Theorem rpdvds 12066

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 1000	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> K e. ZZ )
2		simpl2 1001	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> M e. ZZ )
3		gcddvds 11931	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( ( K gcd M ) || K /\ ( K gcd M ) || M ) )
4	1, 2, 3	syl2anc 411	. . . . 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 112	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) || K )
6	4	simprd 114	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) || M )
7		simprr 531	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> M || N )
8		1ne0 8960	. . . . . . . . . . 11 |- 1 =/= 0
9		simprl 529	. . . . . . . . . . . 12 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd N ) = 1 )
10	9	neeq1d 2363	. . . . . . . . . . 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 168	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd N ) =/= 0 )
12	11	neneqd 2366	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> -. ( K gcd N ) = 0 )
13		simprl 529	. . . . . . . . . . . 12 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> K = 0 )
14		simprr 531	. . . . . . . . . . . . . 14 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> M = 0 )
15		simplrr 536	. . . . . . . . . . . . . 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 4022	. . . . . . . . . . . . 13 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> 0 || N )
17		simpll3 1038	. . . . . . . . . . . . . 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 11786	. . . . . . . . . . . . . 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 147	. . . . . . . . . . . 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 306	. . . . . . . . . . 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 115	. . . . . . . . . 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 1002	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> N e. ZZ )
24		gcdeq0 11945	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) = 0 <-> ( K = 0 /\ N = 0 ) ) )
25	1, 23, 24	syl2anc 411	. . . . . . . . . 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 169	. . . . . . . . 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 663	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> -. ( K = 0 /\ M = 0 ) )
28		gcdn0cl 11930	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ -. ( K = 0 /\ M = 0 ) ) -> ( K gcd M ) e. NN )
29	1, 2, 27, 28	syl21anc 1237	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) e. NN )
30	29	nnzd 9347	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) e. ZZ )
31		dvdstr 11803	. . . . . 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 1238	. . . . 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 433	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) || N )
34	12, 25	mtbid 672	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> -. ( K = 0 /\ N = 0 ) )
35		dvdslegcd 11932	. . . . 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 1241	. . . 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 433	. . 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 4024	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) <_ 1 )
39		nnle1eq1 8916	. . 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 147	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 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2146   =/= wne 2345   class class class wbr 3998  (class class class)co 5865   0cc0 7786   1c1 7787   <_ cle 7967   NNcn 8892   ZZcz 9226   || cdvds 11762   gcd cgcd 11910

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-n0 9150  df-z 9227  df-uz 9502  df-q 9593  df-rp 9625  df-fz 9980  df-fzo 10113  df-fl 10240  df-mod 10293  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821  df-rsqrt 10975  df-abs 10976  df-dvds 11763  df-gcd 11911

This theorem is referenced by: (None)