Theorem nnnn0modprm0 12764

Description: For a positive integer and a nonnegative integer both less than a given prime number there is always a second nonnegative integer (less than the given prime number) so that the sum of this second nonnegative integer multiplied with the positive integer and the first nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 8-Nov-2018.)

Assertion

Ref	Expression
nnnn0modprm0	|- ( ( P e. Prime /\ N e. ( 1..^ P ) /\ I e. ( 0..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 )

Distinct variable groups:   j, I   j, N   P, j

Proof of Theorem nnnn0modprm0

Step	Hyp	Ref	Expression
1		prmnn 12618	. . . . . 6 |- ( P e. Prime -> P e. NN )
2	1	adantr 276	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> P e. NN )
3		fzo0sn0fzo1 10414	. . . . 5 |- ( P e. NN -> ( 0..^ P ) = ( { 0 } u. ( 1..^ P ) ) )
4	2, 3	syl 14	. . . 4 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( 0..^ P ) = ( { 0 } u. ( 1..^ P ) ) )
5	4	eleq2d 2299	. . 3 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( I e. ( 0..^ P ) <-> I e. ( { 0 } u. ( 1..^ P ) ) ) )
6		elun 3345	. . . . 5 |- ( I e. ( { 0 } u. ( 1..^ P ) ) <-> ( I e. { 0 } \/ I e. ( 1..^ P ) ) )
7		elsni 3684	. . . . . . 7 |- ( I e. { 0 } -> I = 0 )
8		lbfzo0 10369	. . . . . . . . . . . 12 |- ( 0 e. ( 0..^ P ) <-> P e. NN )
9	1, 8	sylibr 134	. . . . . . . . . . 11 |- ( P e. Prime -> 0 e. ( 0..^ P ) )
10		elfzoelz 10331	. . . . . . . . . . . . . . 15 |- ( N e. ( 1..^ P ) -> N e. ZZ )
11		zcn 9439	. . . . . . . . . . . . . . 15 |- ( N e. ZZ -> N e. CC )
12		mul02 8521	. . . . . . . . . . . . . . . . 17 |- ( N e. CC -> ( 0 x. N ) = 0 )
13	12	oveq2d 6010	. . . . . . . . . . . . . . . 16 |- ( N e. CC -> ( 0 + ( 0 x. N ) ) = ( 0 + 0 ) )
14		00id 8275	. . . . . . . . . . . . . . . 16 |- ( 0 + 0 ) = 0
15	13, 14	eqtrdi 2278	. . . . . . . . . . . . . . 15 |- ( N e. CC -> ( 0 + ( 0 x. N ) ) = 0 )
16	10, 11, 15	3syl 17	. . . . . . . . . . . . . 14 |- ( N e. ( 1..^ P ) -> ( 0 + ( 0 x. N ) ) = 0 )
17	16	adantl 277	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( 0 + ( 0 x. N ) ) = 0 )
18	17	oveq1d 6009	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( ( 0 + ( 0 x. N ) ) mod P ) = ( 0 mod P ) )
19		nnq 9816	. . . . . . . . . . . . . . 15 |- ( P e. NN -> P e. QQ )
20	1, 19	syl 14	. . . . . . . . . . . . . 14 |- ( P e. Prime -> P e. QQ )
21	1	nngt0d 9142	. . . . . . . . . . . . . 14 |- ( P e. Prime -> 0 < P )
22		q0mod 10564	. . . . . . . . . . . . . 14 |- ( ( P e. QQ /\ 0 < P ) -> ( 0 mod P ) = 0 )
23	20, 21, 22	syl2anc 411	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( 0 mod P ) = 0 )
24	23	adantr 276	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( 0 mod P ) = 0 )
25	18, 24	eqtrd 2262	. . . . . . . . . . 11 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( ( 0 + ( 0 x. N ) ) mod P ) = 0 )
26		oveq1 6001	. . . . . . . . . . . . . . 15 |- ( j = 0 -> ( j x. N ) = ( 0 x. N ) )
27	26	oveq2d 6010	. . . . . . . . . . . . . 14 |- ( j = 0 -> ( 0 + ( j x. N ) ) = ( 0 + ( 0 x. N ) ) )
28	27	oveq1d 6009	. . . . . . . . . . . . 13 |- ( j = 0 -> ( ( 0 + ( j x. N ) ) mod P ) = ( ( 0 + ( 0 x. N ) ) mod P ) )
29	28	eqeq1d 2238	. . . . . . . . . . . 12 |- ( j = 0 -> ( ( ( 0 + ( j x. N ) ) mod P ) = 0 <-> ( ( 0 + ( 0 x. N ) ) mod P ) = 0 ) )
30	29	rspcev 2907	. . . . . . . . . . 11 |- ( ( 0 e. ( 0..^ P ) /\ ( ( 0 + ( 0 x. N ) ) mod P ) = 0 ) -> E. j e. ( 0..^ P ) ( ( 0 + ( j x. N ) ) mod P ) = 0 )
31	9, 25, 30	syl2an2r 597	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E. j e. ( 0..^ P ) ( ( 0 + ( j x. N ) ) mod P ) = 0 )
