Theorem prmdivdiv 12827

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 10352	. . 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 3225	. 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 10289	. . . . . . 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 9601	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ZZ )
8		prmnn 12700	. . . . . 6 |- ( P e. Prime -> P e. NN )
9		fzm1ndvds 12435	. . . . . 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 12825	. . . . 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 9157	. . . . 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 10289	. . . . . . 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 9157	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. CC )
20	15, 19	mulcomd 8201	. . . 4 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( A x. R ) = ( R x. A ) )
21	20	oveq1d 6033	. . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( ( A x. R ) - 1 ) = ( ( R x. A ) - 1 ) )
22	14, 21	breqtrd 4114	. 2 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P || ( ( R x. A ) - 1 ) )
23		elfzelz 10260	. . . 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 12435	. . . 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 2231	. . . 4 |- ( ( R ^ ( P - 2 ) ) mod P ) = ( ( R ^ ( P - 2 ) ) mod P )
28	27	prmdiveq 12826	. . 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 2202   class class class wbr 4088  (class class class)co 6018   0cc0 8032   1c1 8033   x. cmul 8037   - cmin 8350   NNcn 9143   2c2 9194   ZZcz 9479   ...cfz 10243   mod cmo 10585   ^cexp 10801   || cdvds 12366   Primecprime 12697

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-ihash 11039  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-clim 11857  df-proddc 12130  df-dvds 12367  df-gcd 12543  df-prm 12698  df-phi 12801

This theorem is referenced by: (None)