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Theorem prmdiveq 12854
Description: The modular inverse of  A  mod  P is unique. (Contributed by Mario Carneiro, 24-Jan-2015.)
Hypothesis
Ref Expression
prmdiv.1  |-  R  =  ( ( A ^
( P  -  2 ) )  mod  P
)
Assertion
Ref Expression
prmdiveq  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  (
( A  x.  S
)  -  1 ) )  <->  S  =  R
) )

Proof of Theorem prmdiveq
StepHypRef Expression
1 simprr 733 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  ||  ( ( A  x.  S )  - 
1 ) )
2 prmdiv.1 . . . . . . . . . . 11  |-  R  =  ( ( A ^
( P  -  2 ) )  mod  P
)
32prmdiv 12853 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )
43adantr 451 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
54simprd 449 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  ||  ( ( A  x.  R )  - 
1 ) )
6 simpl1 958 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  Prime )
7 prmz 12762 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
86, 7syl 15 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  ZZ )
9 simpl2 959 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  A  e.  ZZ )
10 elfzelz 10798 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  e.  ZZ )
1110ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  ZZ )
129, 11zmulcld 10123 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( A  x.  S
)  e.  ZZ )
13 1z 10053 . . . . . . . . . 10  |-  1  e.  ZZ
14 zsubcl 10061 . . . . . . . . . 10  |-  ( ( ( A  x.  S
)  e.  ZZ  /\  1  e.  ZZ )  ->  ( ( A  x.  S )  -  1 )  e.  ZZ )
1512, 13, 14sylancl 643 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( ( A  x.  S )  -  1 )  e.  ZZ )
164simpld 445 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  R  e.  ( 1 ... ( P  - 
1 ) ) )
17 elfzelz 10798 . . . . . . . . . . . 12  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  ZZ )
1816, 17syl 15 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  R  e.  ZZ )
199, 18zmulcld 10123 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( A  x.  R
)  e.  ZZ )
20 zsubcl 10061 . . . . . . . . . 10  |-  ( ( ( A  x.  R
)  e.  ZZ  /\  1  e.  ZZ )  ->  ( ( A  x.  R )  -  1 )  e.  ZZ )
2119, 13, 20sylancl 643 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( ( A  x.  R )  -  1 )  e.  ZZ )
22 dvds2sub 12561 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( ( A  x.  S )  -  1 )  e.  ZZ  /\  ( ( A  x.  R )  -  1 )  e.  ZZ )  ->  ( ( P 
||  ( ( A  x.  S )  - 
1 )  /\  P  ||  ( ( A  x.  R )  -  1 ) )  ->  P  ||  ( ( ( A  x.  S )  - 
1 )  -  (
( A  x.  R
)  -  1 ) ) ) )
238, 15, 21, 22syl3anc 1182 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( ( P  ||  ( ( A  x.  S )  -  1 )  /\  P  ||  ( ( A  x.  R )  -  1 ) )  ->  P  ||  ( ( ( A  x.  S )  - 
1 )  -  (
( A  x.  R
)  -  1 ) ) ) )
241, 5, 23mp2and 660 . . . . . . 7  |-  ( ( ( 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 ) ) )
2512zcnd 10118 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( A  x.  S
)  e.  CC )
2619zcnd 10118 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( A  x.  R
)  e.  CC )
27 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
2827a1i 10 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
1  e.  CC )
2925, 26, 28nnncan2d 9192 . . . . . . . 8  |-  ( ( ( 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 ) ) )
309zcnd 10118 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  A  e.  CC )
31 elfznn0 10822 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  e.  NN0 )
3231ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  NN0 )
3332nn0red 10019 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  RR )
3433recnd 8861 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  CC )
3518zcnd 10118 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  R  e.  CC )
3630, 34, 35subdid 9235 . . . . . . . 8  |-  ( ( ( 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 ) ) )
3729, 36eqtr4d 2318 . . . . . . 7  |-  ( ( ( 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 ) ) )
3824, 37breqtrd 4047 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  ||  ( A  x.  ( S  -  R
) ) )
39 simpl3 960 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  -.  P  ||  A )
40 coprm 12779 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
416, 9, 40syl2anc 642 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( -.  P  ||  A 
<->  ( P  gcd  A
)  =  1 ) )
4239, 41mpbid 201 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( P  gcd  A
)  =  1 )
4311, 18zsubcld 10122 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( S  -  R
)  e.  ZZ )
44 coprmdvds 12781 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  A  e.  ZZ  /\  ( S  -  R )  e.  ZZ )  ->  (
( P  ||  ( A  x.  ( S  -  R ) )  /\  ( P  gcd  A )  =  1 )  ->  P  ||  ( S  -  R ) ) )
458, 9, 43, 44syl3anc 1182 . . . . . 6  |-  ( ( ( 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 )
) )
4638, 42, 45mp2and 660 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  ||  ( S  -  R ) )
47 prmnn 12761 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
486, 47syl 15 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  NN )
49 moddvds 12538 . . . . . 6  |-  ( ( P  e.  NN  /\  S  e.  ZZ  /\  R  e.  ZZ )  ->  (
( S  mod  P
)  =  ( R  mod  P )  <->  P  ||  ( S  -  R )
) )
5048, 11, 18, 49syl3anc 1182 . . . . 5  |-  ( ( ( 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 )
) )
5146, 50mpbird 223 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( S  mod  P
)  =  ( R  mod  P ) )
5248nnrpd 10389 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  RR+ )
53 elfzle1 10799 . . . . . 6  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  0  <_  S )
5453ad2antrl 708 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
0  <_  S )
55 elfzle2 10800 . . . . . . 7  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  <_  ( P  -  1 ) )
5655ad2antrl 708 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  <_  ( P  - 
1 ) )
57 zltlem1 10070 . . . . . . 7  |-  ( ( S  e.  ZZ  /\  P  e.  ZZ )  ->  ( S  <  P  <->  S  <_  ( P  - 
1 ) ) )
5811, 8, 57syl2anc 642 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( S  <  P  <->  S  <_  ( P  - 
1 ) ) )
5956, 58mpbird 223 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  <  P )
60 modid 10993 . . . . 5  |-  ( ( ( S  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  S  /\  S  <  P ) )  ->  ( S  mod  P )  =  S )
6133, 52, 54, 59, 60syl22anc 1183 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( S  mod  P
)  =  S )
62 prmuz2 12776 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
63 uznn0sub 10259 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( P  -  2 )  e. 