32	31	adantl 277	. . . . . . . . 9 |- ( ( I = 0 /\ ( P e. Prime /\ N e. ( 1..^ P ) ) ) -> E. j e. ( 0..^ P ) ( ( 0 + ( j x. N ) ) mod P ) = 0 )
33		oveq1 6001	. . . . . . . . . . . . 13 |- ( I = 0 -> ( I + ( j x. N ) ) = ( 0 + ( j x. N ) ) )
34	33	oveq1d 6009	. . . . . . . . . . . 12 |- ( I = 0 -> ( ( I + ( j x. N ) ) mod P ) = ( ( 0 + ( j x. N ) ) mod P ) )
35	34	eqeq1d 2238	. . . . . . . . . . 11 |- ( I = 0 -> ( ( ( I + ( j x. N ) ) mod P ) = 0 <-> ( ( 0 + ( j x. N ) ) mod P ) = 0 ) )
36	35	adantr 276	. . . . . . . . . 10 |- ( ( I = 0 /\ ( P e. Prime /\ N e. ( 1..^ P ) ) ) -> ( ( ( I + ( j x. N ) ) mod P ) = 0 <-> ( ( 0 + ( j x. N ) ) mod P ) = 0 ) )
37	36	rexbidv 2531	. . . . . . . . 9 |- ( ( I = 0 /\ ( P e. Prime /\ N e. ( 1..^ P ) ) ) -> ( E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 <-> E. j e. ( 0..^ P ) ( ( 0 + ( j x. N ) ) mod P ) = 0 ) )
38	32, 37	mpbird 167	. . . . . . . 8 |- ( ( I = 0 /\ ( P e. Prime /\ N e. ( 1..^ P ) ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 )
39	38	ex 115	. . . . . . 7 |- ( I = 0 -> ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 ) )
40	7, 39	syl 14	. . . . . 6 |- ( I e. { 0 } -> ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 ) )
41		simpl 109	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> P e. Prime )
42	41	adantl 277	. . . . . . . 8 |- ( ( I e. ( 1..^ P ) /\ ( P e. Prime /\ N e. ( 1..^ P ) ) ) -> P e. Prime )
43		simprr 531	. . . . . . . 8 |- ( ( I e. ( 1..^ P ) /\ ( P e. Prime /\ N e. ( 1..^ P ) ) ) -> N e. ( 1..^ P ) )
44		simpl 109	. . . . . . . 8 |- ( ( I e. ( 1..^ P ) /\ ( P e. Prime /\ N e. ( 1..^ P ) ) ) -> I e. ( 1..^ P ) )
45		modprm0 12763	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1..^ P ) /\ I e. ( 1..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 )
46	42, 43, 44, 45	syl3anc 1271	. . . . . . 7 |- ( ( I e. ( 1..^ P ) /\ ( P e. Prime /\ N e. ( 1..^ P ) ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 )
47	46	ex 115	. . . . . 6 |- ( I e. ( 1..^ P ) -> ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 ) )
48	40, 47	jaoi 721	. . . . 5 |- ( ( I e. { 0 } \/ I e. ( 1..^ P ) ) -> ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 ) )
49	6, 48	sylbi 121	. . . 4 |- ( I e. ( { 0 } u. ( 1..^ P ) ) -> ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 ) )
50	49	com12 30	. . 3 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( I e. ( { 0 } u. ( 1..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 ) )
51	5, 50	sylbid 150	. 2 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( I e. ( 0..^ P ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 ) )
52	51	3impia 1224	1 |- ( ( P e. Prime /\ N e. ( 1..^ P ) /\ I e. ( 0..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   u. cun 3195   {csn 3666   class class class wbr 4082  (class class class)co 5994   CCcc 7985   0cc0 7987   1c1 7988   + caddc 7990   x. cmul 7992   < clt 8169   NNcn 9098   ZZcz 9434   QQcq 9802  ..^cfzo 10326   mod cmo 10531   Primecprime 12615

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-isom 5323  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-frec 6527  df-1o 6552  df-2o 6553  df-oadd 6556  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-sup 7139  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-fz 10193  df-fzo 10327  df-fl 10477  df-mod 10532  df-seqfrec 10657  df-exp 10748  df-ihash 10985  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-clim 11776  df-proddc 12048  df-dvds 12285  df-gcd 12461  df-prm 12616  df-phi 12719

This theorem is referenced by:  modprmn0modprm0  12765