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Theorem prmdiv 13103
Description: Show an explicit expression for the modular inverse of  A  mod  P. (Contributed by Mario Carneiro, 24-Jan-2015.)
Hypothesis
Ref Expression
prmdiv.1  |-  R  =  ( ( A ^
( P  -  2 ) )  mod  P
)
Assertion
Ref Expression
prmdiv  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )

Proof of Theorem prmdiv
StepHypRef Expression
1 nprmdvds1 13040 . . . . . 6  |-  ( P  e.  Prime  ->  -.  P  ||  1 )
213ad2ant1 978 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  -.  P  ||  1 )
3 prmnn 13011 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  P  e.  NN )
433ad2ant1 978 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  NN )
5 simp2 958 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  A  e.  ZZ )
6 prmz 13012 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  P  e.  ZZ )
763ad2ant1 978 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  ZZ )
8 gcdcom 12949 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( A  gcd  P
)  =  ( P  gcd  A ) )
95, 7, 8syl2anc 643 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  ( P  gcd  A
) )
10 coprm 13029 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
1110biimp3a 1283 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  gcd  A )  =  1 )
129, 11eqtrd 2421 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  1 )
13 eulerth 13101 . . . . . . . . . . 11  |-  ( ( P  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
144, 5, 12, 13syl3anc 1184 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
15 phiprm 13095 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1 ) )
16153ad2ant1 978 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( P  -  1 ) )
17 nnm1nn0 10195 . . . . . . . . . . . . . 14  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
184, 17syl 16 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  -  1 )  e.  NN0 )
1916, 18eqeltrd 2463 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  e. 
NN0 )
20 zexpcl 11325 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( phi `  P )  e.  NN0 )  -> 
( A ^ ( phi `  P ) )  e.  ZZ )
215, 19, 20syl2anc 643 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  e.  ZZ )
22 1z 10245 . . . . . . . . . . . 12  |-  1  e.  ZZ
2322a1i 11 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  1  e.  ZZ )
24 moddvds 12788 . . . . . . . . . . 11  |-  ( ( P  e.  NN  /\  ( A ^ ( phi `  P ) )  e.  ZZ  /\  1  e.  ZZ )  ->  (
( ( A ^
( phi `  P
) )  mod  P
)  =  ( 1  mod  P )  <->  P  ||  (
( A ^ ( phi `  P ) )  -  1 ) ) )
254, 21, 23, 24syl3anc 1184 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( ( A ^
( phi `  P
) )  mod  P
)  =  ( 1  mod  P )  <->  P  ||  (
( A ^ ( phi `  P ) )  -  1 ) ) )
2614, 25mpbid 202 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  ||  ( ( A ^
( phi `  P
) )  -  1 ) )
27 prmuz2 13026 . . . . . . . . . . . . . . . . 17  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
28273ad2ant1 978 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  ( ZZ>= `  2 )
)
29 uznn0sub 10451 . . . . . . . . . . . . . . . 16  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( P  -  2 )  e. 
NN0 )
3028, 29syl 16 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  -  2 )  e.  NN0 )
31 zexpcl 11325 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  ( P  -  2
)  e.  NN0 )  ->  ( A ^ ( P  -  2 ) )  e.  ZZ )
325, 30, 31syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( P  - 
2 ) )  e.  ZZ )
3332zred 10309 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( P  - 
2 ) )  e.  RR )
3433, 4nndivred 9982 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( P  -  2 ) )  /  P )  e.  RR )
3534flcld 11136 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) )  e.  ZZ )
365, 35zmulcld 10315 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) )  e.  ZZ )
37 dvdsmul1 12800 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  ( A  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) )  e.  ZZ )  ->  P  ||  ( P  x.  ( A  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) )
387, 36, 37syl2anc 643 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  ||  ( P  x.  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) ) ) )
39 zsubcl 10253 . . . . . . . . . . 11  |-  ( ( ( A ^ ( phi `  P ) )  e.  ZZ  /\  1  e.  ZZ )  ->  (
( A ^ ( phi `  P ) )  -  1 )  e.  ZZ )
4021, 22, 39sylancl 644 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  e.  ZZ )
417, 36zmulcld 10315 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) )  e.  ZZ )
