MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lgslem1 Unicode version

Theorem lgslem1 20551
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 3311 . . . . . . . . 9  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
213ad2ant2 977 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  Prime )
3 prmnn 12777 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
42, 3syl 15 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  NN )
5 simp1 955 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  A  e.  ZZ )
6 prmz 12778 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
72, 6syl 15 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  ZZ )
8 gcdcom 12715 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( A  gcd  P
)  =  ( P  gcd  A ) )
95, 7, 8syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  ( P  gcd  A
) )
10 simp3 957 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  -.  P  ||  A )
11 coprm 12795 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
122, 5, 11syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
1310, 12mpbid 201 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  gcd  A )  =  1 )
149, 13eqtrd 2328 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  1 )
15 eulerth 12867 . . . . . . 7  |-  ( ( P  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
164, 5, 14, 15syl3anc 1182 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
17 phiprm 12861 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1 ) )
182, 17syl 15 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( P  -  1 ) )
19 nnm1nn0 10021 . . . . . . . . . 10  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
204, 19syl 15 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  -  1 )  e.  NN0 )
2118, 20eqeltrd 2370 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( phi `  P )  e. 
NN0 )
22 zexpcl 11134 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( phi `  P )  e.  NN0 )  -> 
( A ^ ( phi `  P ) )  e.  ZZ )
235, 21, 22syl2anc 642 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  e.  ZZ )
24 1z 10069 . . . . . . . 8  |-  1  e.  ZZ
2524a1i 10 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  1  e.  ZZ )
26 moddvds 12554 . . . . . . 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 1182 . . . . . 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 201 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  ||  ( ( A ^
( phi `  P
) )  -  1 ) )
2920nn0cnd 10036 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  -  1 )  e.  CC )
30 2cn 9832 . . . . . . . . . . . . 13  |-  2  e.  CC
3130a1i 10 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  CC )
32 2ne0 9845 . . . . . . . . . . . . 13  |-  2  =/=  0
3332a1i 10 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  =/=  0 )
3429, 31, 33divcan1d 9553 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( P  - 
1 )  /  2
)  x.  2 )  =  ( P  - 
1 ) )
3518, 34eqtr4d 2331 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( ( ( P  -  1 )  / 
2 )  x.  2 ) )
3635oveq2d 5890 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( A ^ (
( ( P  - 
1 )  /  2
)  x.  2 ) ) )
375zcnd 10134 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  A  e.  CC )
38 2nn0 9998 . . . . . . . . . . 11  |-  2  e.  NN0
3938a1i 10 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  NN0 )
40 oddprm 12884 . . . . . . . . . . . 12  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
41403ad2ant2 977 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN )
4241nnnn0d 10034 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN0 )
4337, 39, 42expmuld 11264 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( ( P  -  1 )  /  2 )  x.  2 ) )  =  ( ( A ^
( ( P  - 
1 )  /  2
) ) ^ 2 ) )
4436, 43eqtrd 2328 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( ( A ^
( ( P  - 
1 )  /  2
) ) ^ 2 ) )
4544oveq1d 5889 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  1 ) )
46 sq1 11214 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
4746oveq2i 5885 . . . . . . 7  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) ) ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  1 )
4845, 47syl6eqr 2346 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  (
1 ^ 2 ) ) )
49 zexpcl 11134 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( ( P  - 
1 )  /  2
)  e.  NN0 )  ->  ( A ^ (
( P  -  1 )  /  2 ) )  e.  ZZ )
505, 42, 49syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  ZZ )
5150zcnd 10134 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  CC )
52 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
53 subsq 11226 . . . . . . 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 643 . . . . . 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 2328 . . . . 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 4063 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  ||  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  x.  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 ) ) )
5750peano2zd 10136 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  e.  ZZ )
58 peano2zm 10078 . . . . . 6  |-  ( ( A ^ ( ( P  -  1 )  /  2 ) )  e.  ZZ  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 )  e.  ZZ )
5950, 58syl 15 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 )  e.  ZZ )
60 euclemma 12803 . . . . 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 1182 . . . 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 201 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  \/  P  ||  ( ( A ^
( ( P  - 
1 )  /  2
) )  -  1 ) ) )
63 dvdsval3 12551 . . . . 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 642 . . . 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 10070 . . . . . . 7  |-  2  e.  ZZ
6665a1i 10 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  ZZ )
67 moddvds 12554 . . . . . 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 1182 . . . . 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 9831 . . . . . . . 8  |-  2  e.  RR
7069a1i 10 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  RR )
714nnrpd 10405 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  RR+ )
7238nn0ge0i 10009 . . . . . . . 8  |-  0  <_  2
7372a1i 10 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  0  <_  2 )
74 prmuz2 12792 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
752, 74syl 15 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  ( ZZ>= `  2 )
)
76 eluzle 10256 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  2  <_  P )
7775, 76syl 15 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  <_  P )
78 eldifsni 3763 . . . . . . . . 9  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
79783ad2ant2 977 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  =/=  2 )
804nnred 9777 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  RR )
8170, 80ltlend 8980 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
2  <  P  <->  ( 2  <_  P  /\  P  =/=  2 ) ) )
8277, 79, 81mpbir2and 888 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  <  P )
83 modid 11009 . . . . . . 7  |-  ( ( ( 2  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  2  /\  2  <  P ) )  ->  ( 2  mod  P )  =  2 )
8470, 71, 73, 82, 83syl22anc 1183 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
2  mod  P )  =  2 )
8584eqeq2d 2307 . . . . 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 9820 . . . . . . . 8  |-  2  =  ( 1  +  1 )
8786oveq2i 5885 . . . . . . 7  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  -  2 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  -  (
1  +  1 ) )
8852a1i 10 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  1  e.  CC )
8951, 88, 88pnpcan2d 9211 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  -  ( 1  +  1 ) )  =  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 ) )
9087, 89syl5eq 2340 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  -  2 )  =  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 ) )
9190breq2d 4051 . . . . 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 275 . . . 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 690 . . 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 201 . 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 5899 . . 3  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  e. 
_V
9695elpr 3671 . 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 203 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 176    \/ wo 357    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   {csn 3653   {cpr 3654   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370    mod cmo 10989   ^cexp 11120    || cdivides 12547    gcd cgcd 12701   Primecprime 12774   phicphi 12848
This theorem is referenced by:  lgslem4  20554
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
  Copyright terms: Public domain W3C validator