Theorem reumodprminv 12244

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 10141	. . . . 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 12101	. . . . 5 |- ( P e. Prime -> P e. NN )
5		prmz 12102	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
6		fzoval 10142	. . . . . . . 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 2247	. . . . . 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 11853	. . . . 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 2177	. . . . . . 7 |- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P )
13	12	modprminv 12240	. . . . . 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 9569	. . . . . . . . . . 11 |- 1 e. ( ZZ>= ` 0 )
17		fzss1 10057	. . . . . . . . . . 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 3154	. . . . . . . . 9 |- ( P e. Prime -> ( s e. ( 1 ... ( P - 1 ) ) -> s e. ( 0 ... ( P - 1 ) ) ) )
20	19	3ad2ant1 1018	. . . . . . . 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 12241	. . . . . . . . . 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 2183	. . . . . . . 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 2550	. . . . 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 1238	. . 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 5879	. . . . . . 7 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( N x. i ) = ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) )
31	30	oveq1d 5886	. . . . . 6 |- ( i = ( ( N ^ ( P - 2 ) ) mod P ) -> ( ( N x. i ) mod P ) = ( ( N x. ( ( N ^ ( P - 2 ) ) mod P ) ) mod P ) )
32	31	eqeq1d 2186	. . . . 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 2184	. . . . . . 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 2477	. . . . 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 2841	. . 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 5879	. . . . 5 |- ( i = s -> ( N x. i ) = ( N x. s ) )
40	39	oveq1d 5886	. . . 4 |- ( i = s -> ( ( N x. i ) mod P ) = ( ( N x. s ) mod P ) )
41	40	eqeq1d 2186	. . 3 |- ( i = s -> ( ( ( N x. i ) mod P ) = 1 <-> ( ( N x. s ) mod P ) = 1 ) )
42	41	reu8 2933	. 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 978   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   E!wreu 2457   C_ wss 3129   class class class wbr 4002   ` cfv 5214  (class class class)co 5871   0cc0 7807   1c1 7808   x. cmul 7812   - cmin 8123   NNcn 8914   2c2 8965   ZZcz 9248   ZZ>=cuz 9523   ...cfz 10003  ..^cfzo 10136   mod cmo 10316   ^cexp 10513   || cdvds 11786   Primecprime 12098

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7898  ax-resscn 7899  ax-1cn 7900  ax-1re 7901  ax-icn 7902  ax-addcl 7903  ax-addrcl 7904  ax-mulcl 7905  ax-mulrcl 7906  ax-addcom 7907  ax-mulcom 7908  ax-addass 7909  ax-mulass 7910  ax-distr 7911  ax-i2m1 7912  ax-0lt1 7913  ax-1rid 7914  ax-0id 7915  ax-rnegex 7916  ax-precex 7917  ax-cnre 7918  ax-pre-ltirr 7919  ax-pre-ltwlin 7920  ax-pre-lttrn 7921  ax-pre-apti 7922  ax-pre-ltadd 7923  ax-pre-mulgt0 7924  ax-pre-mulext 7925  ax-arch 7926  ax-caucvg 7927

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-isom 5223  df-riota 5827  df-ov 5874  df-oprab 5875  df-mpo 5876  df-1st 6137  df-2nd 6138  df-recs 6302  df-irdg 6367  df-frec 6388  df-1o 6413  df-2o 6414  df-oadd 6417  df-er 6531  df-en 6737  df-dom 6738  df-fin 6739  df-sup 6979  df-pnf 7989  df-mnf 7990  df-xr 7991  df-ltxr 7992  df-le 7993  df-sub 8125  df-neg 8126  df-reap 8527  df-ap 8534  df-div 8625  df-inn 8915  df-2 8973  df-3 8974  df-4 8975  df-n0 9172  df-z 9249  df-uz 9524  df-q 9615  df-rp 9649  df-fz 10004  df-fzo 10137  df-fl 10264  df-mod 10317  df-seqfrec 10440  df-exp 10514  df-ihash 10748  df-cj 10843  df-re 10844  df-im 10845  df-rsqrt 10999  df-abs 11000  df-clim 11279  df-proddc 11551  df-dvds 11787  df-gcd 11935  df-prm 12099  df-phi 12202

This theorem is referenced by:  modprm0  12245