Theorem prmdivdiv 12832

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

Step	Hyp	Ref	Expression
1		fz1ssfz0 10357	. . 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 ) ) )
3	1, 2	sselid 3224	. 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 10294	. . . . . . 7 |- ( A e. ( 1 ... ( P - 1 ) ) -> A e. NN )
6	5	adantl 277	. . . . . 6 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. NN )
7	6	nnzd 9606	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ZZ )
8		prmnn 12705	. . . . . 6 |- ( P e. Prime -> P e. NN )
9		fzm1ndvds 12440	. . . . . 6 |- ( ( P e. NN /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || A )
10	8, 9	sylan 283	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || A )
11		prmdiv.1	. . . . . 6 |- R = ( ( A ^ ( P - 2 ) ) mod P )
12	11	prmdiv 12830	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
13	4, 7, 10, 12	syl3anc 1273	. . . 4 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
14	13	simprd 114	. . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P || ( ( A x. R ) - 1 ) )
15	6	nncnd 9162	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. CC )
16	13	simpld 112	. . . . . . 7 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. ( 1 ... ( P - 1 ) ) )
17		elfznn 10294	. . . . . . 7 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. NN )
18	16, 17	syl 14	. . . . . 6 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. NN )
19	18	nncnd 9162	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. CC )
20	15, 19	mulcomd 8206	. . . 4 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( A x. R ) = ( R x. A ) )
21	20	oveq1d 6038	. . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( ( A x. R ) - 1 ) = ( ( R x. A ) - 1 ) )
22	14, 21	breqtrd 4115	. 2 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P || ( ( R x. A ) - 1 ) )
23		elfzelz 10265	. . . 4 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ZZ )
24	16, 23	syl 14	. . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. ZZ )
25		fzm1ndvds 12440	. . . 4 |- ( ( P e. NN /\ R e. ( 1 ... ( P - 1 ) ) ) -> -. P || R )
26	8, 16, 25	syl2an2r 599	. . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || R )
27		eqid 2230	. . . 4 |- ( ( R ^ ( P - 2 ) ) mod P ) = ( ( R ^ ( P - 2 ) ) mod P )
28	27	prmdiveq 12831	. . 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 ) ) )
29	4, 24, 26, 28	syl3anc 1273	. 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 ) ) )
30	3, 22, 29	mpbi2and 951	1 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A = ( ( R ^ ( P - 2 ) ) mod P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2201   class class class wbr 4089  (class class class)co 6023   0cc0 8037   1c1 8038   x. cmul 8042   - cmin 8355   NNcn 9148   2c2 9199   ZZcz 9484   ...cfz 10248   mod cmo 10590   ^cexp 10806   || cdvds 12371   Primecprime 12702

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155  ax-arch 8156  ax-caucvg 8157

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-isom 5337  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-irdg 6541  df-frec 6562  df-1o 6587  df-2o 6588  df-oadd 6591  df-er 6707  df-en 6915  df-dom 6916  df-fin 6917  df-sup 7188  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-n0 9408  df-z 9485  df-uz 9761  df-q 9859  df-rp 9894  df-fz 10249  df-fzo 10383  df-fl 10536  df-mod 10591  df-seqfrec 10716  df-exp 10807  df-ihash 11044  df-cj 11425  df-re 11426  df-im 11427  df-rsqrt 11581  df-abs 11582  df-clim 11862  df-proddc 12135  df-dvds 12372  df-gcd 12548  df-prm 12703  df-phi 12806

This theorem is referenced by: (None)