Theorem modprmn0modprm0 12962

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 12815	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
3		zmodfzo 10716	. . . . . . . . 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 10529	. . . . . . . 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 12961	. . . 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 10488	. . . . . . . . . 10 |- ( j e. ( 0..^ P ) -> j e. ZZ )
16	15	zcnd 9707	. . . . . . . . 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 10713	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ P e. NN ) -> ( N mod P ) e. NN0 )
19		nn0cn 9511	. . . . . . . . . . . 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 8261	. . . . . . . . 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 6068	. . . . . . 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 6067	. . . . . 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 10488	. . . . . . . . 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 9964	. . . . . . . 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 9964	. . . . . . . 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 9971	. . . . . . . 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 10033	. . . . . . . . . 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 10038	. . . . . . 7 |- ( ( ( ( P e. Prime /\ N e. ZZ /\ ( N mod P ) =/= 0 ) /\ I e. ( 0..^ P ) ) /\ j e. ( 0..^ P ) ) -> 0 < P )
43		modqaddmulmod 10760	. . . . . . 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 9587	. . . . . . . . . . . . . 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 8300	. . . . . . . . . . . 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 6068	. . . . . . 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 6067	. . . . . 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 4111  (class class class)co 6052   CCcc 8130   0cc0 8132   1c1 8133   + caddc 8135   x. cmul 8137   < clt 8313   NNcn 9242   NN0cn0 9501   ZZcz 9582   QQcq 9957   RR+crp 9992  ..^cfzo 10483   mod cmo 10691   Primecprime 12812

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 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-sup 7277  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-fz 10349  df-fzo 10484  df-fl 10637  df-mod 10692  df-seqfrec 10817  df-exp 10908  df-ihash 11147  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-clim 11972  df-proddc 12245  df-dvds 12482  df-gcd 12658  df-prm 12813  df-phi 12916

This theorem is referenced by: (None)