ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reumodprminv Unicode version

Theorem reumodprminv 12273
Description: For any prime number and for any positive integer less than this prime number, there is a unique modular inverse of this positive integer. (Contributed by Alexander van der Vekens, 12-May-2018.)
Assertion
Ref Expression
reumodprminv  |-  ( ( P  e.  Prime  /\  N  e.  ( 1..^ P ) )  ->  E! i  e.  ( 1 ... ( P  -  1 ) ) ( ( N  x.  i )  mod 
P )  =  1 )
Distinct variable groups:    i, N    P, i

Proof of Theorem reumodprminv
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 simpl 109 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1..^ P ) )  ->  P  e.  Prime )
2 elfzoelz 10167 . . . . 5  |-  ( N  e.  ( 1..^ P )  ->  N  e.  ZZ )
32adantl 277 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1..^ P ) )  ->  N  e.  ZZ )
4 prmnn 12130 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
5 prmz 12131 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ZZ )
6 fzoval 10168 . . . . . . . 8  |-  ( P  e.  ZZ  ->  (
1..^ P )  =  ( 1 ... ( P  -  1 ) ) )
75, 6syl 14 . . . . . . 7  |-  ( P  e.  Prime  ->  ( 1..^ P )  =  ( 1 ... ( P  -  1 ) ) )
87eleq2d 2259 . . . . . 6  |-  ( P  e.  Prime  ->  ( N  e.  ( 1..^ P )  <->  N  e.  (
1 ... ( P  - 
1 ) ) ) )
98biimpa 296 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ( 1..^ P ) )  ->  N  e.  ( 1 ... ( P  -  1 ) ) )
10 fzm1ndvds 11882 . . . . 5  |-  ( ( P  e.  NN  /\  N  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  N
)
114, 9, 10syl2an2r 595 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1..^ P ) )  ->  -.  P  ||  N )
12 eqid 2189 . . . . . . 7  |-  ( ( N ^ ( P  -  2 ) )  mod  P )  =  ( ( N ^
( P  -  2 ) )  mod  P
)
1312modprminv 12269 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  ->  (
( ( N ^
( P  -  2 ) )  mod  P
)  e.  ( 1 ... ( P  - 
1 ) )  /\  ( ( N  x.  ( ( N ^
( P  -  2 ) )  mod  P
) )  mod  P
)  =  1 ) )
1413simpld 112 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  ->  (
( N ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) )
1513simprd 114 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  ->  (
( N  x.  (
( N ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  1 )
16 1eluzge0 9594 . . . . . . . . . . 11  |-  1  e.  ( ZZ>= `  0 )
17 fzss1 10083 . . . . . . . . . . 11  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) ) )
1816, 17mp1i 10 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) ) )
1918sseld 3169 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( s  e.  ( 1 ... ( P  -  1 ) )  ->  s  e.  ( 0 ... ( P  -  1 ) ) ) )
20193ad2ant1 1020 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  ->  (
s  e.  ( 1 ... ( P  - 
1 ) )  -> 
s  e.  ( 0 ... ( P  - 
1 ) ) ) )
2120imdistani 445 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  /\  s  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  /\  s  e.  ( 0 ... ( P  -  1 ) ) ) )
2212modprminveq 12270 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  ->  (
( s  e.  ( 0 ... ( P  -  1 ) )  /\  ( ( N  x.  s )  mod 
P )  =  1 )  <->  s  =  ( ( N ^ ( P  -  2 ) )  mod  P ) ) )
