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Theorem lgslem1 23327
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 3626 . . . . . . . . 9  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
213ad2ant2 1018 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  Prime )
3 prmnn 14079 . . . . . . . 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 996 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  A  e.  ZZ )
6 prmz 14080 . . . . . . . . . 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 14017 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( A  gcd  P
)  =  ( P  gcd  A ) )
95, 7, 8syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  ( P  gcd  A
) )
10 simp3 998 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  -.  P  ||  A )
11 coprm 14100 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
122, 5, 11syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
1310, 12mpbid 210 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  gcd  A )  =  1 )
149, 13eqtrd 2508 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  1 )
15 eulerth 14172 . . . . . . 7  |-  ( ( P  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
164, 5, 14, 15syl3anc 1228 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
17 phiprm 14166 . . . . . . . . . 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 10837 . . . . . . . . . 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 2555 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( phi `  P )  e. 
NN0 )
22 zexpcl 12149 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( phi `  P )  e.  NN0 )  -> 
( A ^ ( phi `  P ) )  e.  ZZ )
235, 21, 22syl2anc 661 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  e.  ZZ )
24 1zzd 10895 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  1  e.  ZZ )
25 moddvds 13854 . . . . . . 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 ) ) )
264, 23, 24, 25syl3anc 1228 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( phi `  P
) )  mod  P
)  =  ( 1  mod  P )  <->  P  ||  (
( A ^ ( phi `  P ) )  -  1 ) ) )
2716, 26mpbid 210 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  ||  ( ( A ^
( phi `  P
) )  -  1 ) )
2820nn0cnd 10854 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  -  1 )  e.  CC )
29 2cnd 10608 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  CC )
30 2ne0 10628 . . . . . . . . . . . . 13  |-  2  =/=  0
3130a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  =/=  0 )
3228, 29, 31divcan1d 10321 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( P  - 
1 )  /  2
)  x.  2 )  =  ( P  - 
1 ) )
3318, 32eqtr4d 2511 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( ( ( P  -  1 )  / 
2 )  x.  2 ) )
3433oveq2d 6300 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( A ^ (
( ( P  - 
1 )  /  2
)  x.  2 ) ) )
355zcnd 10967 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  A  e.  CC )
36 2nn0 10812 . . . . . . . . . . 11  |-  2  e.  NN0
3736a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  NN0 )
38 oddprm 14198 . . . . . . . . . . . 12  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
39383ad2ant2 1018 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN )
4039nnnn0d 10852 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN0 )
4135, 37, 40expmuld 12281 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( ( P  -  1 )  /  2 )  x.  2 ) )  =  ( ( A ^
( ( P  - 
1 )  /  2
) ) ^ 2 ) )
4234, 41eqtrd 2508 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( ( A ^
( ( P  - 
1 )  /  2
) ) ^ 2 ) )
4342oveq1d 6299 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  1 ) )
44 sq1 12230 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
4544oveq2i 6295 . . . . . . 7  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) ) ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  1 )
4643, 45syl6eqr 2526 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  (
1 ^ 2 ) ) )
47 zexpcl 12149 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( ( P  - 
1 )  /  2
)  e.  NN0 )  ->  ( A ^ (
( P  -  1 )  /  2 ) )  e.  ZZ )
485, 40, 47syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  ZZ )
4948zcnd 10967 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  CC )
50 ax-1cn 9550 . . . . . . 7  |-  1  e.  CC
51 subsq 12243 . . . . . . 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 ) ) )
5249, 50, 51sylancl 662 . . . . . 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 ) ) )
5346, 52eqtrd 2508 . . . . 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 ) ) )
5427, 53breqtrd 4471 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  ||  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  x.  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 ) ) )
5548peano2zd 10969 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  e.  ZZ )
56 peano2zm 10906 . . . . . 6  |-  ( ( A ^ ( ( P  -  1 )  /  2 ) )  e.  ZZ  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 )  e.  ZZ )
5748, 56syl 16 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 )  e.  ZZ )
58 euclemma 14108 . . . . 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 ) ) ) )
592, 55, 57, 58syl3anc 1228 . . . 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 ) ) ) )
6054, 59mpbid 210 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  \/  P  ||  ( ( A ^
( ( P  - 
1 )  /  2
) )  -  1 ) ) )
61 dvdsval3 13851 . . . . 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 ) )
624, 55, 61syl2anc 661 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  <->  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  =  0 ) )
63 2z 10896 . . . . . . 7  |-  2  e.  ZZ
6463a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  ZZ )
65 moddvds 13854 . . . . . 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 ) ) )
664, 55, 64, 65syl3anc 1228 . . . . 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 ) ) )
67 2re 10605 . . . . . . . 8  |-  2  e.  RR
6867a1i 11 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  RR )
694nnrpd 11255 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  RR+ )
70 0le2 10626 . . . . . . . 8  |-  0  <_  2
7170a1i 11 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  0  <_  2 )
72 prmuz2 14094 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
732, 72syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  ( ZZ>= `  2 )
)
74 eluzle 11094 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  2  <_  P )
7573, 74syl 16 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  <_  P )
76 eldifsni 4153 . . . . . . . . 9  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
77763ad2ant2 1018 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  =/=  2 )
784nnred 10551 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  RR )
7968, 78ltlend 9729 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
2  <  P  <->  ( 2  <_  P  /\  P  =/=  2 ) ) )
8075, 77, 79mpbir2and 920 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  <  P )
81 modid 11988 . . . . . . 7  |-  ( ( ( 2  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  2  /\  2  <  P ) )  ->  ( 2  mod  P )  =  2 )
8268, 69, 71, 80, 81syl22anc 1229 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
2  mod  P )  =  2 )
8382eqeq2d 2481 . . . . 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 ) )
84 df-2 10594 . . . . . . . 8  |-  2  =  ( 1  +  1 )
8584oveq2i 6295 . . . . . . 7  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  -  2 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  -  (
1  +  1 ) )
8650a1i 11 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  1  e.  CC )
8749, 86, 86pnpcan2d 9968 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  -  ( 1  +  1 ) )  =  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 ) )
8885, 87syl5eq 2520 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  -  2 )  =  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 ) )
8988breq2d 4459 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  ||  ( ( ( A ^ ( ( P  -  1 )  /  2 ) )  +  1 )  - 
2 )  <->  P  ||  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 ) ) )
9066, 83, 893bitr3rd 284 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 )  <->  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  =  2 ) )
9162, 90orbi12d 709 . . 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 ) ) )
9260, 91mpbid 210 . 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 ) )
93 ovex 6309 . . 3  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  e. 
_V
9493elpr 4045 . 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 ) )
9592, 94sylibr 212 1  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  e.  { 0 ,  2 } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473   {csn 4027   {cpr 4029   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805    / cdiv 10206   NNcn 10536   2c2 10585   NN0cn0 10795   ZZcz 10864   ZZ>=cuz 11082   RR+crp 11220    mod cmo 11964   ^cexp 12134    || cdivides 13847    gcd cgcd 14003   Primecprime 14076   phicphi 14153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-gcd 14004  df-prm 14077  df-phi 14155
This theorem is referenced by:  lgslem4  23330
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