Theorem modprmn0modprm0 12425

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 1002	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> P e. Prime )
2		prmnn 12278	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
3		zmodfzo 10439	. . . . . . . . 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 1019	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( N mod P ) e. ( 0..^ P ) )
7		fzo1fzo0n0 10259	. . . . . . . 8 |- ( ( N mod P ) e. ( 1..^ P ) <-> ( ( N mod P ) e. ( 0..^ P ) /\ ( N mod P ) =/= 0 ) )
8	7	simplbi2com 1455	. . . . . . 7 |- ( ( N mod P ) =/= 0 -> ( ( N mod P ) e. ( 0..^ P ) -> ( N mod P ) e. ( 1..^ P ) ) )
9	8	3ad2ant3 1022	. . . . . 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 12424	. . . 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 1249	. . 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 10222	. . . . . . . . . 10 |- ( j e. ( 0..^ P ) -> j e. ZZ )
16	15	zcnd 9449	. . . . . . . . 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 10436	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ P e. NN ) -> ( N mod P ) e. NN0 )
19		nn0cn 9259	. . . . . . . . . . . 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 1019	. . . . . . . . . 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 8008	. . . . . . . . 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 5938	. . . . . . 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 5937	. . . . . 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 10222	. . . . . . . . 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 9700	. . . . . . . 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 1039	. . . . . . . 8 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> N e. ZZ )
32		zq 9700	. . . . . . . 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 1020	. . . . . . . . 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 9707	. . . . . . . 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 9769	. . . . . . . . . 10 |- ( P e. Prime -> P e. RR+ )
40	39	3ad2ant1 1020	. . . . . . . . 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 9774	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> 0 < P )
43		modqaddmulmod 10483	. . . . . . 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 1257	. . . . . 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 9331	. . . . . . . . . . . . . 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 8048	. . . . . . . . . . . 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 1021	. . . . . . . . . 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 5938	. . . . . . 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 5937	. . . . . 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 2234	. . . . 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 2205	. . . 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 2494	. . 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 980   = wceq 1364   e. wcel 2167   =/= wne 2367   E.wrex 2476   class class class wbr 4033  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   < clt 8061   NNcn 8990   NN0cn0 9249   ZZcz 9326   QQcq 9693   RR+crp 9728  ..^cfzo 10217   mod cmo 10414   Primecprime 12275

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7050  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-proddc 11716  df-dvds 11953  df-gcd 12121  df-prm 12276  df-phi 12379

This theorem is referenced by: (None)