Theorem nnnn0modprm0 12815

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 12669	. . . . . 6 |- ( P e. Prime -> P e. NN )
2	1	adantr 276	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> P e. NN )
3		fzo0sn0fzo1 10454	. . . . 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 3346	. . . . 5 |- ( I e. ( { 0 } u. ( 1..^ P ) ) <-> ( I e. { 0 } \/ I e. ( 1..^ P ) ) )
7		elsni 3685	. . . . . . 7 |- ( I e. { 0 } -> I = 0 )
8		lbfzo0 10408	. . . . . . . . . . . 12 |- ( 0 e. ( 0..^ P ) <-> P e. NN )
9	1, 8	sylibr 134	. . . . . . . . . . 11 |- ( P e. Prime -> 0 e. ( 0..^ P ) )
10		elfzoelz 10370	. . . . . . . . . . . . . . 15 |- ( N e. ( 1..^ P ) -> N e. ZZ )
11		zcn 9472	. . . . . . . . . . . . . . 15 |- ( N e. ZZ -> N e. CC )
12		mul02 8554	. . . . . . . . . . . . . . . . 17 |- ( N e. CC -> ( 0 x. N ) = 0 )
13	12	oveq2d 6027	. . . . . . . . . . . . . . . 16 |- ( N e. CC -> ( 0 + ( 0 x. N ) ) = ( 0 + 0 ) )
14		00id 8308	. . . . . . . . . . . . . . . 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 6026	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ N e. ( 1..^ P ) ) -> ( ( 0 + ( 0 x. N ) ) mod P ) = ( 0 mod P ) )
19		nnq 9855	. . . . . . . . . . . . . . 15 |- ( P e. NN -> P e. QQ )
20	1, 19	syl 14	. . . . . . . . . . . . . 14 |- ( P e. Prime -> P e. QQ )
21	1	nngt0d 9175	. . . . . . . . . . . . . 14 |- ( P e. Prime -> 0 < P )
22		q0mod 10605	. . . . . . . . . . . . . 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 6018	. . . . . . . . . . . . . . 15 |- ( j = 0 -> ( j x. N ) = ( 0 x. N ) )
27	26	oveq2d 6027	. . . . . . . . . . . . . 14 |- ( j = 0 -> ( 0 + ( j x. N ) ) = ( 0 + ( 0 x. N ) ) )
28	27	oveq1d 6026	. . . . . . . . . . . . 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 2908	. . . . . . . . . . 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 6018	. . . . . . . . . . . . 13 |- ( I = 0 -> ( I + ( j x. N ) ) = ( 0 + ( j x. N ) ) )
34	33	oveq1d 6026	. . . . . . . . . . . 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 12814	. . . . . . . 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 3196   {csn 3667   class class class wbr 4084  (class class class)co 6011   CCcc 8018   0cc0 8020   1c1 8021   + caddc 8023   x. cmul 8025   < clt 8202   NNcn 9131   ZZcz 9467   QQcq 9841  ..^cfzo 10365   mod cmo 10572   Primecprime 12666

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 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137  ax-pre-mulext 8138  ax-arch 8139  ax-caucvg 8140

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-po 4389  df-iso 4390  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-isom 5331  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-irdg 6529  df-frec 6550  df-1o 6575  df-2o 6576  df-oadd 6579  df-er 6695  df-en 6903  df-dom 6904  df-fin 6905  df-sup 7172  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-div 8841  df-inn 9132  df-2 9190  df-3 9191  df-4 9192  df-n0 9391  df-z 9468  df-uz 9744  df-q 9842  df-rp 9877  df-fz 10232  df-fzo 10366  df-fl 10518  df-mod 10573  df-seqfrec 10698  df-exp 10789  df-ihash 11026  df-cj 11390  df-re 11391  df-im 11392  df-rsqrt 11546  df-abs 11547  df-clim 11827  df-proddc 12099  df-dvds 12336  df-gcd 12512  df-prm 12667  df-phi 12770

This theorem is referenced by:  modprmn0modprm0  12816