Theorem reumodprminv 12207

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 108	. . . 4 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> P e. Prime )
2		elfzoelz 10103	. . . . 5 |- ( N e. ( 1..^ P ) -> N e. ZZ )
3	2	adantl 275	. . . 4 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> N e. ZZ )
4		prmnn 12064	. . . . 5 |- ( P e. Prime -> P e. NN )
5		prmz 12065	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
6		fzoval 10104	. . . . . . . 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 2240	. . . . . 6 |- ( P e. Prime -> ( N e. ( 1..^ P ) <-> N e. ( 1 ... ( P - 1 ) ) ) )
9	8	biimpa 294	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> N e. ( 1 ... ( P - 1 ) ) )
10		fzm1ndvds 11816	. . . . 5 |- ( ( P e. NN /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N )
11	4, 9, 10	syl2an2r 590	. . . 4 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> -. P || N )
12		eqid 2170	. . . . . . 7 |- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P )
13	12	modprminv 12203	. . . . . 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 111	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( ( N ^ ( P - 2 ) ) mod P ) e. ( 1 ... ( P - 1 ) ) )
15	13	simprd 113	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) = 1 )
16		1eluzge0 9533	. . . . . . . . . . 11 |- 1 e. ( ZZ>= ` 0 )
17		fzss1 10019	. . . . . . . . . . 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 3146	. . . . . . . . 9 |- ( P e. Prime -> ( s e. ( 1 ... ( P - 1 ) ) -> s e. ( 0 ... ( P - 1 ) ) ) )
20	19	3ad2ant1 1013	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( s e. ( 1 ... ( P - 1 ) ) -> s e. ( 0 ... ( P - 1 ) ) ) )
21	20	imdistani 443	. . . . . . 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 12204	. . . . . . . . . 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 294	. . . . . . . . 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 2176	. . . . . . . 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 373	. . . . . . 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 2543	. . . . 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 308	. . . 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 1233	. . 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 5861	. . . . . . 7 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( N x. i ) = ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) )
31	30	oveq1d 5868	. . . . . 6 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( ( N x. i ) mod P ) = ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) )
32	31	eqeq1d 2179	. . . . 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 2177	. . . . . . 7 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( i = s <-> ( ( N ^ ( P - 2 ) ) mod P ) = s ) )
34	33	imbi2d 229	. . . . . 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 2470	. . . . 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 470	. . . 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 2834	. . 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 5861	. . . . 5 |- ( i = s -> ( N x. i ) = ( N x. s ) )
40	39	oveq1d 5868	. . . 4 |- ( i = s -> ( ( N x. i ) mod P ) = ( ( N x. s ) mod P ) )
41	40	eqeq1d 2179	. . 3 |- ( i = s -> ( ( ( N x. i ) mod P ) = 1 <-> ( ( N x. s ) mod P ) = 1 ) )
42	41	reu8 2926	. 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 133	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 103   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   E!wreu 2450   C_ wss 3121   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   0cc0 7774   1c1 7775   x. cmul 7779   - cmin 8090   NNcn 8878   2c2 8929   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965  ..^cfzo 10098   mod cmo 10278   ^cexp 10475   || cdvds 11749   Primecprime 12061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-2o 6396  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-proddc 11514  df-dvds 11750  df-gcd 11898  df-prm 12062  df-phi 12165

This theorem is referenced by:  modprm0  12208