Theorem rpdvds 12031

Description: If K is relatively prime to N then it is also relatively prime to any divisor M of N. (Contributed by Mario Carneiro, 19-Jun-2015.)

Assertion

Ref	Expression
rpdvds	|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) = 1 )

Proof of Theorem rpdvds

Step	Hyp	Ref	Expression
1		simpl1 990	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> K e. ZZ )
2		simpl2 991	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> M e. ZZ )
3		gcddvds 11896	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( ( K gcd M ) || K /\ ( K gcd M ) || M ) )
4	1, 2, 3	syl2anc 409	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( ( K gcd M ) || K /\ ( K gcd M ) || M ) )
5	4	simpld 111	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) || K )
6	4	simprd 113	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) || M )
7		simprr 522	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> M || N )
8		1ne0 8925	. . . . . . . . . . 11 |- 1 =/= 0
9		simprl 521	. . . . . . . . . . . 12 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd N ) = 1 )
10	9	neeq1d 2354	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( ( K gcd N ) =/= 0 <-> 1 =/= 0 ) )
11	8, 10	mpbiri 167	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd N ) =/= 0 )
12	11	neneqd 2357	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> -. ( K gcd N ) = 0 )
13		simprl 521	. . . . . . . . . . . 12 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> K = 0 )
14		simprr 522	. . . . . . . . . . . . . 14 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> M = 0 )
15		simplrr 526	. . . . . . . . . . . . . 14 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> M || N )
16	14, 15	eqbrtrrd 4006	. . . . . . . . . . . . 13 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> 0 || N )
17		simpll3 1028	. . . . . . . . . . . . . 14 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> N e. ZZ )
18		0dvds 11751	. . . . . . . . . . . . . 14 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
19	17, 18	syl 14	. . . . . . . . . . . . 13 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> ( 0 || N <-> N = 0 ) )
20	16, 19	mpbid 146	. . . . . . . . . . . 12 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> N = 0 )
21	13, 20	jca 304	. . . . . . . . . . 11 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> ( K = 0 /\ N = 0 ) )
22	21	ex 114	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( ( K = 0 /\ M = 0 ) -> ( K = 0 /\ N = 0 ) ) )
23		simpl3 992	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> N e. ZZ )
24		gcdeq0 11910	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) = 0 <-> ( K = 0 /\ N = 0 ) ) )
25	1, 23, 24	syl2anc 409	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( ( K gcd N ) = 0 <-> ( K = 0 /\ N = 0 ) ) )
26	22, 25	sylibrd 168	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( ( K = 0 /\ M = 0 ) -> ( K gcd N ) = 0 ) )
27	12, 26	mtod 653	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> -. ( K = 0 /\ M = 0 ) )
28		gcdn0cl 11895	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ -. ( K = 0 /\ M = 0 ) ) -> ( K gcd M ) e. NN )
29	1, 2, 27, 28	syl21anc 1227	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) e. NN )
30	29	nnzd 9312	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) e. ZZ )
31		dvdstr 11768	. . . . . 6 |- ( ( ( K gcd M ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K gcd M ) || M /\ M || N ) -> ( K gcd M ) || N ) )
32	30, 2, 23, 31	syl3anc 1228	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( ( ( K gcd M ) || M /\ M || N ) -> ( K gcd M ) || N ) )
33	6, 7, 32	mp2and 430	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) || N )
34	12, 25	mtbid 662	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> -. ( K = 0 /\ N = 0 ) )
35		dvdslegcd 11897	. . . . 5 |- ( ( ( ( K gcd M ) e. ZZ /\ K e. ZZ /\ N e. ZZ ) /\ -. ( K = 0 /\ N = 0 ) ) -> ( ( ( K gcd M ) || K /\ ( K gcd M ) || N ) -> ( K gcd M ) <_ ( K gcd N ) ) )
36	30, 1, 23, 34, 35	syl31anc 1231	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( ( ( K gcd M ) || K /\ ( K gcd M ) || N ) -> ( K gcd M ) <_ ( K gcd N ) ) )
37	5, 33, 36	mp2and 430	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) <_ ( K gcd N ) )
38	37, 9	breqtrd 4008	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) <_ 1 )
39		nnle1eq1 8881	. . 3 |- ( ( K gcd M ) e. NN -> ( ( K gcd M ) <_ 1 <-> ( K gcd M ) = 1 ) )
40	29, 39	syl 14	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( ( K gcd M ) <_ 1 <-> ( K gcd M ) = 1 ) )
41	38, 40	mpbid 146	1 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) = 1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982  (class class class)co 5842   0cc0 7753   1c1 7754   <_ cle 7934   NNcn 8857   ZZcz 9191   || cdvds 11727   gcd cgcd 11875

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876

This theorem is referenced by: (None)