42 dvds2sub 12811 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  ( ( A ^
( phi `  P
) )  -  1 )  e.  ZZ  /\  ( P  x.  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) ) )  e.  ZZ )  -> 
( ( P  ||  ( ( A ^
( phi `  P
) )  -  1 )  /\  P  ||  ( P  x.  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) ) ) )  ->  P  ||  (
( ( A ^
( phi `  P
) )  -  1 )  -  ( P  x.  ( A  x.  ( |_ `  ( ( A ^ ( P  -  2 ) )  /  P ) ) ) ) ) ) )
437, 40, 41, 42syl3anc 1184 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( P  ||  (
( A ^ ( phi `  P ) )  -  1 )  /\  P  ||  ( P  x.  ( A  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) )  ->  P  ||  ( ( ( A ^ ( phi `  P ) )  - 
1 )  -  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) ) ) ) )
4426, 38, 43mp2and 661 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  ||  ( ( ( A ^ ( phi `  P ) )  - 
1 )  -  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) ) ) )
455zcnd 10310 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  A  e.  CC )
4632zcnd 10310 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( P  - 
2 ) )  e.  CC )
477, 35zmulcld 10315 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) )  e.  ZZ )
4847zcnd 10310 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) )  e.  CC )
4945, 46, 48subdid 9423 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  ( ( A ^ ( P  - 
2 ) )  -  ( P  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) )  =  ( ( A  x.  ( A ^ ( P  - 
2 ) ) )  -  ( A  x.  ( P  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) ) )
50 prmdiv.1 . . . . . . . . . . . . 13  |-  R  =  ( ( A ^
( P  -  2 ) )  mod  P
)
514nnrpd 10581 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  RR+ )
52 modval 11181 . . . . . . . . . . . . . 14  |-  ( ( ( A ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ )  -> 
( ( A ^
( P  -  2 ) )  mod  P
)  =  ( ( A ^ ( P  -  2 ) )  -  ( P  x.  ( |_ `  ( ( A ^ ( P  -  2 ) )  /  P ) ) ) ) )
5333, 51, 52syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( P  -  2 ) )  mod  P )  =  ( ( A ^ ( P  - 
2 ) )  -  ( P  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) )
5450, 53syl5eq 2433 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  R  =  ( ( A ^ ( P  - 
2 ) )  -  ( P  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) )
5554oveq2d 6038 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  R )  =  ( A  x.  ( ( A ^
( P  -  2 ) )  -  ( P  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) ) ) ) )
56 2m1e1 10029 . . . . . . . . . . . . . . . . 17  |-  ( 2  -  1 )  =  1
5756oveq2i 6033 . . . . . . . . . . . . . . . 16  |-  ( P  -  ( 2  -  1 ) )  =  ( P  -  1 )
5816, 57syl6eqr 2439 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( P  -  (
2  -  1 ) ) )
594nncnd 9950 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  CC )
60 2cn 10004 . . . . . . . . . . . . . . . . 17  |-  2  e.  CC
6160a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  2  e.  CC )
62 ax-1cn 8983 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
6362a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  1  e.  CC )
6459, 61, 63subsubd 9373 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  -  ( 2  -  1 ) )  =  ( ( P  -  2 )  +  1 ) )
6558, 64eqtrd 2421 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( ( P  - 
2 )  +  1 ) )
6665oveq2d 6038 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( A ^ (
( P  -  2 )  +  1 ) ) )
6745, 30expp1d 11453 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  2 )  +  1 ) )  =  ( ( A ^
( P  -  2 ) )  x.  A
) )
6846, 45mulcomd 9044 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( P  -  2 ) )  x.  A )  =  ( A  x.  ( A ^ ( P  -  2 ) ) ) )
6966, 67, 683eqtrd 2425 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( A  x.  ( A ^ ( P  - 
2 ) ) ) )
7035zcnd 10310 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) )  e.  CC )
7159, 45, 70mul12d 9209 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) )  =  ( A  x.  ( P  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) ) ) )
7269, 71oveq12d 6040 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  ( P  x.  ( A  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) )  =  ( ( A  x.  ( A ^ ( P  - 
2 ) ) )  -  ( A  x.  ( P  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) ) )
7349, 55, 723eqtr4d 2431 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  R )  =  ( ( A ^ ( phi `  P ) )  -  ( P  x.  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) ) ) ) )
