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Theorem prmdivdiv 14194
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 11012 . . . . 5  |-  1  =  ( 0  +  1 )
21oveq1i 6291 . . . 4  |-  ( 1 ... ( P  - 
1 ) )  =  ( ( 0  +  1 ) ... ( P  -  1 ) )
3 0z 10881 . . . . 5  |-  0  e.  ZZ
4 fzp1ss 11740 . . . . 5  |-  ( 0  e.  ZZ  ->  (
( 0  +  1 ) ... ( P  -  1 ) ) 
C_  ( 0 ... ( P  -  1 ) ) )
53, 4ax-mp 5 . . . 4  |-  ( ( 0  +  1 ) ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
62, 5eqsstri 3519 . . 3  |-  ( 1 ... ( P  - 
1 ) )  C_  ( 0 ... ( P  -  1 ) )
7 simpr 461 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  e.  ( 1 ... ( P  -  1 ) ) )
86, 7sseldi 3487 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  e.  ( 0 ... ( P  -  1 ) ) )
9 simpl 457 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
10 elfznn 11723 . . . . . . 7  |-  ( A  e.  ( 1 ... ( P  -  1 ) )  ->  A  e.  NN )
1110adantl 466 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  e.  NN )
1211nnzd 10973 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  e.  ZZ )
13 prmnn 14097 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
14 fzm1ndvds 13915 . . . . . 6  |-  ( ( P  e.  NN  /\  A  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  A
)
1513, 14sylan 471 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  A )
16 prmdiv.1 . . . . . 6  |-  R  =  ( ( A ^
( P  -  2 ) )  mod  P
)
1716prmdiv 14192 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )
189, 12, 15, 17syl3anc 1229 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( R  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( A  x.  R )  - 
1 ) ) )
1918simprd 463 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  ||  ( ( A  x.  R )  -  1 ) )
2011nncnd 10558 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  e.  CC )
2118simpld 459 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  R  e.  ( 1 ... ( P  -  1 ) ) )
22 elfznn 11723 . . . . . . 7  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  NN )
2321, 22syl 16 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  R  e.  NN )
2423nncnd 10558 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  R  e.  CC )
2520, 24mulcomd 9620 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  ( A  x.  R )  =  ( R  x.  A ) )
2625oveq1d 6296 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( A  x.  R
)  -  1 )  =  ( ( R  x.  A )  - 
1 ) )
2719, 26breqtrd 4461 . 2  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  ||  ( ( R  x.  A )  -  1 ) )
28 elfzelz 11697 . . . 4  |-  ( R  e.  ( 1 ... ( P  -  1 ) )  ->  R  e.  ZZ )
2921, 28syl 16 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  R  e.  ZZ )
3013adantr 465 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  NN )
31 fzm1ndvds 13915 . . . 4  |-  ( ( P  e.  NN  /\  R  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  R
)
3230, 21, 31syl2anc 661 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  R )
33 eqid 2443 . . . 4  |-  ( ( R ^ ( P  -  2 ) )  mod  P )  =  ( ( R ^
( P  -  2 ) )  mod  P
)
3433prmdiveq 14193 . . 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 1229 . 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 921 1  |-  ( ( P  e.  Prime  /\  A  e.  ( 1 ... ( P  -  1 ) ) )  ->  A  =  ( ( R ^ ( P  - 
2 ) )  mod 
P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    C_ wss 3461   class class class wbr 4437  (class class class)co 6281   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500    - cmin 9810   NNcn 10542   2c2 10591   ZZcz 10870   ...cfz 11681    mod cmo 11975   ^cexp 12145    || cdvds 13863   Primecprime 14094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-dvds 13864  df-gcd 14022  df-prm 14095  df-phi 14173
This theorem is referenced by:  wilthlem2  23215
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