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Theorem prmdiveq 14813
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 774 . . . . . . . 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 14812 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )
43adantr 472 . . . . . . . . 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 470 . . . . . . . 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 1033 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  Prime )
7 prmz 14705 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
86, 7syl 17 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  ZZ )
9 simpl2 1034 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  A  e.  ZZ )
10 elfzelz 11826 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  e.  ZZ )
1110ad2antrl 742 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  ZZ )
129, 11zmulcld 11069 . . . . . . . . . 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 10991 . . . . . . . . . 10  |-  1  e.  ZZ
14 zsubcl 11003 . . . . . . . . . 10  |-  ( ( ( A  x.  S
)  e.  ZZ  /\  1  e.  ZZ )  ->  ( ( A  x.  S )  -  1 )  e.  ZZ )
1512, 13, 14sylancl 675 . . . . . . . . 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 466 . . . . . . . . . . . 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 11826 . . . . . . . . . . . 12  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  ZZ )
1816, 17syl 17 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  R  e.  ZZ )
199, 18zmulcld 11069 . . . . . . . . . 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 11003 . . . . . . . . . 10  |-  ( ( ( A  x.  R
)  e.  ZZ  /\  1  e.  ZZ )  ->  ( ( A  x.  R )  -  1 )  e.  ZZ )
2119, 13, 20sylancl 675 . . . . . . . . 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 14412 . . . . . . . . 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 1292 . . . . . . . 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 693 . . . . . . 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 11064 . . . . . . . . 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 11064 . . . . . . . . 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 1cnd 9677 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
1  e.  CC )
2825, 26, 27nnncan2d 10040 . . . . . . . 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 ) ) )
299zcnd 11064 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  A  e.  CC )
30 elfznn0 11913 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  e.  NN0 )
3130ad2antrl 742 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  NN0 )
3231nn0red 10950 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  RR )
3332recnd 9687 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  CC )
3418zcnd 11064 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  R  e.  CC )
3529, 33, 34subdid 10095 . . . . . . . 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 ) ) )
3628, 35eqtr4d 2508 . . . . . . 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 ) ) )
3724, 36breqtrd 4420 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  ||  ( A  x.  ( S  -  R
) ) )
38 simpl3 1035 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  -.  P  ||  A )
39 coprm 14736 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
406, 9, 39syl2anc 673 . . . . . . 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 ) )
4138, 40mpbid 215 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( P  gcd  A
)  =  1 )
4211, 18zsubcld 11068 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( S  -  R
)  e.  ZZ )
43 coprmdvds 14738 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  A  e.  ZZ  /\  ( S  -  R )  e.  ZZ )  ->  (
( P  ||  ( A  x.  ( S  -  R ) )  /\  ( P  gcd  A )  =  1 )  ->  P  ||  ( S  -  R ) ) )
448, 9, 42, 43syl3anc 1292 . . . . . 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 )
) )
4537, 41, 44mp2and 693 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  ||  ( S  -  R ) )
46 prmnn 14704 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
476, 46syl 17 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  NN )
48 moddvds 14389 . . . . . 6  |-  ( ( P  e.  NN  /\  S  e.  ZZ  /\  R  e.  ZZ )  ->  (
( S  mod  P
)  =  ( R  mod  P )  <->  P  ||  ( S  -  R )
) )
4947, 11, 18, 48syl3anc 1292 . . . . 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 )
) )
5045, 49mpbird 240 . . . 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 ) )
5147nnrpd 11362 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  RR+ )
52 elfzle1 11828 . . . . . 6  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  0  <_  S )
5352ad2antrl 742 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
0  <_  S )
54 elfzle2 11829 . . . . . . 7  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  <_  ( P  -  1 ) )
5554ad2antrl 742 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  <_  ( P  - 
1 ) )
56 zltlem1 11013 . . . . . . 7  |-  ( ( S  e.  ZZ  /\  P  e.  ZZ )  ->  ( S  <  P  <->  S  <_  ( P  - 
1 ) ) )
5711, 8, 56syl2anc 673 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( S  <  P  <->  S  <_  ( P  - 
1 ) ) )
5855, 57mpbird 240 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  <  P )
59 modid 12154 . . . . 5  |-  ( ( ( S  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  S  /\  S  <  P ) )  ->  ( S  mod  P )  =  S )
6032, 51, 53, 58, 59syl22anc 1293 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( S  mod  P
)  =  S )
61 prmuz2 14721 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
62 uznn0sub 11214 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( P  -  2 )  e. 
