Step |
Hyp |
Ref |
Expression |
1 |
|
elfzelz |
|- ( N e. ( 1 ... ( P - 1 ) ) -> N e. ZZ ) |
2 |
1
|
adantl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ZZ ) |
3 |
|
peano2zm |
|- ( N e. ZZ -> ( N - 1 ) e. ZZ ) |
4 |
2 3
|
syl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - 1 ) e. ZZ ) |
5 |
4
|
zcnd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - 1 ) e. CC ) |
6 |
2
|
peano2zd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. ZZ ) |
7 |
6
|
zcnd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. CC ) |
8 |
5 7
|
mulcomd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) ) |
9 |
2
|
zcnd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. CC ) |
10 |
|
ax-1cn |
|- 1 e. CC |
11 |
|
subsq |
|- ( ( N e. CC /\ 1 e. CC ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) ) |
12 |
9 10 11
|
sylancl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) ) |
13 |
9
|
sqvald |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N ^ 2 ) = ( N x. N ) ) |
14 |
|
sq1 |
|- ( 1 ^ 2 ) = 1 |
15 |
14
|
a1i |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 ^ 2 ) = 1 ) |
16 |
13 15
|
oveq12d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N x. N ) - 1 ) ) |
17 |
8 12 16
|
3eqtr2d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N x. N ) - 1 ) ) |
18 |
17
|
breq2d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> P || ( ( N x. N ) - 1 ) ) ) |
19 |
|
fz1ssfz0 |
|- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) ) |
20 |
|
simpr |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ( 1 ... ( P - 1 ) ) ) |
21 |
19 20
|
sselid |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ( 0 ... ( P - 1 ) ) ) |
22 |
21
|
biantrurd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N x. N ) - 1 ) <-> ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) ) ) |
23 |
18 22
|
bitrd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) ) ) |
24 |
|
simpl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. Prime ) |
25 |
|
euclemma |
|- ( ( P e. Prime /\ ( N - 1 ) e. ZZ /\ ( N + 1 ) e. ZZ ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) ) |
26 |
24 4 6 25
|
syl3anc |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) ) |
27 |
|
prmnn |
|- ( P e. Prime -> P e. NN ) |
28 |
|
fzm1ndvds |
|- ( ( P e. NN /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N ) |
29 |
27 28
|
sylan |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N ) |
30 |
|
eqid |
|- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P ) |
31 |
30
|
prmdiveq |
|- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) <-> N = ( ( N ^ ( P - 2 ) ) mod P ) ) ) |
32 |
24 2 29 31
|
syl3anc |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) <-> N = ( ( N ^ ( P - 2 ) ) mod P ) ) ) |
33 |
23 26 32
|
3bitr3rd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N = ( ( N ^ ( P - 2 ) ) mod P ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) ) |
34 |
24 27
|
syl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. NN ) |
35 |
|
1zzd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 1 e. ZZ ) |
36 |
|
moddvds |
|- ( ( P e. NN /\ N e. ZZ /\ 1 e. ZZ ) -> ( ( N mod P ) = ( 1 mod P ) <-> P || ( N - 1 ) ) ) |
37 |
34 2 35 36
|
syl3anc |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( 1 mod P ) <-> P || ( N - 1 ) ) ) |
38 |
|
elfznn |
|- ( N e. ( 1 ... ( P - 1 ) ) -> N e. NN ) |
39 |
38
|
adantl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. NN ) |
40 |
39
|
nnred |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. RR ) |
41 |
34
|
nnrpd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. RR+ ) |
42 |
39
|
nnnn0d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. NN0 ) |
43 |
42
|
nn0ge0d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ N ) |
44 |
|
elfzle2 |
|- ( N e. ( 1 ... ( P - 1 ) ) -> N <_ ( P - 1 ) ) |
45 |
44
|
adantl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N <_ ( P - 1 ) ) |
46 |
|
prmz |
|- ( P e. Prime -> P e. ZZ ) |
47 |
|
zltlem1 |
|- ( ( N e. ZZ /\ P e. ZZ ) -> ( N < P <-> N <_ ( P - 1 ) ) ) |
48 |
1 46 47
|
syl2anr |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N < P <-> N <_ ( P - 1 ) ) ) |
