Theorem reumodprminv 12181

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 10078	. . . . 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 12038	. . . . 5 |- ( P e. Prime -> P e. NN )
5		prmz 12039	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
6		fzoval 10079	. . . . . . . 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 2235	. . . . . 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 11790	. . . . 5 |- ( ( P e. NN /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N )
11	4, 9, 10	syl2an2r 585	. . . 4 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> -. P || N )
12		eqid 2165	. . . . . . 7 |- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P )
13	12	modprminv 12177	. . . . . 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 9508	. . . . . . . . . . 11 |- 1 e. ( ZZ>= ` 0 )
17		fzss1 9994	. . . . . . . . . . 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 3140	. . . . . . . . 9 |- ( P e. Prime -> ( s e. ( 1 ... ( P - 1 ) ) -> s e. ( 0 ... ( P - 1 ) ) ) )
20	19	3ad2ant1 1008	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( s e. ( 1 ... ( P - 1 ) ) -> s e. ( 0 ... ( P - 1 ) ) ) )
21	20	imdistani 442	. . . . . . 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 12178	. . . . . . . . . 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 2171	. . . . . . . 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 2538	. . . . 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 1228	. . 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 5849	. . . . . . 7 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( N x. i ) = ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) )
31	30	oveq1d 5856	. . . . . 6 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( ( N x. i ) mod P ) = ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) )
32	31	eqeq1d 2174	. . . . 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 2172	. . . . . . 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 2465	. . . . 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 465	. . . 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 2829	. . 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 5849	. . . . 5 |- ( i = s -> ( N x. i ) = ( N x. s ) )
40	39	oveq1d 5856	. . . 4 |- ( i = s -> ( ( N x. i ) mod P ) = ( ( N x. s ) mod P ) )
41	40	eqeq1d 2174	. . 3 |- ( i = s -> ( ( ( N x. i ) mod P ) = 1 <-> ( ( N x. s ) mod P ) = 1 ) )
42	41	reu8 2921	. 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 968   = wceq 1343   e. wcel 2136   A.wral 2443   E.wrex 2444   E!wreu 2445   C_ wss 3115   class class class wbr 3981   ` cfv 5187  (class class class)co 5841   0cc0 7749   1c1 7750   x. cmul 7754   - cmin 8065   NNcn 8853   2c2 8904   ZZcz 9187   ZZ>=cuz 9462   ...cfz 9940  ..^cfzo 10073   mod cmo 10253   ^cexp 10450   || cdvds 11723   Primecprime 12035

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-irdg 6334  df-frec 6355  df-1o 6380  df-2o 6381  df-oadd 6384  df-er 6497  df-en 6703  df-dom 6704  df-fin 6705  df-sup 6945  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-ihash 10685  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-clim 11216  df-proddc 11488  df-dvds 11724  df-gcd 11872  df-prm 12036  df-phi 12139

This theorem is referenced by:  modprm0  12182