Step |
Hyp |
Ref |
Expression |
1 |
|
prmdiv.1 |
|- R = ( ( A ^ ( P - 2 ) ) mod P ) |
2 |
|
simpl1 |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. Prime ) |
3 |
|
prmz |
|- ( P e. Prime -> P e. ZZ ) |
4 |
2 3
|
syl |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. ZZ ) |
5 |
|
simpl2 |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> A e. ZZ ) |
6 |
|
elfzelz |
|- ( S e. ( 0 ... ( P - 1 ) ) -> S e. ZZ ) |
7 |
6
|
ad2antrl |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. ZZ ) |
8 |
5 7
|
zmulcld |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. S ) e. ZZ ) |
9 |
|
1z |
|- 1 e. ZZ |
10 |
|
zsubcl |
|- ( ( ( A x. S ) e. ZZ /\ 1 e. ZZ ) -> ( ( A x. S ) - 1 ) e. ZZ ) |
11 |
8 9 10
|
sylancl |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( A x. S ) - 1 ) e. ZZ ) |
12 |
1
|
prmdiv |
|- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) ) |
13 |
12
|
adantr |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) ) |
14 |
13
|
simpld |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> R e. ( 1 ... ( P - 1 ) ) ) |
15 |
|
elfzelz |
|- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ZZ ) |
16 |
14 15
|
syl |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> R e. ZZ ) |
17 |
5 16
|
zmulcld |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. R ) e. ZZ ) |
18 |
|
zsubcl |
|- ( ( ( A x. R ) e. ZZ /\ 1 e. ZZ ) -> ( ( A x. R ) - 1 ) e. ZZ ) |
19 |
17 9 18
|
sylancl |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( A x. R ) - 1 ) e. ZZ ) |
20 |
|
simprr |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( ( A x. S ) - 1 ) ) |
21 |
13
|
simprd |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( ( A x. R ) - 1 ) ) |
22 |
4 11 19 20 21
|
dvds2subd |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( ( ( A x. S ) - 1 ) - ( ( A x. R ) - 1 ) ) ) |
23 |
8
|
zcnd |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. S ) e. CC ) |
24 |
17
|
zcnd |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. R ) e. CC ) |
25 |
|
1cnd |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> 1 e. CC ) |
26 |
23 24 25
|
nnncan2d |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( ( A x. S ) - 1 ) - ( ( A x. R ) - 1 ) ) = ( ( A x. S ) - ( A x. R ) ) ) |
27 |
5
|
zcnd |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> A e. CC ) |
28 |
|
elfznn0 |
|- ( S e. ( 0 ... ( P - 1 ) ) -> S e. NN0 ) |
29 |
28
|
ad2antrl |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. NN0 ) |
30 |
29
|
nn0red |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. RR ) |
31 |
30
|
recnd |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. CC ) |
32 |
16
|
zcnd |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> R e. CC ) |
33 |
27 31 32
|
subdid |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. ( S - R ) ) = ( ( A x. S ) - ( A x. R ) ) ) |
34 |
26 33
|
eqtr4d |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( ( A x. S ) - 1 ) - ( ( A x. R ) - 1 ) ) = ( A x. ( S - R ) ) ) |
35 |
22 34
|
breqtrd |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( A x. ( S - R ) ) ) |
36 |
|
simpl3 |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> -. P || A ) |
37 |
|
coprm |
|- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) ) |
38 |
2 5 37
|
syl2anc |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( -. P || A <-> ( P gcd A ) = 1 ) ) |
39 |
36 38
|
mpbid |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( P gcd A ) = 1 ) |
40 |
7 16
|
zsubcld |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( S - R ) e. ZZ ) |
41 |
|
coprmdvds |
|- ( ( P e. ZZ /\ A e. ZZ /\ ( S - R ) e. ZZ ) -> ( ( P || ( A x. ( S - R ) ) /\ ( P gcd A ) = 1 ) -> P || ( S - R ) ) ) |
42 |
4 5 40 41
|