NN0 )
636, 61, 623syl 18 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( P  -  2 )  e.  NN0 )
64 zexpcl 12325 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( P  -  2
)  e.  NN0 )  ->  ( A ^ ( P  -  2 ) )  e.  ZZ )
659, 63, 64syl2anc 673 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( A ^ ( P  -  2 ) )  e.  ZZ )
6665zred 11063 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( A ^ ( P  -  2 ) )  e.  RR )
67 modabs2 12164 . . . . . 6  |-  ( ( ( A ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ )  -> 
( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)  =  ( ( A ^ ( P  -  2 ) )  mod  P ) )
6866, 51, 67syl2anc 673 . . . . 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 ) )
692oveq1i 6318 . . . . 5  |-  ( R  mod  P )  =  ( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)
7068, 69, 23eqtr4g 2530 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
( R  mod  P
)  =  R )
7150, 60, 703eqtr3d 2513 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  =  R )
7271ex 441 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  (
( A  x.  S
)  -  1 ) )  ->  S  =  R ) )
73 1e0p1 11102 . . . . . . . 8  |-  1  =  ( 0  +  1 )
7473oveq1i 6318 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
75 0z 10972 . . . . . . . 8  |-  0  e.  ZZ
76 fzp1ss 11873 . . . . . . . 8  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... ( P  -  1 ) ) 
C_  ( 0 ... ( P  -  1 ) ) )
7775, 76ax-mp 5 . . . . . . 7  |-  ( ( 0  +  1 ) ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
7874, 77eqsstri 3448 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
7978sseli 3414 . . . . 5  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  ( 0 ... ( P  -  1 ) ) )
80 eleq1 2537 . . . . 5  |-  ( S  =  R  ->  ( S  e.  ( 0 ... ( P  - 
1 ) )  <->  R  e.  ( 0 ... ( P  -  1 ) ) ) )
8179, 80syl5ibr 229 . . . 4  |-  ( S  =  R  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  ->  S  e.  ( 0 ... ( P  - 
1 ) ) ) )
82 oveq2 6316 . . . . . . 7  |-  ( S  =  R  ->  ( A  x.  S )  =  ( A  x.  R ) )
8382oveq1d 6323 . . . . . 6  |-  ( S  =  R  ->  (
( A  x.  S
)  -  1 )  =  ( ( A  x.  R )  - 
1 ) )
8483breq2d 4407 . . . . 5  |-  ( S  =  R  ->  ( P  ||  ( ( A  x.  S )  - 
1 )  <->  P  ||  (
( A  x.  R
)  -  1 ) ) )
8584biimprd 231 . . . 4  |-  ( S  =  R  ->  ( P  ||  ( ( A  x.  R )  - 
1 )  ->  P  ||  ( ( A  x.  S )  -  1 ) ) )
8681, 85anim12d 572 . . 3  |-  ( S  =  R  ->  (
( R  e.  ( 1 ... ( P  -  1 ) )  /\  P  ||  (
( A  x.  R
)  -  1 ) )  ->  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) ) )
873, 86syl5com 30 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( S  =  R  ->  ( S  e.  ( 0 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  S )  - 
1 ) ) ) )
8872, 87impbid 195 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    C_ wss 3390   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   ...cfz 11810    mod cmo 12129   ^cexp 12310    || cdvds 14382    gcd cgcd 14547   Primecprime 14701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793
This theorem is referenced by:  prmdivdiv  14814  modprminveq  14830  wilthlem1  24072  wilthlem2  24073
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