Theorem rpdvds 12101

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 11966	. . . . . 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 8989	. . . . . . . . . . 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 2365	. . . . . . . . . . 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 2368	. . . . . . . . 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 4029	. . . . . . . . . . . . 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 11820	. . . . . . . . . . . . . 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 11980	. . . . . . . . . . 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 11965	. . . . . . . 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 9376	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) e. ZZ )
31		dvdstr 11837	. . . . . 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 11967	. . . . 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 4031	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) <_ 1 )
39		nnle1eq1 8945	. . 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 2148   =/= wne 2347   class class class wbr 4005  (class class class)co 5877   0cc0 7813   1c1 7814   <_ cle 7995   NNcn 8921   ZZcz 9255   || cdvds 11796   gcd cgcd 11945

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946

This theorem is referenced by: (None)