2322biimpa 296 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  /\  ( s  e.  ( 0 ... ( P  -  1 ) )  /\  ( ( N  x.  s )  mod  P )  =  1 ) )  -> 
s  =  ( ( N ^ ( P  -  2 ) )  mod  P ) )
2423eqcomd 2195 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  /\  ( s  e.  ( 0 ... ( P  -  1 ) )  /\  ( ( N  x.  s )  mod  P )  =  1 ) )  -> 
( ( N ^
( P  -  2 ) )  mod  P
)  =  s )
2524expr 375 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  /\  s  e.  ( 0 ... ( P  -  1 ) ) )  ->  ( (
( N  x.  s
)  mod  P )  =  1  ->  (
( N ^ ( P  -  2 ) )  mod  P )  =  s ) )
2621, 25syl 14 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  /\  s  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( (
( N  x.  s
)  mod  P )  =  1  ->  (
( N ^ ( P  -  2 ) )  mod  P )  =  s ) )
2726ralrimiva 2563 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  ->  A. s  e.  ( 1 ... ( P  -  1 ) ) ( ( ( N  x.  s )  mod  P )  =  1  ->  ( ( N ^ ( P  - 
2 ) )  mod 
P )  =  s ) )
2814, 15, 27jca32 310 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  -.  P  ||  N )  ->  (
( ( N ^
( P  -  2 ) )  mod  P
)  e.  ( 1 ... ( P  - 
1 ) )  /\  ( ( ( N  x.  ( ( N ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P )  =  1  /\  A. s  e.  ( 1 ... ( P  -  1 ) ) ( ( ( N  x.  s )  mod  P )  =  1  ->  ( ( N ^ ( P  - 
2 ) )  mod 
P )  =  s ) ) ) )
291, 3, 11, 28syl3anc 1249 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ( 1..^ P ) )  ->  ( (
( N ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) )  /\  (
( ( N  x.  ( ( N ^
( P  -  2 ) )  mod  P
) )  mod  P
)  =  1  /\ 
A. s  e.  ( 1 ... ( P  -  1 ) ) ( ( ( N  x.  s )  mod 
P )  =  1  ->  ( ( N ^ ( P  - 
2 ) )  mod 
P )  =  s ) ) ) )
30 oveq2 5900 . . . . . . 7  |-  ( i  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  ->  ( N  x.  i )  =  ( N  x.  ( ( N ^
( P  -  2 ) )  mod  P
) ) )
3130oveq1d 5907 . . . . . 6  |-  ( i  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  ->  (
( N  x.  i
)  mod  P )  =  ( ( N  x.  ( ( N ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P ) )
3231eqeq1d 2198 . . . . 5  |-  ( i  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  ->  (
( ( N  x.  i )  mod  P
)  =  1  <->  (
( N  x.  (
( N ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  1 ) )
33 eqeq1 2196 . . . . . . 7  |-  ( i  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  ->  (
i  =  s  <->  ( ( N ^ ( P  - 
2 ) )  mod 
P )  =  s ) )
3433imbi2d 230 . . . . . 6  |-  ( i  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  ->  (
( ( ( N  x.  s )  mod 
P )  =  1  ->  i  =  s )  <->  ( ( ( N  x.  s )  mod  P )  =  1  ->  ( ( N ^ ( P  - 
2 ) )  mod 
P )  =  s ) ) )
3534ralbidv 2490 . . . . 5  |-  ( i  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  ->  ( A. s  e.  (
1 ... ( P  - 
1 ) ) ( ( ( N  x.  s )  mod  P
)  =  1  -> 
i  =  s )  <->  A. s  e.  (
1 ... ( P  - 
1 ) ) ( ( ( N  x.  s )  mod  P
)  =  1  -> 
( ( N ^
( P  -  2 ) )  mod  P
)  =  s ) ) )
3632, 35anbi12d 473 . . . 4  |-  ( i  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  ->  (
( ( ( N  x.  i )  mod 
P )  =  1  /\  A. s  e.  ( 1 ... ( P  -  1 ) ) ( ( ( N  x.  s )  mod  P )  =  1  ->  i  =  s ) )  <->  ( (
