  |-  2  e.  RR
7069a1i 11 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  RR )
714nnrpd 10603 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  RR+ )
7238nn0ge0i 10205 . . . . . . . 8  |-  0  <_  2
7372a1i 11 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  0  <_  2 )
74 prmuz2 13052 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
752, 74syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  ( ZZ>= `  2 )
)
76 eluzle 10454 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  2  <_  P )
7775, 76syl 16 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  <_  P )
78 eldifsni 3888 . . . . . . . . 9  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
79783ad2ant2 979 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  =/=  2 )
804nnred 9971 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  RR )
8170, 80ltlend 9174 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
2  <  P  <->  ( 2  <_  P  /\  P  =/=  2 ) ) )
8277, 79, 81mpbir2and 889 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  <  P )
83 modid 11225 . . . . . . 7  |-  ( ( ( 2  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  2  /\  2  <  P ) )  ->  ( 2  mod  P )  =  2 )
8470, 71, 73, 82, 83syl22anc 1185 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
2  mod  P )  =  2 )
8584eqeq2d 2415 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  ( 2  mod  P )  <->  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  =  2 ) )
86 df-2 10014 . . . . . . . 8  |-  2  =  ( 1  +  1 )
8786oveq2i 6051 . . . . . . 7  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  -  2 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  -  (
1  +  1 ) )
8852a1i 11 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  1  e.  CC )
8951, 88, 88pnpcan2d 9405 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  -  ( 1  +  1 ) )  =  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 ) )
9087, 89syl5eq 2448 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  -  2 )  =  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 ) )
9190breq2d 4184 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  ||  ( ( ( A ^ ( ( P  -  1 )  /  2 ) )  +  1 )  - 
2 )  <->  P  ||  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 ) ) )
9268, 85, 913bitr3rd 276 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 )  <->  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  =  2 ) )
9364, 92orbi12d 691 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( P  ||  (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  \/  P  ||  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 ) )  <->  ( ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  =  0  \/  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  =  2 ) ) )
9462, 93mpbid 202 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  0  \/  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  2 ) )
95 ovex 6065 . . 3  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  e. 
_V
9695elpr 3792 . 2  |-  ( ( ( ( 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 ) )
9794, 96sylibr 204 1  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  e.  { 0 ,  2 } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277   {csn 3774   {cpr 3775   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568    mod cmo 11205   ^cexp 11337    || cdivides 12807    gcd cgcd 12961   Primecprime 13034   phicphi 13108
This theorem is referenced by:  lgslem4  21036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-phi 13110
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