Theorem dvdsmod 12381

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 1026	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> K e. ZZ )
2		zq 9829	. . . . 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 1025	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> N e. NN )
5		nnq 9836	. . . . 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 9162	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> 0 < N )
8		modqval 10554	. . . 4 |- ( ( K e. QQ /\ N e. QQ /\ 0 < N ) -> ( K mod N ) = ( K - ( N x. ( |_ ` ( K / N ) ) ) ) )
9	3, 6, 7, 8	syl3anc 1271	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K mod N ) = ( K - ( N x. ( |_ ` ( K / N ) ) ) ) )
10	9	breq2d 4095	. 2 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( P || ( K mod N ) <-> P || ( K - ( N x. ( |_ ` ( K / N ) ) ) ) ) )
11		simpl1 1024	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P e. NN )
12	11	nnzd 9576	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P e. ZZ )
13	4	nnzd 9576	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> N e. ZZ )
14		znq 9827	. . . . . . . . 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 10505	. . . . . . 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 12351	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || ( N x. ( |_ ` ( K / N ) ) ) )
19	13, 16	zmulcld 9583	. . . . . . . 8 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( N x. ( |_ ` ( K / N ) ) ) e. ZZ )
20	19	zcnd 9578	. . . . . . 7 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( N x. ( |_ ` ( K / N ) ) ) e. CC )
21	20	subid1d 8454	. . . . . 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 4111	. . . . 5 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> P || ( ( N x. ( |_ ` ( K / N ) ) ) - 0 ) )
23		0zd 9466	. . . . . 6 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> 0 e. ZZ )
24		moddvds 12318	. . . . . 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 1271	. . . . 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 2241	. . 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 12318	. . . 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 1271	. . 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 12318	. . . 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 1271	. . 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 9578	. . . 4 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> K e. CC )
34	33	subid1d 8454	. . 3 |- ( ( ( P e. NN /\ N e. NN /\ K e. ZZ ) /\ P || N ) -> ( K - 0 ) = K )
35	34	breq2d 4095	. 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 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   0cc0 8007   x. cmul 8012   < clt 8189   - cmin 8325   / cdiv 8827   NNcn 9118   ZZcz 9454   QQcq 9822   |_cfl 10496   mod cmo 10552   || cdvds 12306

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-n0 9378  df-z 9455  df-q 9823  df-rp 9858  df-fl 10498  df-mod 10553  df-dvds 12307

This theorem is referenced by:  lgsdir2lem2  15716