Theorem reumodprminv 12273

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 10167	. . . . 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 12130	. . . . 5 |- ( P e. Prime -> P e. NN )
5		prmz 12131	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
6		fzoval 10168	. . . . . . . 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 2259	. . . . . 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 11882	. . . . 5 |- ( ( P e. NN /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N )
11	4, 9, 10	syl2an2r 595	. . . 4 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> -. P || N )
12		eqid 2189	. . . . . . 7 |- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P )
13	12	modprminv 12269	. . . . . 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 9594	. . . . . . . . . . 11 |- 1 e. ( ZZ>= ` 0 )
17		fzss1 10083	. . . . . . . . . . 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 3169	. . . . . . . . 9 |- ( P e. Prime -> ( s e. ( 1 ... ( P - 1 ) ) -> s e. ( 0 ... ( P - 1 ) ) ) )
20	19	3ad2ant1 1020	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( s e. ( 1 ... ( P - 1 ) ) -> s e. ( 0 ... ( P - 1 ) ) ) )
21	20	imdistani 445	. . . . . . 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 12270	. . . . . . . . . 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 2195	. . . . . . . 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 2563	. . . . 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 1249	. . 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 5900	. . . . . . 7 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( N x. i ) = ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) )
31	30	oveq1d 5907	. . . . . 6 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( ( N x. i ) mod P ) = ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) )
32	31	eqeq1d 2198	. . . . 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 2196	. . . . . . 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 2490	. . . . 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 473	. . . 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 2856	. . 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 5900	. . . . 5 |- ( i = s -> ( N x. i ) = ( N x. s ) )
40	39	oveq1d 5907	. . . 4 |- ( i = s -> ( ( N x. i ) mod P ) = ( ( N x. s ) mod P ) )
41	40	eqeq1d 2198	. . 3 |- ( i = s -> ( ( ( N x. i ) mod P ) = 1 <-> ( ( N x. s ) mod P ) = 1 ) )
42	41	reu8 2948	. 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 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   E!wreu 2470   C_ wss 3144   class class class wbr 4018   ` cfv 5232  (class class class)co 5892   0cc0 7831   1c1 7832   x. cmul 7836   - cmin 8148   NNcn 8939   2c2 8990   ZZcz 9273   ZZ>=cuz 9548   ...cfz 10028  ..^cfzo 10162   mod cmo 10342   ^cexp 10539   || cdvds 11814   Primecprime 12127

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-mulrcl 7930  ax-addcom 7931  ax-mulcom 7932  ax-addass 7933  ax-mulass 7934  ax-distr 7935  ax-i2m1 7936  ax-0lt1 7937  ax-1rid 7938  ax-0id 7939  ax-rnegex 7940  ax-precex 7941  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-ltwlin 7944  ax-pre-lttrn 7945  ax-pre-apti 7946  ax-pre-ltadd 7947  ax-pre-mulgt0 7948  ax-pre-mulext 7949  ax-arch 7950  ax-caucvg 7951

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-isom 5241  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-frec 6411  df-1o 6436  df-2o 6437  df-oadd 6440  df-er 6554  df-en 6760  df-dom 6761  df-fin 6762  df-sup 7003  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018  df-sub 8150  df-neg 8151  df-reap 8552  df-ap 8559  df-div 8650  df-inn 8940  df-2 8998  df-3 8999  df-4 9000  df-n0 9197  df-z 9274  df-uz 9549  df-q 9640  df-rp 9674  df-fz 10029  df-fzo 10163  df-fl 10290  df-mod 10343  df-seqfrec 10466  df-exp 10540  df-ihash 10776  df-cj 10871  df-re 10872  df-im 10873  df-rsqrt 11027  df-abs 11028  df-clim 11307  df-proddc 11579  df-dvds 11815  df-gcd 11964  df-prm 12128  df-phi 12231

This theorem is referenced by:  modprm0  12274