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Theorem prmdiv 13129
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 13066 . . . . . 6  |-  ( P  e.  Prime  ->  -.  P  ||  1 )
213ad2ant1 978 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  -.  P  ||  1 )
3 prmnn 13037 . . . . . . . . . . . 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 13038 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  P  e.  ZZ )
763ad2ant1 978 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  ZZ )
8 gcdcom 12975 . . . . . . . . . . . . 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 13055 . . . . . . . . . . . . 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 2436 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  1 )
13 eulerth 13127 . . . . . . . . . . 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 13121 . . . . . . . . . . . . . 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 10217 . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  e. 
NN0 )
20 zexpcl 11351 . . . . . . . . . . . 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 10267 . . . . . . . . . . . 12  |-  1  e.  ZZ
2322a1i 11 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  1  e.  ZZ )
24 moddvds 12814 . . . . . . . . . . 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 13052 . . . . . . . . . . . . . . . . 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 10473 . . . . . . . . . . . . . . . 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 11351 . . . . . . . . . . . . . . 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 10331 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( P  - 
2 ) )  e.  RR )
3433, 4nndivred 10004 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( P  -  2 ) )  /  P )  e.  RR )
3534flcld 11162 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) )  e.  ZZ )
365, 35zmulcld 10337 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) )  e.  ZZ )
37 dvdsmul1 12826 . . . . . . . . . 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 10275 . . . . . . . . . . 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 10337 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) )  e.  ZZ )
42 dvds2sub 12837 . . . . . . . . . 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 10332 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  A  e.  CC )
4632zcnd 10332 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( P  - 
2 ) )  e.  CC )
477, 35zmulcld 10337 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) )  e.  ZZ )
4847zcnd 10332 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) )  e.  CC )
4945, 46, 48subdid 9445 . . . . . . . . . . 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 10603 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  RR+ )
52 modval 11207 . . . . . . . . . . . . . 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 2448 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  R  =  ( ( A ^ ( P  - 
2 ) )  -  ( P  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) )
5554oveq2d 6056 . . . . . . . . . . 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 10051 . . . . . . . . . . . . . . . . 17  |-  ( 2  -  1 )  =  1
5756oveq2i 6051 . . . . . . . . . . . . . . . 16  |-  ( P  -  ( 2  -  1 ) )  =  ( P  -  1 )
5816, 57syl6eqr 2454 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( P  -  (
2  -  1 ) ) )
594nncnd 9972 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  CC )
60 2cn 10026 . . . . . . . . . . . . . . . . 17  |-  2  e.  CC
6160a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  2  e.  CC )
62 ax-1cn 9004 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
6362a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  1  e.  CC )
6459, 61, 63subsubd 9395 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  -  ( 2  -  1 ) )  =  ( ( P  -  2 )  +  1 ) )
6558, 64eqtrd 2436 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( ( P  - 
2 )  +  1 ) )
6665oveq2d 6056 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( A ^ (
( P  -  2 )  +  1 ) ) )
6745, 30expp1d 11479 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  2 )  +  1 ) )  =  ( ( A ^
( P  -  2 ) )  x.  A
) )
6846, 45mulcomd 9065 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( P  -  2 ) )  x.  A )  =  ( A  x.  ( A ^ ( P  -  2 ) ) ) )
6966, 67, 683eqtrd 2440 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( A  x.  ( A ^ ( P  - 
2 ) ) ) )
7035zcnd 10332 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) )  e.  CC )
7159, 45, 70mul12d 9231 . . . . . . . . . . . 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 6058 . . . . . . . . . . 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 2446 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  R )  =  ( ( A ^ ( phi `  P ) )  -  ( P  x.  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) ) ) ) )
7473oveq1d 6055 . . . . . . . . 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 10332 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  e.  CC )
7641zcnd 10332 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) )  e.  CC )
7775, 76, 63sub32d 9399 . . . . . . . . 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 2436 . . . . . . . 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 4198 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  ||  ( ( A  x.  R )  -  1 ) )
80 oveq2 6048 . . . . . . . . 9  |-  ( R  =  0  ->  ( A  x.  R )  =  ( A  x.  0 ) )
8180oveq1d 6055 . . . . . . . 8  |-  ( R  =  0  ->  (
( A  x.  R
)  -  1 )  =  ( ( A  x.  0 )  - 
1 ) )
8281breq2d 4184 . . . . . . 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 9221 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  0 )  =  0 )
8584oveq1d 6055 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A  x.  0 )  -  1 )  =  ( 0  -  1 ) )
86 df-neg 9250 . . . . . . . . 9  |-  -u 1  =  ( 0  -  1 )
8785, 86syl6eqr 2454 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A  x.  0 )  -  1 )  =  -u 1 )
8887breq2d 4184 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A  x.  0 )  - 
1 )  <->  P  ||  -u 1
) )
89 dvdsnegb 12822 . . . . . . . 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 11223 . . . . . . . 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 2488 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  R  e.  ( 0 ... ( P  -  1 ) ) )
97 nn0uz 10476 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
9818, 97syl6eleq 2494 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  -  1 )  e.  ( ZZ>= `  0
) )
99 elfzp12 11081 . . . . . . 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 10366 . . . 4  |-  1  =  ( 0  +  1 )
105104oveq1i 6050 . . 3  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
106103, 105syl6eleqr 2495 . 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 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   ...cfz 10999   |_cfl 11156    mod cmo 11205   ^cexp 11337    || cdivides 12807    gcd cgcd 12961   Primecprime 13034   phicphi 13108
This theorem is referenced by:  prmdiveq  13130  prmdivdiv  13131  wilthlem2  20805  wilthlem3  20806
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-phi 13110
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