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Theorem prmdivdiv 12855
Description: The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.)
Hypothesis
Ref Expression
prmdiv.1  |-  R  =  ( ( A ^
( P  -  2 ) )  mod  P
)
Assertion
Ref Expression
prmdivdiv  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  =  ( ( R ^ ( P  - 
2 ) )  mod 
P ) )

Proof of Theorem prmdivdiv
StepHypRef Expression
1 1e0p1 10152 . . . . 5  |-  1  =  ( 0  +  1 )
21oveq1i 5868 . . . 4  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
3 0z 10035 . . . . 5  |-  0  e.  ZZ
4 fzp1ss 10837 . . . . 5  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... ( P  -  1 ) ) 
C_  ( 0 ... ( P  -  1 ) ) )
53, 4ax-mp 8 . . . 4  |-  ( ( 0  +  1 ) ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
62, 5eqsstri 3208 . . 3  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
7 simpr 447 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  e.  ( 1 ... ( P  -  1 ) ) )
86, 7sseldi 3178 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  e.  ( 0 ... ( P  -  1 ) ) )
9 simpl 443 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
10 elfznn 10819 . . . . . . 7  |-  ( A  e.  ( 1 ... ( P  -  1 ) )  ->  A  e.  NN )
1110adantl 452 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  e.  NN )
1211nnzd 10116 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  e.  ZZ )
13 prmnn 12761 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
14 fzm1ndvds 12580 . . . . . 6  |-  ( ( P  e.  NN  /\  A  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  A
)
1513, 14sylan 457 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  A )
16 prmdiv.1 . . . . . 6  |-  R  =  ( ( A ^
( P  -  2 ) )  mod  P
)
1716prmdiv 12853 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )
189, 12, 15, 17syl3anc 1182 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )
1918simprd 449 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  ||  ( ( A  x.  R )  -  1 ) )
2011nncnd 9762 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  e.  CC )
2118simpld 445 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  R  e.  ( 1 ... ( P  -  1 ) ) )
22 elfznn 10819 . . . . . . 7  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  NN )
2321, 22syl 15 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  R  e.  NN )
2423nncnd 9762 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  R  e.  CC )
2520, 24mulcomd 8856 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( A  x.  R )  =  ( R  x.  A ) )
2625oveq1d 5873 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( A  x.  R
)  -  1 )  =  ( ( R  x.  A )  - 
1 ) )
2719, 26breqtrd 4047 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  ||  ( ( R  x.  A )  -  1 ) )
28 elfzelz 10798 . . . 4  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  ZZ )
2921, 28syl 15 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  R  e.  ZZ )
3013adantr 451 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  NN )
31 fzm1ndvds 12580 . . . 4  |-  ( ( P  e.  NN  /\  R  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  R
)
3230, 21, 31syl2anc 642 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  R )
33 eqid 2283 . . . 4  |-  ( ( R ^ ( P  -  2 ) )  mod  P )  =  ( ( R ^
( P  -  2 ) )  mod  P
)
3433prmdiveq 12854 . . 3  |-  ( ( P  e.  Prime  /\  R  e.  ZZ  /\  -.  P  ||  R )  ->  (
( A  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  (
( R  x.  A
)  -  1 ) )  <->  A  =  (
( R ^ ( P  -  2 ) )  mod  P ) ) )
359, 29, 32, 34syl3anc 1182 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( A  e.  ( 0 ... ( P  -  1 ) )  /\  P  ||  (
( R  x.  A
)  -  1 ) )  <->  A  =  (
( R ^ ( P  -  2 ) )  mod  P ) ) )
368, 27, 35mpbi2and 887 1  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  =  ( ( R ^ ( P  - 
2 ) )  mod 
P ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   2c2 9795   ZZcz 10024   ...cfz 10782    mod cmo 10973   ^cexp 11104    || cdivides 12531   Primecprime 12758
This theorem is referenced by:  wilthlem2  20307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-phi 12834
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