syl3anc |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( P || ( A x. ( S - R ) ) /\ ( P gcd A ) = 1 ) -> P || ( S - R ) ) ) |
43 |
35 39 42
|
mp2and |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( S - R ) ) |
44 |
|
prmnn |
|- ( P e. Prime -> P e. NN ) |
45 |
2 44
|
syl |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. NN ) |
46 |
|
moddvds |
|- ( ( P e. NN /\ S e. ZZ /\ R e. ZZ ) -> ( ( S mod P ) = ( R mod P ) <-> P || ( S - R ) ) ) |
47 |
45 7 16 46
|
syl3anc |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( S mod P ) = ( R mod P ) <-> P || ( S - R ) ) ) |
48 |
43 47
|
mpbird |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( S mod P ) = ( R mod P ) ) |
49 |
45
|
nnrpd |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. RR+ ) |
50 |
|
elfzle1 |
|- ( S e. ( 0 ... ( P - 1 ) ) -> 0 <_ S ) |
51 |
50
|
ad2antrl |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> 0 <_ S ) |
52 |
|
elfzle2 |
|- ( S e. ( 0 ... ( P - 1 ) ) -> S <_ ( P - 1 ) ) |
53 |
52
|
ad2antrl |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S <_ ( P - 1 ) ) |
54 |
|
zltlem1 |
|- ( ( S e. ZZ /\ P e. ZZ ) -> ( S < P <-> S <_ ( P - 1 ) ) ) |
55 |
7 4 54
|
syl2anc |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( S < P <-> S <_ ( P - 1 ) ) ) |
56 |
53 55
|
mpbird |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S < P ) |
57 |
|
modid |
|- ( ( ( S e. RR /\ P e. RR+ ) /\ ( 0 <_ S /\ S < P ) ) -> ( S mod P ) = S ) |
58 |
30 49 51 56 57
|
syl22anc |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( S mod P ) = S ) |
59 |
|
prmuz2 |
|- ( P e. Prime -> P e. ( ZZ>= ` 2 ) ) |
60 |
|
uznn0sub |
|- ( P e. ( ZZ>= ` 2 ) -> ( P - 2 ) e. NN0 ) |
61 |
2 59 60
|
3syl |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( P - 2 ) e. NN0 ) |
62 |
|
zexpcl |
|- ( ( A e. ZZ /\ ( P - 2 ) e. NN0 ) -> ( A ^ ( P - 2 ) ) e. ZZ ) |
63 |
5 61 62
|
syl2anc |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A ^ ( P - 2 ) ) e. ZZ ) |
64 |
63
|
zred |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A ^ ( P - 2 ) ) e. RR ) |
65 |
|
modabs2 |
|- ( ( ( A ^ ( P - 2 ) ) e. RR /\ P e. RR+ ) -> ( ( ( A ^ ( P - 2 ) ) mod P ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P ) ) |
66 |
64 49 65
|
syl2anc |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( ( A ^ ( P - 2 ) ) mod P ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P ) ) |
67 |
1
|
oveq1i |
|- ( R mod P ) = ( ( ( A ^ ( P - 2 ) ) mod P ) mod P ) |
68 |
66 67 1
|
3eqtr4g |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( R mod P ) = R ) |
69 |
48 58 68
|
3eqtr3d |
|- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S = R ) |
70 |
69
|
ex |
|- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) -> S = R ) ) |
71 |
|
fz1ssfz0 |
|- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) ) |
72 |
71
|
sseli |
|- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ( 0 ... ( P - 1 ) ) ) |
73 |
|
eleq1 |
|- ( S = R -> ( S e. ( 0 ... ( P - 1 ) ) <-> R e. ( 0 ... ( P - 1 ) ) ) ) |
74 |
72 73
|
syl5ibr |
|- ( S = R -> ( R e. ( 1 ... ( P - 1 ) ) -> S e. ( 0 ... ( P - 1 ) ) ) ) |
75 |
|
oveq2 |
|- ( S = R -> ( A x. S ) = ( A x. R ) ) |
76 |
75
|
oveq1d |
|- ( S = R -> ( ( A x. S ) - 1 ) = ( ( A x. R ) - 1 ) ) |
77 |
76
|
breq2d |
|- ( S = R -> ( P || ( ( A x. S ) - 1 ) <-> P || ( ( A x. R ) - 1 ) ) ) |
78 |
77
|
biimprd |
|- ( S = R -> ( P || ( ( A x. R ) - 1 ) -> P || ( ( A x. S ) - 1 ) ) ) |
79 |
74 78
|
anim12d |
|- ( S = R -> ( ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) -> ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) ) |
80 |
12 79
|
syl5com |
|- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( S = R -> ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) ) |
81 |
70 80
|
impbid |
|- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) <-> S = R ) ) |