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Theorem prmdiv 14187
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 14124 . . . . . 6  |-  ( P  e.  Prime  ->  -.  P  ||  1 )
213ad2ant1 1016 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  -.  P  ||  1 )
3 prmnn 14092 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  P  e.  NN )
433ad2ant1 1016 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  NN )
5 simp2 996 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  A  e.  ZZ )
6 prmz 14093 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  P  e.  ZZ )
763ad2ant1 1016 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  ZZ )
8 gcdcom 14030 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( A  gcd  P
)  =  ( P  gcd  A ) )
95, 7, 8syl2anc 661 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  ( P  gcd  A
) )
10 coprm 14113 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
1110biimp3a 1327 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  gcd  A )  =  1 )
129, 11eqtrd 2482 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  gcd  P )  =  1 )
13 eulerth 14185 . . . . . . . . . . 11  |-  ( ( P  e.  NN  /\  A  e.  ZZ  /\  ( A  gcd  P )  =  1 )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
144, 5, 12, 13syl3anc 1227 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  mod  P )  =  ( 1  mod  P
) )
15 phiprm 14179 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  ( phi `  P )  =  ( P  -  1 ) )
16153ad2ant1 1016 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( P  -  1 ) )
17 nnm1nn0 10838 . . . . . . . . . . . . . 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 2529 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  e. 
NN0 )
20 zexpcl 12155 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( phi `  P )  e.  NN0 )  -> 
( A ^ ( phi `  P ) )  e.  ZZ )
215, 19, 20syl2anc 661 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  e.  ZZ )
22 1zzd 10896 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  1  e.  ZZ )
23 moddvds 13865 . . . . . . . . . . 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 ) ) )
244, 21, 22, 23syl3anc 1227 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( ( A ^
( phi `  P
) )  mod  P
)  =  ( 1  mod  P )  <->  P  ||  (
( A ^ ( phi `  P ) )  -  1 ) ) )
2514, 24mpbid 210 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  ||  ( ( A ^
( phi `  P
) )  -  1 ) )
26 prmuz2 14107 . . . . . . . . . . . . . . . . 17  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
27263ad2ant1 1016 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  ( ZZ>= `  2 )
)
28 uznn0sub 11116 . . . . . . . . . . . . . . . 16  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( P  -  2 )  e. 
NN0 )
2927, 28syl 16 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  -  2 )  e.  NN0 )
30 zexpcl 12155 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  ( P  -  2
)  e.  NN0 )  ->  ( A ^ ( P  -  2 ) )  e.  ZZ )
315, 29, 30syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( P  - 
2 ) )  e.  ZZ )
3231zred 10969 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( P  - 
2 ) )  e.  RR )
3332, 4nndivred 10585 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( P  -  2 ) )  /  P )  e.  RR )
3433flcld 11909 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) )  e.  ZZ )
355, 34zmulcld 10975 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) )  e.  ZZ )
36 dvdsmul1 13877 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  ( A  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) )  e.  ZZ )  ->  P  ||  ( P  x.  ( A  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) )
377, 35, 36syl2anc 661 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  ||  ( P  x.  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) ) ) )
38 1z 10895 . . . . . . . . . . 11  |-  1  e.  ZZ
