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Theorem prmdivdiv 12934
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 fz1ssfz0 10451 . . 3 |- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
2 simpr 110 . . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ( 1 ... ( P - 1 ) ) )
31, 2sselid 3236 . 2 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ( 0 ... ( P - 1 ) ) )
4 simpl 109 . . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P e. Prime )
5 elfznn 10388 . . . . . . 7 |- ( A e. ( 1 ... ( P - 1 ) ) -> A e. NN )
65adantl 277 . . . . . 6 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. NN )
76nnzd 9699 . . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ZZ )
8 prmnn 12807 . . . . . 6 |- ( P e. Prime -> P e. NN )
9 fzm1ndvds 12542 . . . . . 6 |- ( ( P e. NN /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || A )
108, 9sylan 283 . . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || A )
11 prmdiv.1 . . . . . 6 |- R = ( ( A ^ ( P - 2 ) ) mod P )
1211prmdiv 12932 . . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
134, 7, 10, 12syl3anc 1274 . . . 4 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
1413simprd 114 . . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P || ( ( A x. R ) - 1 ) )
156nncnd 9251 . . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. CC )
1613simpld 112 . . . . . . 7 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. ( 1 ... ( P - 1 ) ) )
17 elfznn 10388 . . . . . . 7 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. NN )
1816, 17syl 14 . . . . . 6 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. NN )
1918nncnd 9251 . . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. CC )
2015, 19mulcomd 8295 . . . 4 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( A x. R ) = ( R x. A ) )
2120oveq1d 6065 . . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( ( A x. R ) - 1 ) = ( ( R x. A ) - 1 ) )
2214, 21breqtrd 4135 . 2 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P || ( ( R x. A ) - 1 ) )
23 elfzelz 10359 . . . 4 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ZZ )
2416, 23syl 14 . . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. ZZ )
25 fzm1ndvds 12542 . . . 4 |- ( ( P e. NN /\ R e. ( 1 ... ( P - 1 ) ) ) -> -. P || R )
268, 16, 25syl2an2r 599 . . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || R )
27 eqid 2232 . . . 4 |- ( ( R ^ ( P - 2 ) ) mod P ) = ( ( R ^ ( P - 2 ) ) mod P )
2827prmdiveq 12933 . . 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 ) ) )
294, 24, 26, 28syl3anc 1274 . 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 ) ) )
303, 22, 29mpbi2and 952 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   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   0cc0 8127   1c1 8128   x. cmul 8132   - cmin 8444   NNcn 9237   2c2 9288   ZZcz 9577   ...cfz 10342   mod cmo 10684   ^cexp 10900   || cdvds 12473   Primecprime 12804
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247
This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-2o 6648  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-proddc 12237  df-dvds 12474  df-gcd 12650  df-prm 12805  df-phi 12908
This theorem is referenced by: (None)
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