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Theorem prmdiveq 14400
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 755 . . . . . . . 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 14399 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )
43adantr 463 . . . . . . . . 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 461 . . . . . . . 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 997 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  Prime )
7 prmz 14305 . . . . . . . . . 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 998 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  A  e.  ZZ )
10 elfzelz 11691 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  e.  ZZ )
1110ad2antrl 725 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  ZZ )
129, 11zmulcld 10971 . . . . . . . . . 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 10890 . . . . . . . . . 10  |-  1  e.  ZZ
14 zsubcl 10902 . . . . . . . . . 10  |-  ( ( ( A  x.  S
)  e.  ZZ  /\  1  e.  ZZ )  ->  ( ( A  x.  S )  -  1 )  e.  ZZ )
1512, 13, 14sylancl 660 . . . . . . . . 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 457 . . . . . . . . . . . 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 11691 . . . . . . . . . . . 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 10971 . . . . . . . . . 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 10902 . . . . . . . . . 10  |-  ( ( ( A  x.  R
)  e.  ZZ  /\  1  e.  ZZ )  ->  ( ( A  x.  R )  -  1 )  e.  ZZ )
2119, 13, 20sylancl 660 . . . . . . . . 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 14100 . . . . . . . . 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 1226 . . . . . . . 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 677 . . . . . . 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 10966 . . . . . . . . 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 10966 . . . . . . . . 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 9601 . . . . . . . . 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 9957 . . . . . . . 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 10966 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  A  e.  CC )
30 elfznn0 11775 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  e.  NN0 )
3130ad2antrl 725 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  NN0 )
3231nn0red 10849 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  RR )
3332recnd 9611 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  e.  CC )
3418zcnd 10966 . . . . . . . . 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 10008 . . . . . . . 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 2498 . . . . . . 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 4463 . . . . . 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 999 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  -.  P  ||  A )
39 coprm 14325 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
406, 9, 39syl2anc 659 . . . . . . 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 210 . . . . . 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 10970 . . . . . . 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 14327 . . . . . . 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 1226 . . . . . 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 677 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  ||  ( S  -  R ) )
46 prmnn 14304 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
476, 46syl 16 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  NN )
48 moddvds 14077 . . . . . 6  |-  ( ( P  e.  NN  /\  S  e.  ZZ  /\  R  e.  ZZ )  ->  (
( S  mod  P
)  =  ( R  mod  P )  <->  P  ||  ( S  -  R )
) )
4947, 11, 18, 48syl3anc 1226 . . . . 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 232 . . . 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 11257 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  P  e.  RR+ )
52 elfzle1 11692 . . . . . 6  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  0  <_  S )
5352ad2antrl 725 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  -> 
0  <_  S )
54 elfzle2 11693 . . . . . . 7  |-  ( S  e.  ( 0 ... ( P  -  1 ) )  ->  S  <_  ( P  -  1 ) )
5554ad2antrl 725 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  <_  ( P  - 
1 ) )
56 zltlem1 10912 . . . . . . 7  |-  ( ( S  e.  ZZ  /\  P  e.  ZZ )  ->  ( S  <  P  <->  S  <_  ( P  - 
1 ) ) )
5711, 8, 56syl2anc 659 . . . . . 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 232 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  <  P )
59 modid 12002 . . . . 5  |-  ( ( ( S  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  S  /\  S  <  P ) )  ->  ( S  mod  P )  =  S )
6032, 51, 53, 58, 59syl22anc 1227 . . . 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 14319 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
62 uznn0sub 11113 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( P  -  2 )  e. 
NN0 )
636, 61, 623syl 20 . . . . . . . 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 12163 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( P  -  2
)  e.  NN0 )  ->  ( A ^ ( P  -  2 ) )  e.  ZZ )
659, 63, 64syl2anc 659 . . . . . . 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 10965 . . . . . 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 12012 . . . . . 6  |-  ( ( ( A ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ )  -> 
( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)  =  ( ( A ^ ( P  -  2 ) )  mod  P ) )
6866, 51, 67syl2anc 659 . . . . 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 6280 . . . . 5  |-  ( R  mod  P )  =  ( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)
7068, 69, 23eqtr4g 2520 . . . 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 2503 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  /\  ( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  ( ( A  x.  S )  -  1 ) ) )  ->  S  =  R )
7271ex 432 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( S  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  (
( A  x.  S
)  -  1 ) )  ->  S  =  R ) )
73 1e0p1 11004 . . . . . . . 8  |-  1  =  ( 0  +  1 )
7473oveq1i 6280 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
75 0z 10871 . . . . . . . 8  |-  0  e.  ZZ
76 fzp1ss 11735 . . . . . . . 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 3519 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
7978sseli 3485 . . . . 5  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  ( 0 ... ( P  -  1 ) ) )
80 eleq1 2526 . . . . 5  |-  ( S  =  R  ->  ( S  e.  ( 0 ... ( P  - 
1 ) )  <->  R  e.  ( 0 ... ( P  -  1 ) ) ) )
8179, 80syl5ibr 221 . . . 4  |-  ( S  =  R  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  ->  S  e.  ( 0 ... ( P  - 
1 ) ) ) )
82 oveq2 6278 . . . . . . 7  |-  ( S  =  R  ->  ( A  x.  S )  =  ( A  x.  R ) )
8382oveq1d 6285 . . . . . 6  |-  ( S  =  R  ->  (
( A  x.  S
)  -  1 )  =  ( ( A  x.  R )  - 
1 ) )
8483breq2d 4451 . . . . 5  |-  ( S  =  R  ->  ( P  ||  ( ( A  x.  S )  - 
1 )  <->  P  ||  (
( A  x.  R
)  -  1 ) ) )
8584biimprd 223 . . . 4  |-  ( S  =  R  ->  ( P  ||  ( ( A  x.  R )  - 
1 )  ->  P  ||  ( ( A  x.  S )  -  1 ) ) )
8681, 85anim12d 561 . . 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 191 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 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    <_ cle 9618    - cmin 9796   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   RR+crp 11221   ...cfz 11675    mod cmo 11978   ^cexp 12148    || cdvds 14070    gcd cgcd 14228   Primecprime 14301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-dvds 14071  df-gcd 14229  df-prm 14302  df-phi 14380
This theorem is referenced by:  prmdivdiv  14401  modprminveq  14411  wilthlem1  23540  wilthlem2  23541
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