39 zsubcl 10907 . . . . . . . . . . 11  |-  ( ( ( A ^ ( phi `  P ) )  e.  ZZ  /\  1  e.  ZZ )  ->  (
( A ^ ( phi `  P ) )  -  1 )  e.  ZZ )
4021, 38, 39sylancl 662 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( phi `  P ) )  -  1 )  e.  ZZ )
417, 35zmulcld 10975 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) )  e.  ZZ )
42 dvds2sub 13888 . . . . . . . . . 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 1227 . . . . . . . . 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 ) ) ) ) ) ) )
4425, 37, 43mp2and 679 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  ||  ( ( ( A ^ ( phi `  P ) )  - 
1 )  -  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) ) ) )
455zcnd 10970 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  A  e.  CC )
4631zcnd 10970 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( P  - 
2 ) )  e.  CC )
477, 34zmulcld 10975 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) )  e.  ZZ )
4847zcnd 10970 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) )  e.  CC )
4945, 46, 48subdid 10013 . . . . . . . . . . 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 11259 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  RR+ )
52 modval 11972 . . . . . . . . . . . . . 14  |-  ( ( ( A ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ )  -> 
( ( A ^
( P  -  2 ) )  mod  P
)  =  ( ( A ^ ( P  -  2 ) )  -  ( P  x.  ( |_ `  ( ( A ^ ( P  -  2 ) )  /  P ) ) ) ) )
5332, 51, 52syl2anc 661 . . . . . . . . . . . . 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 2494 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  R  =  ( ( A ^ ( P  - 
2 ) )  -  ( P  x.  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) ) ) ) )
5554oveq2d 6293 . . . . . . . . . . 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 10651 . . . . . . . . . . . . . . . . 17  |-  ( 2  -  1 )  =  1
5756oveq2i 6288 . . . . . . . . . . . . . . . 16  |-  ( P  -  ( 2  -  1 ) )  =  ( P  -  1 )
5816, 57syl6eqr 2500 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( P  -  (
2  -  1 ) ) )
594nncnd 10553 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  e.  CC )
60 2cnd 10609 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  2  e.  CC )
61 1cnd 9610 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  1  e.  CC )
6259, 60, 61subsubd 9959 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  -  ( 2  -  1 ) )  =  ( ( P  -  2 )  +  1 ) )
6358, 62eqtrd 2482 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( phi `  P )  =  ( ( P  - 
2 )  +  1 ) )
6463oveq2d 6293 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( A ^ (
( P  -  2 )  +  1 ) ) )
6545, 29expp1d 12285 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( ( P  -  2 )  +  1 ) )  =  ( ( A ^
( P  -  2 ) )  x.  A
) )
6646, 45mulcomd 9615 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( P  -  2 ) )  x.  A )  =  ( A  x.  ( A ^ ( P  -  2 ) ) ) )
6764, 65, 663eqtrd 2486 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  =  ( A  x.  ( A ^ ( P  - 
2 ) ) ) )
6834zcnd 10970 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( |_ `  ( ( A ^ ( P  - 
2 ) )  /  P ) )  e.  CC )
6959, 45, 68mul12d 9787 . . . . . . . . . . . 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
) ) ) ) )
7067, 69oveq12d 6295 . . . . . . . . . . 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 ) ) ) ) ) )
7149, 55, 703eqtr4d 2492 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  R )  =  ( ( A ^ ( phi `  P ) )  -  ( P  x.  ( A  x.  ( |_ `  ( ( A ^
( P  -  2 ) )  /  P
) ) ) ) ) )
7271oveq1d 6292 . . . . . . . . 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 ) )
7321zcnd 10970 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A ^ ( phi `  P ) )  e.  CC )
7441zcnd 10970 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  x.  ( A  x.  ( |_ `  (
( A ^ ( P  -  2 ) )  /  P ) ) ) )  e.  CC )
7573, 74, 61sub32d 9963 . . . . . . . . 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 ) ) ) ) ) )
