Theorem modprmn0modprm0 12982

Description: For an integer not being 0 modulo a given prime number and a nonnegative integer less than the 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 integer and the first nonnegative integer is 0 ( modulo the given prime number). (Contributed by Alexander van der Vekens, 10-Nov-2018.)

Assertion

Ref	Expression
modprmn0modprm0	|- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( 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 modprmn0modprm0

Step	Hyp	Ref	Expression
1		simpl1 1027	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> P e. Prime )
2		prmnn 12835	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
3		zmodfzo 10736	. . . . . . . . 9 |- ( ( N e. ZZ /\ P e. NN ) -> ( N mod P ) e. ( 0..^ P ) )
4	2, 3	sylan2 286	. . . . . . . 8 |- ( ( N e. ZZ /\ P e. Prime ) -> ( N mod P ) e. ( 0..^ P ) )
5	4	ancoms 268	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ ) -> ( N mod P ) e. ( 0..^ P ) )
6	5	3adant3 1044	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( N mod P ) e. ( 0..^ P ) )
7		fzo1fzo0n0 10547	. . . . . . . 8 |- ( ( N mod P ) e. ( 1..^ P ) <-> ( ( N mod P ) e. ( 0..^ P ) /\ ( N mod P ) =/= 0 ) )
8	7	simplbi2com 1490	. . . . . . 7 |- ( ( N mod P ) =/= 0 -> ( ( N mod P ) e. ( 0..^ P ) -> ( N mod P ) e. ( 1..^ P ) ) )
9	8	3ad2ant3 1047	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( ( N mod P ) e. ( 0..^ P ) -> ( N mod P ) e. ( 1..^ P ) ) )
10	6, 9	mpd 13	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( N mod P ) e. ( 1..^ P ) )
11	10	adantr 276	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> ( N mod P ) e. ( 1..^ P ) )
12		simpr 110	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> I e. ( 0..^ P ) )
13		nnnn0modprm0 12981	. . . 4 |- ( ( P e. Prime /\ ( N mod P ) e. ( 1..^ P ) /\ I e. ( 0..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. ( N mod P ) ) ) mod P ) = 0 )
14	1, 11, 12, 13	syl3anc 1274	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. ( N mod P ) ) ) mod P ) = 0 )
15		elfzoelz 10506	. . . . . . . . . 10 |- ( j e. ( 0..^ P ) -> j e. ZZ )
16	15	zcnd 9722	. . . . . . . . 9 |- ( j e. ( 0..^ P ) -> j e. CC )
17	2	anim1ci 341	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ N e. ZZ ) -> ( N e. ZZ /\ P e. NN ) )
18		zmodcl 10733	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ P e. NN ) -> ( N mod P ) e. NN0 )
19		nn0cn 9526	. . . . . . . . . . . 12 |- ( ( N mod P ) e. NN0 -> ( N mod P ) e. CC )
20	17, 18, 19	3syl 17	. . . . . . . . . . 11 |- ( ( P e. Prime /\ N e. ZZ ) -> ( N mod P ) e. CC )
21	20	3adant3 1044	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( N mod P ) e. CC )
22	21	adantr 276	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> ( N mod P ) e. CC )
23		mulcom 8272	. . . . . . . . 9 |- ( ( j e. CC /\ ( N mod P ) e. CC ) -> ( j x. ( N mod P ) ) = ( ( N mod P ) x. j ) )
24	16, 22, 23	syl2anr 290	. . . . . . . 8 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> ( j x. ( N mod P ) ) = ( ( N mod P ) x. j ) )
25	24	oveq2d 6074	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> ( I + ( j x. ( N mod P ) ) ) = ( I + ( ( N mod P ) x. j ) ) )
26	25	oveq1d 6073	. . . . . 6 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> ( ( I + ( j x. ( N mod P ) ) ) mod P ) = ( ( I + ( ( N mod P ) x. j ) ) mod P ) )
27		elfzoelz 10506	. . . . . . . . 9 |- ( I e. ( 0..^ P ) -> I e. ZZ )
28	27	ad2antlr 489	. . . . . . . 8 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> I e. ZZ )
29		zq 9979	. . . . . . . 8 |- ( I e. ZZ -> I e. QQ )
30	28, 29	syl 14	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> I e. QQ )
31		simpll2 1064	. . . . . . . 8 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> N e. ZZ )