NN0 )
646, 62, 633syl 18 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( P  -  2 )  e.  NN0 )
65 zexpcl 11118 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( P  -  2
)  e.  NN0 )  ->  ( A ^ ( P  -  2 ) )  e.  ZZ )
669, 64, 65syl2anc 642 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( A ^ ( P  -  2 ) )  e.  ZZ )
6766zred 10117 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( A ^ ( P  -  2 ) )  e.  RR )
68 modabs2 10998 . . . . . 6  |-  ( ( ( A ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ )  -> 
( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)  =  ( ( A ^ ( P  -  2 ) )  mod  P ) )
6967, 52, 68syl2anc 642 . . . . 5  |-  ( ( ( 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 ) )
702oveq1i 5868 . . . . 5  |-  ( R  mod  P )  =  ( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)
7169, 70, 23eqtr4g 2340 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( R  mod  P
)  =  R )
7251, 61, 713eqtr3d 2323 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  =  R )
7372ex 423 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  (
( A  x.  S
)  -  1 ) )  ->  S  =  R ) )
74 1e0p1 10152 . . . . . . . 8  |-  1  =  ( 0  +  1 )
7574oveq1i 5868 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
76 0z 10035 . . . . . . . 8  |-  0  e.  ZZ
77 fzp1ss 10837 . . . . . . . 8  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... ( P  -  1 ) ) 
C_  ( 0 ... ( P  -  1 ) ) )
7876, 77ax-mp 8 . . . . . . 7  |-  ( ( 0  +  1 ) ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
7975, 78eqsstri 3208 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
8079sseli 3176 . . . . 5  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  ( 0 ... ( P  -  1 ) ) )
81 eleq1 2343 . . . . 5  |-  ( S  =  R  ->  ( S  e.  ( 0 ... ( P  - 
1 ) )  <->  R  e.  ( 0 ... ( P  -  1 ) ) ) )
8280, 81syl5ibr 212 . . . 4  |-  ( S  =  R  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  ->  S  e.  ( 0 ... ( P  - 
1 ) ) ) )
83 oveq2 5866 . . . . . . 7  |-  ( S  =  R  ->  ( A  x.  S )  =  ( A  x.  R ) )
8483oveq1d 5873 . . . . . 6  |-  ( S  =  R  ->  (
( A  x.  S
)  -  1 )  =  ( ( A  x.  R )  - 
1 ) )
8584breq2d 4035 . . . . 5  |-  ( S  =  R  ->  ( P  ||  ( ( A  x.  S )  - 
1 )  <->  P  ||  (
( A  x.  R
)  -  1 ) ) )
8685biimprd 214 . . . 4  |-  ( S  =  R  ->  ( P  ||  ( ( A  x.  R )  - 
1 )  ->  P  ||  ( ( A  x.  S )  -  1 ) ) )
8782, 86anim12d 546 . . 3  |-  ( S  =  R  ->  (
( R  e.  ( 1 ... ( P  -  1 ) )  /\  P  ||  (
( A  x.  R
)  -  1 ) )  ->  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) ) )
883, 87syl5com 26 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( S  =  R  ->  ( S  e.  ( 0 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  S )  - 
1 ) ) ) )
8973, 88impbid 183 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  (
( A  x.  S
)  -  1 ) )  <->  S  =  R
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782    mod cmo 10973   ^cexp 11104    || cdivides 12531    gcd cgcd 12685   Primecprime 12758
This theorem is referenced by:  prmdivdiv  12855  wilthlem1  20306  wilthlem2  20307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-phi 12834
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