Theorem dvdsmod 12446

Description: Any number K whose mod base N is divisible by a divisor P of the base is also divisible by P. This means that primes will also be relatively prime to the base when reduced mod N for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)

Assertion

Ref	Expression
dvdsmod	|- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K mod N ) <-> P || K ) )

Proof of Theorem dvdsmod

Step	Hyp	Ref	Expression
1		simpl3 1028	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> K e. ZZ )
2		zq 9865	. . . . 5 |- ( K e. ZZ -> K e. QQ )
3	1, 2	syl 14	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> K e. QQ )
4		simpl2 1027	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> N e. NN )
5		nnq 9872	. . . . 5 |- ( N e. NN -> N e. QQ )
6	4, 5	syl 14	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> N e. QQ )
7	4	nngt0d 9192	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> 0 < N )
8		modqval 10592	. . . 4 |- ( ( K e. QQ /\ N e. QQ /\ 0 < N ) -> ( K mod N ) = ( K - ( N x. ( |_ ` ( K / N ) ) ) ) )
9	3, 6, 7, 8	syl3anc 1273	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K mod N ) = ( K - ( N x. ( |_ ` ( K / N ) ) ) ) )
10	9	breq2d 4101	. 2 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K mod N ) <-> P || ( K - ( N x. ( |_ ` ( K / N ) ) ) ) ) )
11		simpl1 1026	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P e. NN )
12	11	nnzd 9606	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P e. ZZ )
13	4	nnzd 9606	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> N e. ZZ )
14		znq 9863	. . . . . . . . 9 |- ( ( K e. ZZ /\ N e. NN ) -> ( K / N ) e. QQ )
15	1, 4, 14	syl2anc 411	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K / N ) e. QQ )
16	15	flqcld 10543	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( |_ ` ( K / N ) ) e. ZZ )
17		simpr 110	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || N )
18	12, 13, 16, 17	dvdsmultr1d 12416	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || ( N x. ( |_ ` ( K / N ) ) ) )
19	13, 16	zmulcld 9613	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( N x. ( |_ ` ( K / N ) ) ) e. ZZ )
20	19	zcnd 9608	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( N x. ( |_ ` ( K / N ) ) ) e. CC )
21	20	subid1d 8484	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( ( N x. ( |_ ` ( K / N ) ) ) - 0 ) = ( N x. ( |_ ` ( K / N ) ) ) )
22	18, 21	breqtrrd 4117	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || ( ( N x. ( |_ ` ( K / N ) ) ) - 0 ) )
23		0zd 9496	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> 0 e. ZZ )
24		moddvds 12383	. . . . . 6 |- ( ( P e. NN /\ ( N x. ( |_ ` ( K / N ) ) ) e. ZZ /\ 0 e. ZZ ) -> ( ( ( N x. ( |_ ` ( K / N ) ) ) mod P ) = ( 0 mod P ) <-> P || ( ( N x. ( |_ ` ( K / N ) ) ) - 0 ) ) )
25	11, 19, 23, 24	syl3anc 1273	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( ( ( N x. ( |_ ` ( K / N ) ) ) mod P ) = ( 0 mod P ) <-> P || ( ( N x. ( |_ ` ( K / N ) ) ) - 0 ) ) )
26	22, 25	mpbird 167	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( ( N x. ( |_ ` ( K / N ) ) ) mod P ) = ( 0 mod P ) )
27	26	eqeq2d 2242	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( ( K mod P ) = ( ( N x. ( |_ ` ( K / N ) ) ) mod P ) <-> ( K mod P ) = ( 0 mod P ) ) )
28		moddvds 12383	. . . 4 |- ( ( P e. NN /\ K e. ZZ /\ ( N x. ( |_ ` ( K / N ) ) ) e. ZZ ) -> ( ( K mod P ) = ( ( N x. ( |_ ` ( K / N ) ) ) mod P ) <-> P || ( K - ( N x. ( |_ ` ( K / N ) ) ) ) ) )
29	11, 1, 19, 28	syl3anc 1273	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( ( K mod P ) = ( ( N x. ( |_ ` ( K / N ) ) ) mod P ) <-> P || ( K - ( N x. ( |_ ` ( K / N ) ) ) ) ) )
30		moddvds 12383	. . . 4 |- ( ( P e. NN /\ K e. ZZ /\ 0 e. ZZ ) -> ( ( K mod P ) = ( 0 mod P ) <-> P || ( K - 0 ) ) )
31	11, 1, 23, 30	syl3anc 1273	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( ( K mod P ) = ( 0 mod P ) <-> P || ( K - 0 ) ) )
32	27, 29, 31	3bitr3d 218	. 2 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K - ( N x. ( |_ ` ( K / N ) ) ) ) <-> P || ( K - 0 ) ) )
33	1	zcnd 9608	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> K e. CC )
34	33	subid1d 8484	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K - 0 ) = K )
35	34	breq2d 4101	. 2 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K - 0 ) <-> P || K ) )
36	10, 32, 35	3bitrd 214	1 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K mod N ) <-> P || K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2201   class class class wbr 4089   ` cfv 5328  (class class class)co 6023   0cc0 8037   x. cmul 8042   < clt 8219   - cmin 8355   / cdiv 8857   NNcn 9148   ZZcz 9484   QQcq 9858   |_cfl 10534   mod cmo 10590   || cdvds 12371

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155  ax-arch 8156

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-po 4395  df-iso 4396  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-n0 9408  df-z 9485  df-q 9859  df-rp 9894  df-fl 10536  df-mod 10591  df-dvds 12372

This theorem is referenced by:  lgsdir2lem2  15787