Theorem modprmn0modprm0 12394

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 12248	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
3		zmodfzo 10418	. . . . . . . . 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 10250	. . . . . . . 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 12393	. . . 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 10213	. . . . . . . . . 10 |- ( j e. ( 0..^ P ) -> j e. ZZ )
16	15	zcnd 9440	. . . . . . . . 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 10415	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ P e. NN ) -> ( N mod P ) e. NN0 )
19		nn0cn 9250	. . . . . . . . . . . 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 8001	. . . . . . . . 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 5934	. . . . . . 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 5933	. . . . . 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 10213	. . . . . . . . 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 9691	. . . . . . . 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 9691	. . . . . . . 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 9698	. . . . . . . 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 9760	. . . . . . . . . 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 9765	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> 0 < P )
43		modqaddmulmod 10462	. . . . . . 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 9322	. . . . . . . . . . . . . 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 8041	. . . . . . . . . . . 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 5934	. . . . . . 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 5933	. . . . . 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 2231	. . . . 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 2202	. . . 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 2491	. . 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 2164   =/= wne 2364   E.wrex 2473   class class class wbr 4029  (class class class)co 5918   CCcc 7870   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   < clt 8054   NNcn 8982   NN0cn0 9240   ZZcz 9317   QQcq 9684   RR+crp 9719  ..^cfzo 10208   mod cmo 10393   Primecprime 12245

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-2o 6470  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-proddc 11694  df-dvds 11931  df-gcd 12080  df-prm 12246  df-phi 12349

This theorem is referenced by: (None)