32		zq 9979	. . . . . . . 8 |- ( N e. ZZ -> N e. QQ )
33	31, 32	syl 14	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> N e. QQ )
34	15	adantl 277	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> j e. ZZ )
35	2	3ad2ant1 1045	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> P e. NN )
36	35	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> P e. NN )
37		nnq 9986	. . . . . . . 8 |- ( P e. NN -> P e. QQ )
38	36, 37	syl 14	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> P e. QQ )
39	2	nnrpd 10048	. . . . . . . . . 10 |- ( P e. Prime -> P e. RR+ )
40	39	3ad2ant1 1045	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> P e. RR+ )
41	40	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> P e. RR+ )
42	41	rpgt0d 10053	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> 0 < P )
43		modqaddmulmod 10780	. . . . . . 7 |- ( ( ( I e. QQ /\ N e. QQ /\ j e. ZZ ) /\ ( P e. QQ /\ 0 < P ) ) -> ( ( I + ( ( N mod P ) x. j ) ) mod P ) = ( ( I + ( N x. j ) ) mod P ) )
44	30, 33, 34, 38, 42, 43	syl32anc 1282	. . . . . 6 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> ( ( I + ( ( N mod P ) x. j ) ) mod P ) = ( ( I + ( N x. j ) ) mod P ) )
45		zcn 9602	. . . . . . . . . . . . . 14 |- ( N e. ZZ -> N e. CC )
46	45	adantr 276	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ j e. ( 0..^ P ) ) -> N e. CC )
47	16	adantl 277	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ j e. ( 0..^ P ) ) -> j e. CC )
48	46, 47	mulcomd 8311	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ j e. ( 0..^ P ) ) -> ( N x. j ) = ( j x. N ) )
49	48	ex 115	. . . . . . . . . . 11 |- ( N e. ZZ -> ( j e. ( 0..^ P ) -> ( N x. j ) = ( j x. N ) ) )
50	49	3ad2ant2 1046	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( j e. ( 0..^ P ) -> ( N x. j ) = ( j x. N ) ) )
51	50	adantr 276	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> ( j e. ( 0..^ P ) -> ( N x. j ) = ( j x. N ) ) )
52	51	imp 124	. . . . . . . 8 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> ( N x. j ) = ( j x. N ) )
53	52	oveq2d 6074	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> ( I + ( N x. j ) ) = ( I + ( j x. N ) ) )
54	53	oveq1d 6073	. . . . . 6 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> ( ( I + ( N x. j ) ) mod P ) = ( ( I + ( j x. N ) ) mod P ) )
55	26, 44, 54	3eqtrrd 2272	. . . . 5 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> ( ( I + ( j x. N ) ) mod P ) = ( ( I + ( j x. ( N mod P ) ) ) mod P ) )
56	55	eqeq1d 2243	. . . 4 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> ( ( ( I + ( j x. N ) ) mod P ) = 0 <-> ( ( I + ( j x. ( N mod P ) ) ) mod P ) = 0 ) )
57	56	rexbidva 2541	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> ( E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 <-> E. j e. ( 0..^ P ) ( ( I + ( j x. ( N mod P ) ) ) mod P ) = 0 ) )
58	14, 57	mpbird 167	. 2 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 )
59	58	ex 115	1 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( I e. ( 0..^ P ) -> E. j e. ( 0..^ P ) ( ( I + ( j x. N ) ) mod P ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   E.wrex 2523   class class class wbr 4114  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   NNcn 9257   NN0cn0 9516   ZZcz 9597   QQcq 9972   RR+crp 10007  ..^cfzo 10501   mod cmo 10711   Primecprime 12832

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-n0 9517  df-z 9598  df-uz 9875  df-q 9973  df-rp 10008  df-fz 10365  df-fzo 10502  df-fl 10657  df-mod 10712  df-seqfrec 10837  df-exp 10928  df-ihash 11167  df-cj 11555  df-re 11556  df-im 11557  df-rsqrt 11711  df-abs 11712  df-clim 11992  df-proddc 12265  df-dvds 12502  df-gcd 12678  df-prm 12833  df-phi 12936

This theorem is referenced by: (None)