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Theorem lgslem1 23688
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 3540 . . . . . . . . 9  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
213ad2ant2 1016 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  Prime )
3 prmnn 14222 . . . . . . . 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 994 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  A  e.  ZZ )
6 prmz 14223 . . . . . . . . . 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 14160 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( A  gcd  P
)  =  ( P  gcd  A ) )
95, 7, 8syl2anc 659 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  ( P  gcd  A
) )
10 simp3 996 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  -.  P  ||  A )
11 coprm 14243 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
122, 5, 11syl2anc 659 . . . . . . . . 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 2423 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  1 )
15 eulerth 14315 . . . . . . 7  |-  ( ( P  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
164, 5, 14, 15syl3anc 1226 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
17 phiprm 14309 . . . . . . . . . 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 10754 . . . . . . . . . 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 2470 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( phi `  P )  e. 
NN0 )
22 zexpcl 12084 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( phi `  P )  e.  NN0 )  -> 
( A ^ ( phi `  P ) )  e.  ZZ )
235, 21, 22syl2anc 659 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  e.  ZZ )
24 1zzd 10812 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  1  e.  ZZ )
25 moddvds 13995 . . . . . . 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 1226 . . . . . 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 10771 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( P  -  1 )  e.  CC )
29 2cnd 10525 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  CC )
30 2ne0 10545 . . . . . . . . . . . . 13  |-  2  =/=  0
3130a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  =/=  0 )
3228, 29, 31divcan1d 10238 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( P  - 
1 )  /  2
)  x.  2 )  =  ( P  - 
1 ) )
3318, 32eqtr4d 2426 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( ( ( P  -  1 )  / 
2 )  x.  2 ) )
3433oveq2d 6212 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( A ^ (
( ( P  - 
1 )  /  2
)  x.  2 ) ) )
355zcnd 10885 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  A  e.  CC )
36 2nn0 10729 . . . . . . . . . . 11  |-  2  e.  NN0
3736a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  NN0 )
38 oddprm 14341 . . . . . . . . . . . 12  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
39383ad2ant2 1016 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN )
4039nnnn0d 10769 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( P  -  1 )  /  2 )  e.  NN0 )
4135, 37, 40expmuld 12215 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( ( P  -  1 )  /  2 )  x.  2 ) )  =  ( ( A ^
( ( P  - 
1 )  /  2
) ) ^ 2 ) )
4234, 41eqtrd 2423 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( ( A ^
( ( P  - 
1 )  /  2
) ) ^ 2 ) )
4342oveq1d 6211 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  1 ) )
44 sq1 12165 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
4544oveq2i 6207 . . . . . . 7  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) ) ^ 2 )  -  ( 1 ^ 2 ) )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  1 )
4643, 45syl6eqr 2441 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) ) ^
2 )  -  (
1 ^ 2 ) ) )
47 zexpcl 12084 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( ( P  - 
1 )  /  2
)  e.  NN0 )  ->  ( A ^ (
( P  -  1 )  /  2 ) )  e.  ZZ )
485, 40, 47syl2anc 659 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  ZZ )
4948zcnd 10885 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  CC )
50 ax-1cn 9461 . . . . . . 7  |-  1  e.  CC
51 subsq 12178 . . . . . . 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 660 . . . . . 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 2423 . . . . 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 4391 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  ||  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  x.  (
( A ^ (
( P  -  1 )  /  2 ) )  -  1 ) ) )
5548peano2zd 10887 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  e.  ZZ )
56 peano2zm 10824 . . . . . 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 14251 . . . . 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 1226 . . . 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 13992 . . . . 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 659 . . . 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 10813 . . . . . . 7  |-  2  e.  ZZ
6463a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  ZZ )
65 moddvds 13995 . . . . . 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 1226 . . . . 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 10522 . . . . . . . 8  |-  2  e.  RR
6867a1i 11 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  e.  RR )
694nnrpd 11175 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  RR+ )
70 0le2 10543 . . . . . . . 8  |-  0  <_  2
7170a1i 11 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  0  <_  2 )
72 prmuz2 14237 . . . . . . . . . 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 11013 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  2  <_  P )
7573, 74syl 16 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  2  <_  P )
76 eldifsni 4070 . . . . . . . . 9  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
77763ad2ant2 1016 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  =/=  2 )
784nnred 10467 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  P  e.  RR )
7968, 78ltlend 9641 . . . . . . . 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 11921 . . . . . . 7  |-  ( ( ( 2  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  2  /\  2  <  P ) )  ->  ( 2  mod  P )  =  2 )
8268, 69, 71, 80, 81syl22anc 1227 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
2  mod  P )  =  2 )
8382eqeq2d 2396 . . . . 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 10511 . . . . . . . 8  |-  2  =  ( 1  +  1 )
8584oveq2i 6207 . . . . . . 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 9882 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  -  ( 1  +  1 ) )  =  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 ) )
8885, 87syl5eq 2435 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  -.  P  ||  A )  ->  (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  -  2 )  =  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  - 
1 ) )
8988breq2d 4379 . . . . 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 707 . . 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 6224 . . 3  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  e. 
_V
9493elpr 3962 . 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 366    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577    \ cdif 3386   {csn 3944   {cpr 3946   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    x. cmul 9408    < clt 9539    <_ cle 9540    - cmin 9718    / cdiv 10123   NNcn 10452   2c2 10502   NN0cn0 10712   ZZcz 10781   ZZ>=cuz 11001   RR+crp 11139    mod cmo 11896   ^cexp 12069    || cdvds 13988    gcd cgcd 14146   Primecprime 14219   phicphi 14296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-dvds 13989  df-gcd 14147  df-prm 14220  df-phi 14298
This theorem is referenced by:  lgslem4  23691
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