Theorem dvdsmod 11915

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 9668	. . . . 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 9675	. . . . 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 9004	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> 0 < N )
8		modqval 10367	. . . 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 4037	. 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 9415	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P e. ZZ )
13	4	nnzd 9415	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> N e. ZZ )
14		znq 9666	. . . . . . . . 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 10320	. . . . . . 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 11886	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || ( N x. ( |_ ` ( K / N ) ) ) )
19	13, 16	zmulcld 9422	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( N x. ( |_ ` ( K / N ) ) ) e. ZZ )
20	19	zcnd 9417	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( N x. ( |_ ` ( K / N ) ) ) e. CC )
21	20	subid1d 8298	. . . . . 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 4053	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || ( ( N x. ( |_ ` ( K / N ) ) ) - 0 ) )
23		0zd 9306	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> 0 e. ZZ )
24		moddvds 11853	. . . . . 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 2201	. . 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 11853	. . . 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 11853	. . . 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 9417	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> K e. CC )
34	33	subid1d 8298	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K - 0 ) = K )
35	34	breq2d 4037	. 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 2160   class class class wbr 4025   ` cfv 5242  (class class class)co 5904   0cc0 7851   x. cmul 7856   < clt 8033   - cmin 8169   / cdiv 8670   NNcn 8960   ZZcz 9294   QQcq 9661   |_cfl 10311   mod cmo 10365   || cdvds 11841

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-cnex 7942  ax-resscn 7943  ax-1cn 7944  ax-1re 7945  ax-icn 7946  ax-addcl 7947  ax-addrcl 7948  ax-mulcl 7949  ax-mulrcl 7950  ax-addcom 7951  ax-mulcom 7952  ax-addass 7953  ax-mulass 7954  ax-distr 7955  ax-i2m1 7956  ax-0lt1 7957  ax-1rid 7958  ax-0id 7959  ax-rnegex 7960  ax-precex 7961  ax-cnre 7962  ax-pre-ltirr 7963  ax-pre-ltwlin 7964  ax-pre-lttrn 7965  ax-pre-apti 7966  ax-pre-ltadd 7967  ax-pre-mulgt0 7968  ax-pre-mulext 7969  ax-arch 7970

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3150  df-un 3152  df-in 3154  df-ss 3161  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-id 4318  df-po 4321  df-iso 4322  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-fv 5250  df-riota 5859  df-ov 5907  df-oprab 5908  df-mpo 5909  df-1st 6173  df-2nd 6174  df-pnf 8035  df-mnf 8036  df-xr 8037  df-ltxr 8038  df-le 8039  df-sub 8171  df-neg 8172  df-reap 8573  df-ap 8580  df-div 8671  df-inn 8961  df-n0 9218  df-z 9295  df-q 9662  df-rp 9696  df-fl 10313  df-mod 10366  df-dvds 11842

This theorem is referenced by:  lgsdir2lem2  14969