Theorem rpdvds 10987

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 944	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> K e. ZZ )
2		simpl2 945	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> M e. ZZ )
3		gcddvds 10861	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( ( K gcd M ) || K /\ ( K gcd M ) || M ) )
4	1, 2, 3	syl2anc 403	. . . . 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 110	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) || K )
6	4	simprd 112	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) || M )
7		simprr 499	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> M || N )
8		1ne0 8428	. . . . . . . . . . 11 |- 1 =/= 0
9		simprl 498	. . . . . . . . . . . 12 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd N ) = 1 )
10	9	neeq1d 2269	. . . . . . . . . . 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 166	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd N ) =/= 0 )
12	11	neneqd 2272	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> -. ( K gcd N ) = 0 )
13		simprl 498	. . . . . . . . . . . 12 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> K = 0 )
14		simprr 499	. . . . . . . . . . . . . 14 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> M = 0 )
15		simplrr 503	. . . . . . . . . . . . . 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 3844	. . . . . . . . . . . . 13 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> 0 || N )
17		simpll3 982	. . . . . . . . . . . . . 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 10722	. . . . . . . . . . . . . 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 145	. . . . . . . . . . . 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 300	. . . . . . . . . . 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 113	. . . . . . . . . 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 946	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> N e. ZZ )
24		gcdeq0 10874	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) = 0 <-> ( K = 0 /\ N = 0 ) ) )
25	1, 23, 24	syl2anc 403	. . . . . . . . . 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 167	. . . . . . . . 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 622	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> -. ( K = 0 /\ M = 0 ) )
28		gcdn0cl 10860	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ -. ( K = 0 /\ M = 0 ) ) -> ( K gcd M ) e. NN )
29	1, 2, 27, 28	syl21anc 1171	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) e. NN )
30	29	nnzd 8803	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) e. ZZ )
31		dvdstr 10739	. . . . . 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 1172	. . . . 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 424	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) || N )
34	12, 25	mtbid 630	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> -. ( K = 0 /\ N = 0 ) )
35		dvdslegcd 10862	. . . . 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 1175	. . . 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 424	. . 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 3846	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) <_ 1 )
39		nnle1eq1 8384	. . 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 145	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 102   <-> wb 103   /\ w3a 922   = wceq 1287   e. wcel 1436   =/= wne 2251   class class class wbr 3822  (class class class)co 5615   0cc0 7297   1c1 7298   <_ cle 7470   NNcn 8360   ZZcz 8686   || cdvds 10702   gcd cgcd 10844

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378  ax-cnex 7383  ax-resscn 7384  ax-1cn 7385  ax-1re 7386  ax-icn 7387  ax-addcl 7388  ax-addrcl 7389  ax-mulcl 7390  ax-mulrcl 7391  ax-addcom 7392  ax-mulcom 7393  ax-addass 7394  ax-mulass 7395  ax-distr 7396  ax-i2m1 7397  ax-0lt1 7398  ax-1rid 7399  ax-0id 7400  ax-rnegex 7401  ax-precex 7402  ax-cnre 7403  ax-pre-ltirr 7404  ax-pre-ltwlin 7405  ax-pre-lttrn 7406  ax-pre-apti 7407  ax-pre-ltadd 7408  ax-pre-mulgt0 7409  ax-pre-mulext 7410  ax-arch 7411  ax-caucvg 7412

This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-ilim 4172  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-riota 5571  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-frec 6112  df-sup 6626  df-pnf 7471  df-mnf 7472  df-xr 7473  df-ltxr 7474  df-le 7475  df-sub 7602  df-neg 7603  df-reap 7996  df-ap 8003  df-div 8082  df-inn 8361  df-2 8419  df-3 8420  df-4 8421  df-n0 8610  df-z 8687  df-uz 8955  df-q 9040  df-rp 9070  df-fz 9360  df-fzo 9485  df-fl 9608  df-mod 9661  df-iseq 9783  df-iexp 9857  df-cj 10175  df-re 10176  df-im 10177  df-rsqrt 10330  df-abs 10331  df-dvds 10703  df-gcd 10845

This theorem is referenced by: (None)