7473oveq1d 6037 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A  x.  R
)  -  1 )  =  ( ( ( A ^ ( phi `  P ) )  -  ( P  x.  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) ) ) )  -  1 ) )
7521zcnd 10310 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  e.  CC )
7641zcnd 10310 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) )  e.  CC )
7775, 76, 63sub32d 9377 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( ( A ^
( phi `  P
) )  -  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) ) )  -  1 )  =  ( ( ( A ^ ( phi `  P ) )  - 
1 )  -  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) ) ) )
7874, 77eqtrd 2421 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A  x.  R
)  -  1 )  =  ( ( ( A ^ ( phi `  P ) )  - 
1 )  -  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) ) ) )
7944, 78breqtrrd 4181 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  ||  ( ( A  x.  R )  -  1 ) )
80 oveq2 6030 . . . . . . . . 9  |-  ( R  =  0  ->  ( A  x.  R )  =  ( A  x.  0 ) )
8180oveq1d 6037 . . . . . . . 8  |-  ( R  =  0  ->  (
( A  x.  R
)  -  1 )  =  ( ( A  x.  0 )  - 
1 ) )
8281breq2d 4167 . . . . . . 7  |-  ( R  =  0  ->  ( P  ||  ( ( A  x.  R )  - 
1 )  <->  P  ||  (
( A  x.  0 )  -  1 ) ) )
8379, 82syl5ibcom 212 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  =  0  ->  P 
||  ( ( A  x.  0 )  - 
1 ) ) )
8445mul01d 9199 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  0 )  =  0 )
8584oveq1d 6037 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A  x.  0 )  -  1 )  =  ( 0  -  1 ) )
86 df-neg 9228 . . . . . . . . 9  |-  -u 1  =  ( 0  -  1 )
8785, 86syl6eqr 2439 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A  x.  0 )  -  1 )  =  -u 1 )
8887breq2d 4167 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A  x.  0 )  - 
1 )  <->  P  ||  -u 1
) )
89 dvdsnegb 12796 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  1  e.  ZZ )  ->  ( P  ||  1  <->  P 
||  -u 1 ) )
907, 22, 89sylancl 644 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  ||  1  <->  P  ||  -u 1
) )
9188, 90bitr4d 248 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A  x.  0 )  - 
1 )  <->  P  ||  1
) )
9283, 91sylibd 206 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  =  0  ->  P 
||  1 ) )
932, 92mtod 170 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  -.  R  =  0 )
94 zmodfz 11197 . . . . . . . 8  |-  ( ( ( A ^ ( P  -  2 ) )  e.  ZZ  /\  P  e.  NN )  ->  ( ( A ^
( P  -  2 ) )  mod  P
)  e.  ( 0 ... ( P  - 
1 ) ) )
9532, 4, 94syl2anc 643 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( P  -  2 ) )  mod  P )  e.  ( 0 ... ( P  -  1 ) ) )
9650, 95syl5eqel 2473 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  R  e.  ( 0 ... ( P  -  1 ) ) )
97 nn0uz 10454 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
9818, 97syl6eleq 2479 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  -  1 )  e.  ( ZZ>= `  0
) )
99 elfzp12 11058 . . . . . . 7  |-  ( ( P  -  1 )  e.  ( ZZ>= `  0
)  ->  ( R  e.  ( 0 ... ( P  -  1 ) )  <->  ( R  =  0  \/  R  e.  ( ( 0  +  1 ) ... ( P  -  1 ) ) ) ) )
10098, 99syl 16 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 0 ... ( P  - 
1 ) )  <->  ( R  =  0  \/  R  e.  ( ( 0  +  1 ) ... ( P  -  1 ) ) ) ) )
10196, 100mpbid 202 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  =  0  \/  R  e.  ( (
0  +  1 ) ... ( P  - 
1 ) ) ) )
102101ord 367 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( -.  R  =  0  ->  R  e.  ( ( 0  +  1 ) ... ( P  - 
1 ) ) ) )
10393, 102mpd 15 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  R  e.  ( ( 0  +  1 ) ... ( P  -  1 ) ) )
104 1e0p1 10344 . . . 4  |-  1  =  ( 0  +  1 )
105104oveq1i 6032 . . 3  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
106103, 105syl6eleqr 2480 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  R  e.  ( 1 ... ( P  -  1 ) ) )
107106, 79jca 519 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930    - cmin 9225   -ucneg 9226    / cdiv 9611   NNcn 9934   2c2 9983   NN0cn0 10155   ZZcz 10216   ZZ>=cuz 10422   RR+crp 10546   ...cfz 10977   |_cfl 11130    mod cmo 11179   ^cexp 11311    || cdivides 12781    gcd cgcd 12935   Primecprime 13008   phicphi 13082
This theorem is referenced by:  prmdiveq  13104  prmdivdiv  13105  wilthlem2  20721  wilthlem3  20722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-fz 10978  df-fzo 11068  df-fl 11131  df-mod 11180  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-dvds 12782  df-gcd 12936  df-prm 13009  df-phi 13084
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