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Theorem prmdiveq 12870
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 12869 . . . . . . . . . 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 12778 . . . . . . . . . 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 10814 . . . . . . . . . . . 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 10139 . . . . . . . . . 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 10069 . . . . . . . . . 10  |-  1  e.  ZZ
14 zsubcl 10077 . . . . . . . . . 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 10814 . . . . . . . . . . . 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 10139 . . . . . . . . . 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 10077 . . . . . . . . . 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 12577 . . . . . . . . 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 10134 . . . . . . . . 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 10134 . . . . . . . . 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 8811 . . . . . . . . . 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 9208 . . . . . . . 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 10134 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  A  e.  CC )
31 elfznn0 10838 . . . . . . . . . . . 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 10035 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  RR )
3433recnd 8877 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  CC )
3518zcnd 10134 . . . . . . . . 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 9251 . . . . . . . 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 2331 . . . . . . 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 4063 . . . . . 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 12795 . . . . . . . 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 10138 . . . . . . 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 12797 . . . . . . 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 12777 . . . . . . 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 12554 . . . . . 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 10405 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  RR+ )
53 elfzle1 10815 . . . . . 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 10816 . . . . . . 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 10086 . . . . . . 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 11009 . . . . 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 12792 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
63 uznn0sub 10275 . . . . . . . . 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 11134 . . . . . . . 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 10133 . . . . . 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 11014 . . . . . 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 5884 . . . . 5  |-  ( R  mod  P )  =  ( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)
7169, 70, 23eqtr4g 2353 . . . 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 2336 . . 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 10168 . . . . . . . 8  |-  1  =  ( 0  +  1 )
7574oveq1i 5884 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
76 0z 10051 . . . . . . . 8  |-  0  e.  ZZ
77 fzp1ss 10853 . . . . . . . 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 3221 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
8079sseli 3189 . . . . 5  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  ( 0 ... ( P  -  1 ) ) )
81 eleq1 2356 . . . . 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 5882 . . . . . . 7  |-  ( S  =  R  ->  ( A  x.  S )  =  ( A  x.  R ) )
8483oveq1d 5889 . . . . . 6  |-  ( S  =  R  ->  (
( A  x.  S
)  -  1 )  =  ( ( A  x.  R )  - 
1 ) )
8584breq2d 4051 . . . . 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 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   ...cfz 10798    mod cmo 10989   ^cexp 11120    || cdivides 12547    gcd cgcd 12701   Primecprime 12774
This theorem is referenced by:  prmdivdiv  12871  wilthlem1  20322  wilthlem2  20323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-phi 12850
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