Theorem rpmulgcd2 15370

Description: If M is relatively prime to N, then the GCD of K with M x. N is the product of the GCDs with M and N respectively. (Contributed by Mario Carneiro, 2-Jul-2015.)

Assertion

Ref	Expression
rpmulgcd2	|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd ( M x. N ) ) = ( ( K gcd M ) x. ( K gcd N ) ) )

Proof of Theorem rpmulgcd2

Step	Hyp	Ref	Expression
1		simpl1 1064	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> K e. ZZ )
2		simpl2 1065	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> M e. ZZ )
3		simpl3 1066	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> N e. ZZ )
4	2, 3	zmulcld 11488	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( M x. N ) e. ZZ )
5	1, 4	gcdcld 15230	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd ( M x. N ) ) e. NN0 )
6	1, 2	gcdcld 15230	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) e. NN0 )
7	1, 3	gcdcld 15230	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) e. NN0 )
8	6, 7	nn0mulcld 11356	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. ( K gcd N ) ) e. NN0 )
9		mulgcddvds 15369	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) )
10	9	adantr 481	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd ( M x. N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) )
11		gcddvds 15225	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( ( K gcd M ) || K /\ ( K gcd M ) || M ) )
12	1, 2, 11	syl2anc 693	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) || K /\ ( K gcd M ) || M ) )
13	12	simpld 475	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) || K )
14		gcddvds 15225	. . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) || K /\ ( K gcd N ) || N ) )
15	1, 3, 14	syl2anc 693	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd N ) || K /\ ( K gcd N ) || N ) )
16	15	simpld 475	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) || K )
17	6	nn0zd 11480	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) e. ZZ )
18	7	nn0zd 11480	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) e. ZZ )
19		gcddvds 15225	. . . . . . . . . . 11 |- ( ( ( K gcd M ) e. ZZ /\ ( K gcd N ) e. ZZ ) -> ( ( ( K gcd M ) gcd ( K gcd N ) ) || ( K gcd M ) /\ ( ( K gcd M ) gcd ( K gcd N ) ) || ( K gcd N ) ) )
20	17, 18, 19	syl2anc 693	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( ( K gcd M ) gcd ( K gcd N ) ) || ( K gcd M ) /\ ( ( K gcd M ) gcd ( K gcd N ) ) || ( K gcd N ) ) )
21	20	simpld 475	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || ( K gcd M ) )
22	12	simprd 479	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) || M )
23	17, 18	gcdcld 15230	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) e. NN0 )
24	23	nn0zd 11480	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) e. ZZ )
25		dvdstr 15018	. . . . . . . . . 10 |- ( ( ( ( K gcd M ) gcd ( K gcd N ) ) e. ZZ /\ ( K gcd M ) e. ZZ /\ M e. ZZ ) -> ( ( ( ( K gcd M ) gcd ( K gcd N ) ) || ( K gcd M ) /\ ( K gcd M ) || M ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || M ) )
26	24, 17, 2, 25	syl3anc 1326	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( ( ( K gcd M ) gcd ( K gcd N ) ) || ( K gcd M ) /\ ( K gcd M ) || M ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || M ) )
27	21, 22, 26	mp2and 715	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || M )
28	20	simprd 479	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || ( K gcd N ) )
29	15	simprd 479	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) || N )
30		dvdstr 15018	. . . . . . . . . 10 |- ( ( ( ( K gcd M ) gcd ( K gcd N ) ) e. ZZ /\ ( K gcd N ) e. ZZ /\ N e. ZZ ) -> ( ( ( ( K gcd M ) gcd ( K gcd N ) ) || ( K gcd N ) /\ ( K gcd N ) || N ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || N ) )
31	24, 18, 3, 30	syl3anc 1326	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( ( ( K gcd M ) gcd ( K gcd N ) ) || ( K gcd N ) /\ ( K gcd N ) || N ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || N ) )
32	28, 29, 31	mp2and 715	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || N )
33		dvdsgcd 15261	. . . . . . . . 9 |- ( ( ( ( K gcd M ) gcd ( K gcd N ) ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( ( K gcd M ) gcd ( K gcd N ) ) || M /\ ( ( K gcd M ) gcd ( K gcd N ) ) || N ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || ( M gcd N ) ) )
