Step |
Hyp |
Ref |
Expression |
1 |
|
prmdiv.1 |
|- R = ( ( A ^ ( P - 2 ) ) mod P ) |
2 |
|
fz1ssfz0 |
|- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) ) |
3 |
|
simpr |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ( 1 ... ( P - 1 ) ) ) |
4 |
2 3
|
sselid |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ( 0 ... ( P - 1 ) ) ) |
5 |
|
simpl |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P e. Prime ) |
6 |
|
elfznn |
|- ( A e. ( 1 ... ( P - 1 ) ) -> A e. NN ) |
7 |
6
|
adantl |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. NN ) |
8 |
7
|
nnzd |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ZZ ) |
9 |
|
prmnn |
|- ( P e. Prime -> P e. NN ) |
10 |
|
fzm1ndvds |
|- ( ( P e. NN /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || A ) |
11 |
9 10
|
sylan |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || A ) |
12 |
1
|
prmdiv |
|- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) ) |
13 |
5 8 11 12
|
syl3anc |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) ) |
14 |
13
|
simprd |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P || ( ( A x. R ) - 1 ) ) |
15 |
7
|
nncnd |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. CC ) |
16 |
13
|
simpld |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. ( 1 ... ( P - 1 ) ) ) |
17 |
|
elfznn |
|- ( R e. ( 1 ... ( P - 1 ) ) -> R e. NN ) |
18 |
16 17
|
syl |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. NN ) |
19 |
18
|
nncnd |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. CC ) |
20 |
15 19
|
mulcomd |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( A x. R ) = ( R x. A ) ) |
21 |
20
|
oveq1d |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( ( A x. R ) - 1 ) = ( ( R x. A ) - 1 ) ) |
22 |
14 21
|
breqtrd |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P || ( ( R x. A ) - 1 ) ) |
23 |
16
|
elfzelzd |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. ZZ ) |
24 |
|
fzm1ndvds |
|- ( ( P e. NN /\ R e. ( 1 ... ( P - 1 ) ) ) -> -. P || R ) |
25 |
9 16 24
|
syl2an2r |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || R ) |
26 |
|
eqid |
|- ( ( R ^ ( P - 2 ) ) mod P ) = ( ( R ^ ( P - 2 ) ) mod P ) |
27 |
26
|
prmdiveq |
|- ( ( P e. Prime /\ R e. ZZ /\ -. P || R ) -> ( ( A e. ( 0 ... ( P - 1 ) ) /\ P || ( ( R x. A ) - 1 ) ) <-> A = ( ( R ^ ( P - 2 ) ) mod P ) ) ) |
28 |
5 23 25 27
|
syl3anc |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( ( A e. ( 0 ... ( P - 1 ) ) /\ P || ( ( R x. A ) - 1 ) ) <-> A = ( ( R ^ ( P - 2 ) ) mod P ) ) ) |
29 |
4 22 28
|
mpbi2and |
|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A = ( ( R ^ ( P - 2 ) ) mod P ) ) |