Theorem dvdsmod 12044

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 1004	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> K e. ZZ )
2		zq 9717	. . . . 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 1003	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> N e. NN )
5		nnq 9724	. . . . 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 9051	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> 0 < N )
8		modqval 10433	. . . 4 |- ( ( K e. QQ /\ N e. QQ /\ 0 < N ) -> ( K mod N ) = ( K - ( N x. ( |_ ` ( K / N ) ) ) ) )
9	3, 6, 7, 8	syl3anc 1249	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K mod N ) = ( K - ( N x. ( |_ ` ( K / N ) ) ) ) )
10	9	breq2d 4046	. 2 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K mod N ) <-> P || ( K - ( N x. ( |_ ` ( K / N ) ) ) ) ) )
11		simpl1 1002	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P e. NN )
12	11	nnzd 9464	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P e. ZZ )
13	4	nnzd 9464	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> N e. ZZ )
14		znq 9715	. . . . . . . . 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 10384	. . . . . . 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 12014	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || ( N x. ( |_ ` ( K / N ) ) ) )
19	13, 16	zmulcld 9471	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( N x. ( |_ ` ( K / N ) ) ) e. ZZ )
20	19	zcnd 9466	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( N x. ( |_ ` ( K / N ) ) ) e. CC )
21	20	subid1d 8343	. . . . . 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 4062	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || ( ( N x. ( |_ ` ( K / N ) ) ) - 0 ) )
23		0zd 9355	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> 0 e. ZZ )
24		moddvds 11981	. . . . . 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 1249	. . . . 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 2208	. . 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 11981	. . . 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 1249	. . 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 11981	. . . 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 1249	. . 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 9466	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> K e. CC )
34	33	subid1d 8343	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K - 0 ) = K )
35	34	breq2d 4046	. 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 980   = wceq 1364   e. wcel 2167   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   0cc0 7896   x. cmul 7901   < clt 8078   - cmin 8214   / cdiv 8716   NNcn 9007   ZZcz 9343   QQcq 9710   |_cfl 10375   mod cmo 10431   || cdvds 11969

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-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015

This theorem depends on definitions:  df-bi 117  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-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-n0 9267  df-z 9344  df-q 9711  df-rp 9746  df-fl 10377  df-mod 10432  df-dvds 11970

This theorem is referenced by:  lgsdir2lem2  15354