34	24, 2, 3, 33	syl3anc 1326	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( ( ( K gcd M ) gcd ( K gcd N ) ) || M /\ ( ( K gcd M ) gcd ( K gcd N ) ) || N ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || ( M gcd N ) ) )
35	27, 32, 34	mp2and 715	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || ( M gcd N ) )
36		simpr 477	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( M gcd N ) = 1 )
37	35, 36	breqtrd 4679	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || 1 )
38		dvds1 15041	. . . . . . 7 |- ( ( ( K gcd M ) gcd ( K gcd N ) ) e. NN0 -> ( ( ( K gcd M ) gcd ( K gcd N ) ) || 1 <-> ( ( K gcd M ) gcd ( K gcd N ) ) = 1 ) )
39	23, 38	syl 17	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( ( K gcd M ) gcd ( K gcd N ) ) || 1 <-> ( ( K gcd M ) gcd ( K gcd N ) ) = 1 ) )
40	37, 39	mpbid 222	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) = 1 )
41		coprmdvds2 15368	. . . . 5 |- ( ( ( ( K gcd M ) e. ZZ /\ ( K gcd N ) e. ZZ /\ K e. ZZ ) /\ ( ( K gcd M ) gcd ( K gcd N ) ) = 1 ) -> ( ( ( K gcd M ) || K /\ ( K gcd N ) || K ) -> ( ( K gcd M ) x. ( K gcd N ) ) || K ) )
42	17, 18, 1, 40, 41	syl31anc 1329	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( ( K gcd M ) || K /\ ( K gcd N ) || K ) -> ( ( K gcd M ) x. ( K gcd N ) ) || K ) )
43	13, 16, 42	mp2and 715	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. ( K gcd N ) ) || K )
44		dvdscmul 15008	. . . . . 6 |- ( ( ( K gcd N ) e. ZZ /\ N e. ZZ /\ ( K gcd M ) e. ZZ ) -> ( ( K gcd N ) || N -> ( ( K gcd M ) x. ( K gcd N ) ) || ( ( K gcd M ) x. N ) ) )
45	18, 3, 17, 44	syl3anc 1326	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd N ) || N -> ( ( K gcd M ) x. ( K gcd N ) ) || ( ( K gcd M ) x. N ) ) )
46		dvdsmulc 15009	. . . . . 6 |- ( ( ( K gcd M ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd M ) || M -> ( ( K gcd M ) x. N ) || ( M x. N ) ) )
47	17, 2, 3, 46	syl3anc 1326	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) || M -> ( ( K gcd M ) x. N ) || ( M x. N ) ) )
48	17, 18	zmulcld 11488	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. ( K gcd N ) ) e. ZZ )
49	17, 3	zmulcld 11488	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. N ) e. ZZ )
50		dvdstr 15018	. . . . . 6 |- ( ( ( ( K gcd M ) x. ( K gcd N ) ) e. ZZ /\ ( ( K gcd M ) x. N ) e. ZZ /\ ( M x. N ) e. ZZ ) -> ( ( ( ( K gcd M ) x. ( K gcd N ) ) || ( ( K gcd M ) x. N ) /\ ( ( K gcd M ) x. N ) || ( M x. N ) ) -> ( ( K gcd M ) x. ( K gcd N ) ) || ( M x. N ) ) )
51	48, 49, 4, 50	syl3anc 1326	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( ( ( K gcd M ) x. ( K gcd N ) ) || ( ( K gcd M ) x. N ) /\ ( ( K gcd M ) x. N ) || ( M x. N ) ) -> ( ( K gcd M ) x. ( K gcd N ) ) || ( M x. N ) ) )
52	45, 47, 51	syl2and 500	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( ( K gcd N ) || N /\ ( K gcd M ) || M ) -> ( ( K gcd M ) x. ( K gcd N ) ) || ( M x. N ) ) )
53	29, 22, 52	mp2and 715	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. ( K gcd N ) ) || ( M x. N ) )
54		dvdsgcd 15261	. . . 4 |- ( ( ( ( K gcd M ) x. ( K gcd N ) ) e. ZZ /\ K e. ZZ /\ ( M x. N ) e. ZZ ) -> ( ( ( ( K gcd M ) x. ( K gcd N ) ) || K /\ ( ( K gcd M ) x. ( K gcd N ) ) || ( M x. N ) ) -> ( ( K gcd M ) x. ( K gcd N ) ) || ( K gcd ( M x. N ) ) ) )
55	48, 1, 4, 54	syl3anc 1326	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( ( ( K gcd M ) x. ( K gcd N ) ) || K /\ ( ( K gcd M ) x. ( K gcd N ) ) || ( M x. N ) ) -> ( ( K gcd M ) x. ( K gcd N ) ) || ( K gcd ( M x. N ) ) ) )
56	43, 53, 55	mp2and 715	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. ( K gcd N ) ) || ( K gcd ( M x. N ) ) )
57		dvdseq 15036	. 2 |- ( ( ( ( K gcd ( M x. N ) ) e. NN0 /\ ( ( K gcd M ) x. ( K gcd N ) ) e. NN0 ) /\ ( ( K gcd ( M x. N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) /\ ( ( K gcd M ) x. ( K gcd N ) ) || ( K gcd ( M x. N ) ) ) ) -> ( K gcd ( M x. N ) ) = ( ( K gcd M ) x. ( K gcd N ) ) )
58	5, 8, 10, 56, 57	syl22anc 1327	1 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd ( M x. N ) ) = ( ( K gcd M ) x. ( K gcd N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   1c1 9937   x. cmul 9941   NN0cn0 11292   ZZcz 11377   || cdvds 14983   gcd cgcd 15216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217

This theorem is referenced by:  dvdsmulf1o  24920