( N  x.  (
( N ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  1  /\  A. s  e.  ( 1 ... ( P  - 
1 ) ) ( ( ( N  x.  s )  mod  P
)  =  1  -> 
( ( N ^
( P  -  2 ) )  mod  P
)  =  s ) ) ) )
3736rspcev 2856 . . 3  |-  ( ( ( ( N ^
( P  -  2 ) )  mod  P
)  e.  ( 1 ... ( P  - 
1 ) )  /\  ( ( ( N  x.  ( ( N ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P )  =  1  /\  A. s  e.  ( 1 ... ( P  -  1 ) ) ( ( ( N  x.  s )  mod  P )  =  1  ->  ( ( N ^ ( P  - 
2 ) )  mod 
P )  =  s ) ) )  ->  E. i  e.  (
1 ... ( P  - 
1 ) ) ( ( ( N  x.  i )  mod  P
)  =  1  /\ 
A. s  e.  ( 1 ... ( P  -  1 ) ) ( ( ( N  x.  s )  mod 
P )  =  1  ->  i  =  s ) ) )
3829, 37syl 14 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ( 1..^ P ) )  ->  E. i  e.  ( 1 ... ( P  -  1 ) ) ( ( ( N  x.  i )  mod  P )  =  1  /\  A. s  e.  ( 1 ... ( P  -  1 ) ) ( ( ( N  x.  s )  mod  P )  =  1  ->  i  =  s ) ) )
39 oveq2 5900 . . . . 5  |-  ( i  =  s  ->  ( N  x.  i )  =  ( N  x.  s ) )
4039oveq1d 5907 . . . 4  |-  ( i  =  s  ->  (
( N  x.  i
)  mod  P )  =  ( ( N  x.  s )  mod 
P ) )
4140eqeq1d 2198 . . 3  |-  ( i  =  s  ->  (
( ( N  x.  i )  mod  P
)  =  1  <->  (
( N  x.  s
)  mod  P )  =  1 ) )
4241reu8 2948 . 2  |-  ( E! i  e.  ( 1 ... ( P  - 
1 ) ) ( ( N  x.  i
)  mod  P )  =  1  <->  E. i  e.  ( 1 ... ( P  -  1 ) ) ( ( ( N  x.  i )  mod  P )  =  1  /\  A. s  e.  ( 1 ... ( P  -  1 ) ) ( ( ( N  x.  s )  mod  P )  =  1  ->  i  =  s ) ) )
4338, 42sylibr 134 1  |-  ( ( P  e.  Prime  /\  N  e.  ( 1..^ P ) )  ->  E! i  e.  ( 1 ... ( P  -  1 ) ) ( ( N  x.  i )  mod 
P )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469   E!wreu 2470    C_ wss 3144   class class class wbr 4018   ` cfv 5232  (class class class)co 5892   0cc0 7831   1c1 7832    x. cmul 7836    - cmin 8148   NNcn 8939   2c2 8990   ZZcz 9273   ZZ>=cuz 9548   ...cfz 10028  ..^cfzo 10162    mod cmo 10342   ^cexp 10539    || cdvds 11814   Primecprime 12127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-mulrcl 7930  ax-addcom 7931  ax-mulcom 7932  ax-addass 7933  ax-mulass 7934  ax-distr 7935  ax-i2m1 7936  ax-0lt1 7937  ax-1rid 7938  ax-0id 7939  ax-rnegex 7940  ax-precex 7941  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-ltwlin 7944  ax-pre-lttrn 7945  ax-pre-apti 7946  ax-pre-ltadd 7947  ax-pre-mulgt0 7948  ax-pre-mulext 7949  ax-arch 7950  ax-caucvg 7951
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-isom 5241  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-frec 6411  df-1o 6436  df-2o 6437  df-oadd 6440  df-er 6554  df-en 6760  df-dom 6761  df-fin 6762  df-sup 7003  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018  df-sub 8150  df-neg 8151  df-reap 8552  df-ap 8559  df-div 8650  df-inn 8940  df-2 8998  df-3 8999  df-4 9000  df-n0 9197  df-z 9274  df-uz 9549  df-q 9640  df-rp 9674  df-fz 10029  df-fzo 10163  df-fl 10290  df-mod 10343  df-seqfrec 10466  df-exp 10540  df-ihash 10776  df-cj 10871  df-re 10872  df-im 10873  df-rsqrt 11027  df-abs 11028  df-clim 11307  df-proddc 11579  df-dvds 11815  df-gcd 11964  df-prm 12128  df-phi 12231
This theorem is referenced by:  modprm0  12274
  Copyright terms: Public domain W3C validator