Theorem reumodprminv 13032

Description: For any prime number and for any positive integer less than this prime number, there is a unique modular inverse of this positive integer. (Contributed by Alexander van der Vekens, 12-May-2018.)

Assertion

Ref	Expression
reumodprminv	|- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E! i e. ( 1 ... ( P - 1 ) ) ( ( N x. i ) mod P ) = 1 )

Distinct variable groups:   i, N   P, i

Proof of Theorem reumodprminv

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> P e. Prime )
2		elfzoelz 10554	. . . . 5 |- ( N e. ( 1..^ P ) -> N e. ZZ )
3	2	adantl 277	. . . 4 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> N e. ZZ )
4		prmnn 12888	. . . . 5 |- ( P e. Prime -> P e. NN )
5		prmz 12889	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
6		fzoval 10555	. . . . . . . 8 |- ( P e. ZZ -> ( 1..^ P ) = ( 1 ... ( P - 1 ) ) )
7	5, 6	syl 14	. . . . . . 7 |- ( P e. Prime -> ( 1..^ P ) = ( 1 ... ( P - 1 ) ) )
8	7	eleq2d 2308	. . . . . 6 |- ( P e. Prime -> ( N e. ( 1..^ P ) <-> N e. ( 1 ... ( P - 1 ) ) ) )
9	8	biimpa 296	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> N e. ( 1 ... ( P - 1 ) ) )
10		fzm1ndvds 12623	. . . . 5 |- ( ( P e. NN /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N )
11	4, 9, 10	syl2an2r 603	. . . 4 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> -. P || N )
12		eqid 2238	. . . . . . 7 |- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P )
13	12	modprminv 13028	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( ( ( N ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) /\ ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 ) )
14	13	simpld 112	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( ( N ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) )
15	13	simprd 114	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 )
16		1eluzge0 9974	. . . . . . . . . . 11 |- 1 e. ( ZZ>= ` 0 )
17		fzss1 10469	. . . . . . . . . . 11 |- ( 1 e. ( ZZ>= ` 0 ) -> ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) ) )
18	16, 17	mp1i 10	. . . . . . . . . 10 |- ( P e. Prime -> ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) ) )
19	18	sseld 3247	. . . . . . . . 9 |- ( P e. Prime -> ( s e. ( 1 ... ( P - 1 ) ) -> s e. ( 0 ... ( P - 1 ) ) ) )
20	19	3ad2ant1 1049	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( s e. ( 1 ... ( P - 1 ) ) -> s e. ( 0 ... ( P - 1 ) ) ) )
21	20	imdistani 449	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) /\ s e. ( 1 ... ( P - 1 ) ) ) -> ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) /\ s e. ( 0 ... ( P - 1 ) ) ) )
22	12	modprminveq 13029	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( ( s e. ( 0 ... ( P - 1 ) ) /\ ( ( N x. s ) mod P ) = 1 ) <-> s = ( ( N ^ ( P - 2 ) ) mod P ) ) )
23	22	biimpa 296	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) /\ ( s e. ( 0 ... ( P - 1 ) ) /\ ( ( N x. s ) mod P ) = 1 ) ) -> s = ( ( N ^ ( P - 2 ) ) mod P ) )
24	23	eqcomd 2244	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) /\ ( s e. ( 0 ... ( P - 1 ) ) /\ ( ( N x. s ) mod P ) = 1 ) ) -> ( ( N ^ ( P - 2 ) ) mod P ) = s )
25	24	expr 375	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) /\ s e. ( 0 ... ( P - 1 ) ) ) -> ( ( ( N x. s ) mod P ) = 1 -> ( ( N ^ ( P - 2 ) ) mod P ) = s ) )
26	21, 25	syl 14	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) /\ s e. ( 1 ... ( P - 1 ) ) ) -> ( ( ( N x. s ) mod P ) = 1 -> ( ( N ^ ( P - 2 ) ) mod P ) = s ) )
27	26	ralrimiva 2623	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> ( ( N ^ ( P - 2 ) ) mod P ) = s ) )
28	14, 15, 27	jca32 310	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( ( ( N ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) /\ ( ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 /\ A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> ( ( N ^ ( P - 2 ) ) mod P ) = s ) ) ) )
29	1, 3, 11, 28	syl3anc 1278	. . 3 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( ( ( N ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) /\ ( ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 /\ A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> ( ( N ^ ( P - 2 ) ) mod P ) = s ) ) ) )
30		oveq2 6093	. . . . . . 7 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( N x. i ) = ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) )
31	30	oveq1d 6100	. . . . . 6 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( ( N x. i ) mod P ) = ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) )
32	31	eqeq1d 2247	. . . . 5 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( ( ( N x. i ) mod P ) = 1 <-> ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 ) )
33		eqeq1 2245	. . . . . . 7 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( i = s <-> ( ( N ^ ( P - 2 ) ) mod P ) = s ) )
34	33	imbi2d 230	. . . . . 6 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( ( ( ( N x. s ) mod P ) = 1 -> i = s ) <-> ( ( ( N x. s ) mod P ) = 1 -> ( ( N ^ ( P - 2 ) ) mod P ) = s ) ) )
35	34	ralbidv 2550	. . . . 5 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> i = s ) <-> A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> ( ( N ^ ( P - 2 ) ) mod P ) = s ) ) )
36	32, 35	anbi12d 477	. . . 4 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( ( ( ( N x. i ) mod P ) = 1 /\ A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> i = s ) ) <-> ( ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 /\ A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> ( ( N ^ ( P - 2 ) ) mod P ) = s ) ) ) )
37	36	rspcev 2929	. . 3 |- ( ( ( ( N ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) /\ ( ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 /\ A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> ( ( N ^ ( P - 2 ) ) mod P ) = s ) ) ) -> E. i e. ( 1 ... ( P - 1 ) ) ( ( ( N x. i ) mod P ) = 1 /\ A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> i = s ) ) )
38	29, 37	syl 14	. 2 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E. i e. ( 1 ... ( P - 1 ) ) ( ( ( N x. i ) mod P ) = 1 /\ A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> i = s ) ) )
39		oveq2 6093	. . . . 5 |- ( i = s -> ( N x. i ) = ( N x. s ) )
40	39	oveq1d 6100	. . . 4 |- ( i = s -> ( ( N x. i ) mod P ) = ( ( N x. s ) mod P ) )
41	40	eqeq1d 2247	. . 3 |- ( i = s -> ( ( ( N x. i ) mod P ) = 1 <-> ( ( N x. s ) mod P ) = 1 ) )
42	41	reu8 3022	. 2 |- ( E! i e. ( 1 ... ( P - 1 ) ) ( ( N x. i ) mod P ) = 1 <-> E. i e. ( 1 ... ( P - 1 ) ) ( ( ( N x. i ) mod P ) = 1 /\ A. s e. ( 1 ... ( P - 1 ) ) ( ( ( N x. s ) mod P ) = 1 -> i = s ) ) )
43	38, 42	sylibr 134	1 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E! i e. ( 1 ... ( P - 1 ) ) ( ( N x. i ) mod P ) = 1 )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   E!wreu 2530   C_ wss 3220   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   0cc0 8179   1c1 8180   x. cmul 8184   - cmin 8497   NNcn 9304   2c2 9355   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411  ..^cfzo 10549   mod cmo 10759   ^cexp 10975   || cdvds 12554   Primecprime 12885

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-proddc 12318  df-dvds 12555  df-gcd 12731  df-prm 12886  df-phi 12989

This theorem is used by:  modprm0  13033