Theorem dvdsmod 11560

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 986	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> K e. ZZ )
2		zq 9418	. . . . 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 985	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> N e. NN )
5		nnq 9425	. . . . 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 8764	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> 0 < N )
8		modqval 10097	. . . 4 |- ( ( K e. QQ /\ N e. QQ /\ 0 < N ) -> ( K mod N ) = ( K - ( N x. ( |_ ` ( K / N ) ) ) ) )
9	3, 6, 7, 8	syl3anc 1216	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K mod N ) = ( K - ( N x. ( |_ ` ( K / N ) ) ) ) )
10	9	breq2d 3941	. 2 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K mod N ) <-> P || ( K - ( N x. ( |_ ` ( K / N ) ) ) ) ) )
11		simpl1 984	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P e. NN )
12	11	nnzd 9172	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P e. ZZ )
13	4	nnzd 9172	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> N e. ZZ )
14		znq 9416	. . . . . . . . 9 |- ( ( K e. ZZ /\ N e. NN ) -> ( K / N ) e. QQ )
15	1, 4, 14	syl2anc 408	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K / N ) e. QQ )
16	15	flqcld 10050	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( |_ ` ( K / N ) ) e. ZZ )
17		simpr 109	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || N )
18	12, 13, 16, 17	dvdsmultr1d 11532	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || ( N x. ( |_ ` ( K / N ) ) ) )
19	13, 16	zmulcld 9179	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( N x. ( |_ ` ( K / N ) ) ) e. ZZ )
20	19	zcnd 9174	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( N x. ( |_ ` ( K / N ) ) ) e. CC )
21	20	subid1d 8062	. . . . . 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 3956	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || ( ( N x. ( |_ ` ( K / N ) ) ) - 0 ) )
23		0zd 9066	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> 0 e. ZZ )
24		moddvds 11502	. . . . . 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 1216	. . . . 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 166	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( ( N x. ( |_ ` ( K / N ) ) ) mod P ) = ( 0 mod P ) )
27	26	eqeq2d 2151	. . 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 11502	. . . 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 1216	. . 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 11502	. . . 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 1216	. . 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 217	. 2 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K - ( N x. ( |_ ` ( K / N ) ) ) ) <-> P || ( K - 0 ) ) )
33	1	zcnd 9174	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> K e. CC )
34	33	subid1d 8062	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K - 0 ) = K )
35	34	breq2d 3941	. 2 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K - 0 ) <-> P || K ) )
36	10, 32, 35	3bitrd 213	1 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K mod N ) <-> P || K ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   0cc0 7620   x. cmul 7625   < clt 7800   - cmin 7933   / cdiv 8432   NNcn 8720   ZZcz 9054   QQcq 9411   |_cfl 10041   mod cmo 10095   || cdvds 11493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-q 9412  df-rp 9442  df-fl 10043  df-mod 10096  df-dvds 11494

This theorem is referenced by: (None)