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Theorem prmdiveq 13175
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 734 . . . . . . . 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 13174 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )
43adantr 452 . . . . . . . . 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 450 . . . . . . . 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 960 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  Prime )
7 prmz 13083 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
86, 7syl 16 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  ZZ )
9 simpl2 961 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  A  e.  ZZ )
10 elfzelz 11059 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  e.  ZZ )
1110ad2antrl 709 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  ZZ )
129, 11zmulcld 10381 . . . . . . . . . 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 10311 . . . . . . . . . 10  |-  1  e.  ZZ
14 zsubcl 10319 . . . . . . . . . 10  |-  ( ( ( A  x.  S
)  e.  ZZ  /\  1  e.  ZZ )  ->  ( ( A  x.  S )  -  1 )  e.  ZZ )
1512, 13, 14sylancl 644 . . . . . . . . 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 446 . . . . . . . . . . . 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 11059 . . . . . . . . . . . 12  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  ZZ )
1816, 17syl 16 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  R  e.  ZZ )
199, 18zmulcld 10381 . . . . . . . . . 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 10319 . . . . . . . . . 10  |-  ( ( ( A  x.  R
)  e.  ZZ  /\  1  e.  ZZ )  ->  ( ( A  x.  R )  -  1 )  e.  ZZ )
2119, 13, 20sylancl 644 . . . . . . . . 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 12882 . . . . . . . . 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 1184 . . . . . . . 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 661 . . . . . . 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 10376 . . . . . . . . 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 10376 . . . . . . . . 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 9048 . . . . . . . . . 10  |-  1  e.  CC
2827a1i 11 . . . . . . . . 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 9446 . . . . . . . 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 10376 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  A  e.  CC )
31 elfznn0 11083 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  e.  NN0 )
3231ad2antrl 709 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  NN0 )
3332nn0red 10275 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  RR )
3433recnd 9114 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  CC )
3518zcnd 10376 . . . . . . . . 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 9489 . . . . . . . 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 2471 . . . . . . 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 4236 . . . . . 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 962 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  -.  P  ||  A )
40 coprm 13100 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
416, 9, 40syl2anc 643 . . . . . . 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 202 . . . . . 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 10380 . . . . . . 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 13102 . . . . . . 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 1184 . . . . . 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 661 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  ||  ( S  -  R ) )
47 prmnn 13082 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
486, 47syl 16 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  NN )
49 moddvds 12859 . . . . . 6  |-  ( ( P  e.  NN  /\  S  e.  ZZ  /\  R  e.  ZZ )  ->  (
( S  mod  P
)  =  ( R  mod  P )  <->  P  ||  ( S  -  R )
) )
5048, 11, 18, 49syl3anc 1184 . . . . 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 224 . . . 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 10647 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  RR+ )
53 elfzle1 11060 . . . . . 6  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  0  <_  S )
5453ad2antrl 709 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
0  <_  S )
55 elfzle2 11061 . . . . . . 7  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  <_  ( P  -  1 ) )
5655ad2antrl 709 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  <_  ( P  - 
1 ) )
57 zltlem1 10328 . . . . . . 7  |-  ( ( S  e.  ZZ  /\  P  e.  ZZ )  ->  ( S  <  P  <->  S  <_  ( P  - 
1 ) ) )
5811, 8, 57syl2anc 643 . . . . . 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 224 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  <  P )
60 modid 11270 . . . . 5  |-  ( ( ( S  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  S  /\  S  <  P ) )  ->  ( S  mod  P )  =  S )
6133, 52, 54, 59, 60syl22anc 1185 . . . 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 13097 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
63 uznn0sub 10517 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( P  -  2 )  e. 
NN0 )
646, 62, 633syl 19 . . . . . . . 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 11396 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( P  -  2
)  e.  NN0 )  ->  ( A ^ ( P  -  2 ) )  e.  ZZ )
669, 64, 65syl2anc 643 . . . . . . 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 10375 . . . . . 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 11275 . . . . . 6  |-  ( ( ( A ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ )  -> 
( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)  =  ( ( A ^ ( P  -  2 ) )  mod  P ) )
6967, 52, 68syl2anc 643 . . . . 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 6091 . . . . 5  |-  ( R  mod  P )  =  ( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)
7169, 70, 23eqtr4g 2493 . . . 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 2476 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  =  R )
7372ex 424 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  (
( A  x.  S
)  -  1 ) )  ->  S  =  R ) )
74 1e0p1 10410 . . . . . . . 8  |-  1  =  ( 0  +  1 )
7574oveq1i 6091 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
76 0z 10293 . . . . . . . 8  |-  0  e.  ZZ
77 fzp1ss 11098 . . . . . . . 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 3378 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
8079sseli 3344 . . . . 5  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  ( 0 ... ( P  -  1 ) ) )
81 eleq1 2496 . . . . 5  |-  ( S  =  R  ->  ( S  e.  ( 0 ... ( P  - 
1 ) )  <->  R  e.  ( 0 ... ( P  -  1 ) ) ) )
8280, 81syl5ibr 213 . . . 4  |-  ( S  =  R  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  ->  S  e.  ( 0 ... ( P  - 
1 ) ) ) )
83 oveq2 6089 . . . . . . 7  |-  ( S  =  R  ->  ( A  x.  S )  =  ( A  x.  R ) )
8483oveq1d 6096 . . . . . 6  |-  ( S  =  R  ->  (
( A  x.  S
)  -  1 )  =  ( ( A  x.  R )  - 
1 ) )
8584breq2d 4224 . . . . 5  |-  ( S  =  R  ->  ( P  ||  ( ( A  x.  S )  - 
1 )  <->  P  ||  (
( A  x.  R
)  -  1 ) ) )
8685biimprd 215 . . . 4  |-  ( S  =  R  ->  ( P  ||  ( ( A  x.  R )  - 
1 )  ->  P  ||  ( ( A  x.  S )  -  1 ) ) )
8782, 86anim12d 547 . . 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 28 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( S  =  R  ->  ( S  e.  ( 0 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  S )  - 
1 ) ) ) )
8973, 88impbid 184 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    <_ cle 9121    - cmin 9291   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   RR+crp 10612   ...cfz 11043    mod cmo 11250   ^cexp 11382    || cdivides 12852    gcd cgcd 13006   Primecprime 13079
This theorem is referenced by:  prmdivdiv  13176  wilthlem1  20851  wilthlem2  20852  modprminveq  28226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080  df-phi 13155
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