49 |
45 48
|
mpbird |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N < P ) |
50 |
|
modid |
|- ( ( ( N e. RR /\ P e. RR+ ) /\ ( 0 <_ N /\ N < P ) ) -> ( N mod P ) = N ) |
51 |
40 41 43 49 50
|
syl22anc |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N mod P ) = N ) |
52 |
34
|
nnred |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. RR ) |
53 |
|
prmuz2 |
|- ( P e. Prime -> P e. ( ZZ>= ` 2 ) ) |
54 |
24 53
|
syl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. ( ZZ>= ` 2 ) ) |
55 |
|
eluz2gt1 |
|- ( P e. ( ZZ>= ` 2 ) -> 1 < P ) |
56 |
54 55
|
syl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 1 < P ) |
57 |
|
1mod |
|- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 ) |
58 |
52 56 57
|
syl2anc |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 mod P ) = 1 ) |
59 |
51 58
|
eqeq12d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( 1 mod P ) <-> N = 1 ) ) |
60 |
37 59
|
bitr3d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( N - 1 ) <-> N = 1 ) ) |
61 |
35
|
znegcld |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. ZZ ) |
62 |
|
moddvds |
|- ( ( P e. NN /\ N e. ZZ /\ -u 1 e. ZZ ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> P || ( N - -u 1 ) ) ) |
63 |
34 2 61 62
|
syl3anc |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> P || ( N - -u 1 ) ) ) |
64 |
34
|
nncnd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. CC ) |
65 |
64
|
mulid2d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 x. P ) = P ) |
66 |
65
|
oveq2d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( -u 1 + P ) ) |
67 |
|
neg1cn |
|- -u 1 e. CC |
68 |
|
addcom |
|- ( ( -u 1 e. CC /\ P e. CC ) -> ( -u 1 + P ) = ( P + -u 1 ) ) |
69 |
67 64 68
|
sylancr |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + P ) = ( P + -u 1 ) ) |
70 |
|
negsub |
|- ( ( P e. CC /\ 1 e. CC ) -> ( P + -u 1 ) = ( P - 1 ) ) |
71 |
64 10 70
|
sylancl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P + -u 1 ) = ( P - 1 ) ) |
72 |
66 69 71
|
3eqtrd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( P - 1 ) ) |
73 |
72
|
oveq1d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( ( P - 1 ) mod P ) ) |
74 |
|
neg1rr |
|- -u 1 e. RR |
75 |
74
|
a1i |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. RR ) |
76 |
|
modcyc |
|- ( ( -u 1 e. RR /\ P e. RR+ /\ 1 e. ZZ ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( -u 1 mod P ) ) |
77 |
75 41 35 76
|
syl3anc |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( -u 1 mod P ) ) |
78 |
|
peano2rem |
|- ( P e. RR -> ( P - 1 ) e. RR ) |
79 |
52 78
|
syl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) e. RR ) |
80 |
|
nnm1nn0 |
|- ( P e. NN -> ( P - 1 ) e. NN0 ) |
81 |
34 80
|
syl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) e. NN0 ) |
82 |
81
|
nn0ge0d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ ( P - 1 ) ) |
83 |
52
|
ltm1d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) < P ) |
84 |
|
modid |
|- ( ( ( ( P - 1 ) e. RR /\ P e. RR+ ) /\ ( 0 <_ ( P - 1 ) /\ ( P - 1 ) < P ) ) -> ( ( P - 1 ) mod P ) = ( P - 1 ) ) |
85 |
79 41 82 83 84
|
syl22anc |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( P - 1 ) mod P ) = ( P - 1 ) ) |
86 |
73 77 85
|
3eqtr3d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 mod P ) = ( P - 1 ) ) |
87 |
51 86
|
eqeq12d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> N = ( P - 1 ) ) ) |
88 |
|
subneg |
|- ( ( N e. CC /\ 1 e. CC ) -> ( N - -u 1 ) = ( N + 1 ) ) |
89 |
9 10 88
|
sylancl |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - -u 1 ) = ( N + 1 ) ) |
90 |
89
|
breq2d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( N - -u 1 ) <-> P || ( N + 1 ) ) ) |
91 |
63 87 90
|
3bitr3rd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( N + 1 ) <-> N = ( P - 1 ) ) ) |
92 |
60 91
|
orbi12d |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( P || ( N - 1 ) \/ P || ( N + 1 ) ) <-> ( N = 1 \/ N = ( P - 1 ) ) ) ) |
93 |
33 92
|
bitrd |
|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N = ( ( N ^ ( P - 2 ) ) mod P ) <-> ( N = 1 \/ N = ( P - 1 ) ) ) ) |