7672, 75eqtrd 2482 . . . . . . . 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 ) ) ) ) ) )
7744, 76breqtrrd 4459 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  P  ||  ( ( A  x.  R )  -  1 ) )
78 oveq2 6285 . . . . . . . . 9  |-  ( R  =  0  ->  ( A  x.  R )  =  ( A  x.  0 ) )
7978oveq1d 6292 . . . . . . . 8  |-  ( R  =  0  ->  (
( A  x.  R
)  -  1 )  =  ( ( A  x.  0 )  - 
1 ) )
8079breq2d 4445 . . . . . . 7  |-  ( R  =  0  ->  ( P  ||  ( ( A  x.  R )  - 
1 )  <->  P  ||  (
( A  x.  0 )  -  1 ) ) )
8177, 80syl5ibcom 220 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  =  0  ->  P 
||  ( ( A  x.  0 )  - 
1 ) ) )
8245mul01d 9777 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( A  x.  0 )  =  0 )
8382oveq1d 6292 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A  x.  0 )  -  1 )  =  ( 0  -  1 ) )
84 df-neg 9808 . . . . . . . . 9  |-  -u 1  =  ( 0  -  1 )
8583, 84syl6eqr 2500 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A  x.  0 )  -  1 )  =  -u 1 )
8685breq2d 4445 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A  x.  0 )  - 
1 )  <->  P  ||  -u 1
) )
87 dvdsnegb 13873 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  1  e.  ZZ )  ->  ( P  ||  1  <->  P 
||  -u 1 ) )
887, 38, 87sylancl 662 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  ||  1  <->  P  ||  -u 1
) )
8986, 88bitr4d 256 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  ||  ( ( A  x.  0 )  - 
1 )  <->  P  ||  1
) )
9081, 89sylibd 214 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  =  0  ->  P 
||  1 ) )
912, 90mtod 177 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  -.  R  =  0 )
92 zmodfz 11991 . . . . . . . 8  |-  ( ( ( A ^ ( P  -  2 ) )  e.  ZZ  /\  P  e.  NN )  ->  ( ( A ^
( P  -  2 ) )  mod  P
)  e.  ( 0 ... ( P  - 
1 ) ) )
9331, 4, 92syl2anc 661 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( A ^ ( P  -  2 ) )  mod  P )  e.  ( 0 ... ( P  -  1 ) ) )
9450, 93syl5eqel 2533 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  R  e.  ( 0 ... ( P  -  1 ) ) )
95 nn0uz 11119 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
9618, 95syl6eleq 2539 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( P  -  1 )  e.  ( ZZ>= `  0
) )
97 elfzp12 11761 . . . . . . 7  |-  ( ( P  -  1 )  e.  ( ZZ>= `  0
)  ->  ( R  e.  ( 0 ... ( P  -  1 ) )  <->  ( R  =  0  \/  R  e.  ( ( 0  +  1 ) ... ( P  -  1 ) ) ) ) )
9896, 97syl 16 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 0 ... ( P  - 
1 ) )  <->  ( R  =  0  \/  R  e.  ( ( 0  +  1 ) ... ( P  -  1 ) ) ) ) )
9994, 98mpbid 210 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  =  0  \/  R  e.  ( (
0  +  1 ) ... ( P  - 
1 ) ) ) )
10099ord 377 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( -.  R  =  0  ->  R  e.  ( ( 0  +  1 ) ... ( P  - 
1 ) ) ) )
10191, 100mpd 15 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  R  e.  ( ( 0  +  1 ) ... ( P  -  1 ) ) )
102 1e0p1 11007 . . . 4  |-  1  =  ( 0  +  1 )
103102oveq1i 6287 . . 3  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
104101, 103syl6eleqr 2540 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  R  e.  ( 1 ... ( P  -  1 ) ) )
105104, 77jca 532 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495    - cmin 9805   -ucneg 9806    / cdiv 10207   NNcn 10537   2c2 10586   NN0cn0 10796   ZZcz 10865   ZZ>=cuz 11085   RR+crp 11224   ...cfz 11676   |_cfl 11901    mod cmo 11970   ^cexp 12140    || cdvds 13858    gcd cgcd 14016   Primecprime 14089   phicphi 14166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-fz 11677  df-fzo 11799  df-fl 11903  df-mod 11971  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-dvds 13859  df-gcd 14017  df-prm 14090  df-phi 14168
This theorem is referenced by:  prmdiveq  14188  prmdivdiv  14189  modprminv  14198  wilthlem2  23208  wilthlem3  23209
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