Theorem modprmn0modprm0 15512

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 1064	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> P e. Prime )
2		prmnn 15388	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
3		zmodfzo 12693	. . . . . . . . 9 |- ( ( N e. ZZ /\ P e. NN ) -> ( N mod P ) e. ( 0..^ P ) )
4	2, 3	sylan2 491	. . . . . . . 8 |- ( ( N e. ZZ /\ P e. Prime ) -> ( N mod P ) e. ( 0..^ P ) )
5	4	ancoms 469	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ ) -> ( N mod P ) e. ( 0..^ P ) )
6	5	3adant3 1081	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( N mod P ) e. ( 0..^ P ) )
7		fzo1fzo0n0 12518	. . . . . . . 8 |- ( ( N mod P ) e. ( 1..^ P ) <-> ( ( N mod P ) e. ( 0..^ P ) /\ ( N mod P ) =/= 0 ) )
8	7	simplbi2com 657	. . . . . . 7 |- ( ( N mod P ) =/= 0 -> ( ( N mod P ) e. ( 0..^ P ) -> ( N mod P ) e. ( 1..^ P ) ) )
9	8	3ad2ant3 1084	. . . . . 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 15	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( N mod P ) e. ( 1..^ P ) )
11	10	adantr 481	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> ( N mod P ) e. ( 1..^ P ) )
12		simpr 477	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> I e. ( 0..^ P ) )
13		nnnn0modprm0 15511	. . . 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 1326	. . 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 12470	. . . . . . . . . 10 |- ( j e. ( 0..^ P ) -> j e. ZZ )
16	15	zcnd 11483	. . . . . . . . 9 |- ( j e. ( 0..^ P ) -> j e. CC )
17	2	anim1i 592	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P e. NN /\ N e. ZZ ) )
18	17	ancomd 467	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ N e. ZZ ) -> ( N e. ZZ /\ P e. NN ) )
19		zmodcl 12690	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ P e. NN ) -> ( N mod P ) e. NN0 )
20		nn0cn 11302	. . . . . . . . . . . 12 |- ( ( N mod P ) e. NN0 -> ( N mod P ) e. CC )
21	18, 19, 20	3syl 18	. . . . . . . . . . 11 |- ( ( P e. Prime /\ N e. ZZ ) -> ( N mod P ) e. CC )
22	21	3adant3 1081	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( N mod P ) e. CC )
23	22	adantr 481	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> ( N mod P ) e. CC )
24		mulcom 10022	. . . . . . . . 9 |- ( ( j e. CC /\ ( N mod P ) e. CC ) -> ( j x. ( N mod P ) ) = ( ( N mod P ) x. j ) )
25	16, 23, 24	syl2anr 495	. . . . . . . 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 ) )
26	25	oveq2d 6666	. . . . . . 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 ) ) )
27	26	oveq1d 6665	. . . . . 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 ) )
28		elfzoelz 12470	. . . . . . . . . 10 |- ( I e. ( 0..^ P ) -> I e. ZZ )
29	28	zred 11482	. . . . . . . . 9 |- ( I e. ( 0..^ P ) -> I e. RR )
30	29	adantl 482	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> I e. RR )
31	30	adantr 481	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> I e. RR )
32		zre 11381	. . . . . . . . . 10 |- ( N e. ZZ -> N e. RR )
33	32	3ad2ant2 1083	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> N e. RR )
34	33	adantr 481	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> N e. RR )
35	34	adantr 481	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> N e. RR )
36	15	adantl 482	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> j e. ZZ )
37	2	nnrpd 11870	. . . . . . . . . 10 |- ( P e. Prime -> P e. RR+ )
38	37	3ad2ant1 1082	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> P e. RR+ )
39	38	adantr 481	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) -> P e. RR+ )
40	39	adantr 481	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> P e. RR+ )
41		modaddmulmod 12737	. . . . . . 7 |- ( ( ( I e. RR /\ N e. RR /\ j e. ZZ ) /\ P e. RR+ ) -> ( ( I + ( ( N mod P ) x. j ) ) mod P ) = ( ( I + ( N x. j ) ) mod P ) )
42	31, 35, 36, 40, 41	syl31anc 1329	. . . . . 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 ) )
43		zcn 11382	. . . . . . . . . . . . . 14 |- ( N e. ZZ -> N e. CC )
44	43	adantr 481	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ j e. ( 0..^ P ) ) -> N e. CC )
45	16	adantl 482	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ j e. ( 0..^ P ) ) -> j e. CC )
46	44, 45	mulcomd 10061	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ j e. ( 0..^ P ) ) -> ( N x. j ) = ( j x. N ) )
47	46	ex 450	. . . . . . . . . . 11 |- ( N e. ZZ -> ( j e. ( 0..^ P ) -> ( N x. j ) = ( j x. N ) ) )
48	47	3ad2ant2 1083	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) -> ( j e. ( 0..^ P ) -> ( N x. j ) = ( j x. N ) ) )
49	48	adantr 481	. . . . . . . . 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 ) ) )
50	49	imp 445	. . . . . . . 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 ) )
51	50	oveq2d 6666	. . . . . . 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 ) ) )
52	51	oveq1d 6665	. . . . . 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 ) )
53	27, 42, 52	3eqtrrd 2661	. . . . 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 ) )
54	53	eqeq1d 2624	. . . 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 ) )
55	54	rexbidva 3049	. . 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 ) )
56	14, 55	mpbird 247	. 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 )
57	56	ex 450	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   NN0cn0 11292   ZZcz 11377   RR+crp 11832  ..^cfzo 12465   mod cmo 12668   Primecprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471

This theorem is referenced by:  cshwsidrepsw  15800