Theorem nnnn0modprm0 12953

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 12807	. . . . . 6 |- ( P e. Prime -> P e. NN )
2	1	adantr 276	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> P e. NN )
3		fzo0sn0fzo1 10566	. . . . 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 2302	. . 3 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( I e. ( 0..^ P ) <-> I e. ( { 0 } u. ( 1..^ P ) ) ) )
6		elun 3360	. . . . 5 |- ( I e. ( { 0 } u. ( 1..^ P ) ) <-> ( I e. { 0 } \/ I e. ( 1..^ P ) ) )
7		elsni 3707	. . . . . . 7 |- ( I e. { 0 } -> I = 0 )
8		lbfzo0 10519	. . . . . . . . . . . 12 |- ( 0 e. ( 0..^ P ) <-> P e. NN )
9	1, 8	sylibr 134	. . . . . . . . . . 11 |- ( P e. Prime -> 0 e. ( 0..^ P ) )
10		elfzoelz 10481	. . . . . . . . . . . . . . 15 |- ( N e. ( 1..^ P ) -> N e. ZZ )
11		zcn 9582	. . . . . . . . . . . . . . 15 |- ( N e. ZZ -> N e. CC )
12		mul02 8660	. . . . . . . . . . . . . . . . 17 |- ( N e. CC -> ( 0 x. N ) = 0 )
13	12	oveq2d 6066	. . . . . . . . . . . . . . . 16 |- ( N e. CC -> ( 0 + ( 0 x. N ) ) = ( 0 + 0 ) )
14		00id 8414	. . . . . . . . . . . . . . . 16 |- ( 0 + 0 ) = 0
15	13, 14	eqtrdi 2281	. . . . . . . . . . . . . . 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 6065	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( ( 0 + ( 0 x. N ) ) mod P ) = ( 0 mod P ) )
19		nnq 9965	. . . . . . . . . . . . . . 15 |- ( P e. NN -> P e. QQ )
20	1, 19	syl 14	. . . . . . . . . . . . . 14 |- ( P e. Prime -> P e. QQ )
21	1	nngt0d 9281	. . . . . . . . . . . . . 14 |- ( P e. Prime -> 0 < P )
22		q0mod 10717	. . . . . . . . . . . . . 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 2265	. . . . . . . . . . 11 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( ( 0 + ( 0 x. N ) ) mod P ) = 0 )
26		oveq1 6057	. . . . . . . . . . . . . . 15 |- ( j = 0 -> ( j x. N ) = ( 0 x. N ) )
27	26	oveq2d 6066	. . . . . . . . . . . . . 14 |- ( j = 0 -> ( 0 + ( j x. N ) ) = ( 0 + ( 0 x. N ) ) )
28	27	oveq1d 6065	. . . . . . . . . . . . 13 |- ( j = 0 -> ( ( 0 + ( j x. N ) ) mod P ) = ( ( 0 + ( 0 x. N ) ) mod P ) )
29	28	eqeq1d 2241	. . . . . . . . . . . 12 |- ( j = 0 -> ( ( ( 0 + ( j x. N ) ) mod P ) = 0 <-> ( ( 0 + ( 0 x. N ) ) mod P ) = 0 ) )
30	29	rspcev 2921	. . . . . . . . . . 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 599	. . . . . . . . . 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 6057	. . . . . . . . . . . . 13 |- ( I = 0 -> ( I + ( j x. N ) ) = ( 0 + ( j x. N ) ) )
34	33	oveq1d 6065	. . . . . . . . . . . 12 |- ( I = 0 -> ( ( I + ( j x. N ) ) mod P ) = ( ( 0 + ( j x. N ) ) mod P ) )
35	34	eqeq1d 2241	. . . . . . . . . . 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 2543	. . . . . . . . 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 533	. . . . . . . 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 12952	. . . . . . . 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 1274	. . . . . . 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 724	. . . . 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 1227	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   u. cun 3209   {csn 3689   class class class wbr 4109  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   < clt 8308   NNcn 9237   ZZcz 9577   QQcq 9951  ..^cfzo 10476   mod cmo 10684   Primecprime 12804

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-2o 6648  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-proddc 12237  df-dvds 12474  df-gcd 12650  df-prm 12805  df-phi 12908

This theorem is referenced by:  modprmn0modprm0  12954