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Theorem reumodprminv 12267
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 10161 . . . . 5  |-  ( N  e.  ( 1..^ P )  ->  N  e.  ZZ )
32adantl 277 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ( 1..^ P ) )  ->  N  e.  ZZ )
4 prmnn 12124 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
5 prmz 12125 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ZZ )
6 fzoval 10162 . . . . . . . 8  |-  ( P  e.  ZZ  ->  (
1..^ P )  =  ( 1 ... ( P  -  1 ) ) )
75, 6syl 14 . . . . . . 7  |-  ( P  e.  Prime  ->  ( 1..^ P )  =  ( 1 ... ( P  -  1 ) ) )
87eleq2d 2257 . . . . . 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 11876 . . . . 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 2187 . . . . . . 7  |-  ( ( N ^ ( P  -  2 ) )  mod  P )  =  ( ( N ^
( P  -  2 ) )  mod  P
)
1312modprminv 12263 . . . . . 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 9588 . . . . . . . . . . 11  |-  1  e.  ( ZZ>= `  0 )
17 fzss1 10077 . . . . . . . . . . 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 3166 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( s  e.  ( 1 ... ( P  -  1 ) )  ->  s  e.  ( 0 ... ( P  -  1 ) ) ) )
20193ad2ant1 1019 . . . . . . . 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 12264 . . . . . . . . . 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 2193 . . . . . . . 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 2560 . . . . 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 1248 . . 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 5896 . . . . . . 7  |-  ( i  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  ->  ( N  x.  i )  =  ( N  x.  ( ( N ^
( P  -  2 ) )  mod  P
) ) )
3130oveq1d 5903 . . . . . 6  |-  ( i  =  ( ( N ^ ( P  - 
2 ) )  mod 
P )  ->  (
( N  x.  i
)  mod  P )  =  ( ( N  x.  ( ( N ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P ) )
3231eqeq1d 2196 . . . . 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 2194 . . . . . . 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 2487 . . . . 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 2853 . . 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 5896 . . . . 5  |-  ( i  =  s  ->  ( N  x.  i )  =  ( N  x.  s ) )
4039oveq1d 5903 . . . 4  |-  ( i  =  s  ->  (
( N  x.  i
)  mod  P )  =  ( ( N  x.  s )  mod 
P ) )
4140eqeq1d 2196 . . 3  |-  ( i  =  s  ->  (
( ( N  x.  i )  mod  P
)  =  1  <->  (
( N  x.  s
)  mod  P )  =  1 ) )
4241reu8 2945 . 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 979    = wceq 1363    e. wcel 2158   A.wral 2465   E.wrex 2466   E!wreu 2467    C_ wss 3141   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   0cc0 7825   1c1 7826    x. cmul 7830    - cmin 8142   NNcn 8933   2c2 8984   ZZcz 9267   ZZ>=cuz 9542   ...cfz 10022  ..^cfzo 10156    mod cmo 10336   ^cexp 10533    || cdvds 11808   Primecprime 12121
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-frec 6406  df-1o 6431  df-2o 6432  df-oadd 6435  df-er 6549  df-en 6755  df-dom 6756  df-fin 6757  df-sup 6997  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-fl 10284  df-mod 10337  df-seqfrec 10460  df-exp 10534  df-ihash 10770  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-clim 11301  df-proddc 11573  df-dvds 11809  df-gcd 11958  df-prm 12122  df-phi 12225
This theorem is referenced by:  modprm0  12268
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