Theorem rpdvds 12010

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 989	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> K e. ZZ )
2		simpl2 990	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> M e. ZZ )
3		gcddvds 11881	. . . . . 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 8916	. . . . . . . . . . 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 2352	. . . . . . . . . . 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 2355	. . . . . . . . 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 4000	. . . . . . . . . . . . 13 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) /\ ( K = 0 /\ M = 0 ) ) -> 0 || N )
17		simpll3 1027	. . . . . . . . . . . . . 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 11737	. . . . . . . . . . . . . 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 991	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> N e. ZZ )
24		gcdeq0 11895	. . . . . . . . . . 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 11880	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ ) /\ -. ( K = 0 /\ M = 0 ) ) -> ( K gcd M ) e. NN )
29	1, 2, 27, 28	syl21anc 1226	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) e. NN )
30	29	nnzd 9303	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) e. ZZ )
31		dvdstr 11754	. . . . . 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 1227	. . . . 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 11882	. . . . 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 1230	. . . 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 4002	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( ( K gcd N ) = 1 /\ M || N ) ) -> ( K gcd M ) <_ 1 )
39		nnle1eq1 8872	. . 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 967   = wceq 1342   e. wcel 2135   =/= wne 2334   class class class wbr 3976  (class class class)co 5836   0cc0 7744   1c1 7745   <_ cle 7925   NNcn 8848   ZZcz 9182   || cdvds 11713   gcd cgcd 11860

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-sup 6940  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-q 9549  df-rp 9581  df-fz 9936  df-fzo 10068  df-fl 10195  df-mod 10248  df-seqfrec 10371  df-exp 10445  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927  df-dvds 11714  df-gcd 11861

This theorem is referenced by: (None)