Theorem rpmulgcd2 11765

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 984	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> K e. ZZ )
2		simpl2 985	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> M e. ZZ )
3		simpl3 986	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> N e. ZZ )
4	2, 3	zmulcld 9172	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( M x. N ) e. ZZ )
5	1, 4	gcdcld 11646	. 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 11646	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) e. NN0 )
7	1, 3	gcdcld 11646	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) e. NN0 )
8	6, 7	nn0mulcld 9028	. 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 11764	. . 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 274	. 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 11641	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( ( K gcd M ) || K /\ ( K gcd M ) || M ) )
12	1, 2, 11	syl2anc 408	. . . . 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 111	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) || K )
14		gcddvds 11641	. . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) || K /\ ( K gcd N ) || N ) )
15	1, 3, 14	syl2anc 408	. . . . 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 111	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) || K )
17	6	nn0zd 9164	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) e. ZZ )
18	7	nn0zd 9164	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) e. ZZ )
19		gcddvds 11641	. . . . . . . . . . 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 408	. . . . . . . . . 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 111	. . . . . . . . 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 113	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) || M )
23	17, 18	gcdcld 11646	. . . . . . . . . . 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 9164	. . . . . . . . . 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 11519	. . . . . . . . . 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 1216	. . . . . . . . 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 429	. . . . . . . 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 113	. . . . . . . . 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 113	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) || N )
30		dvdstr 11519	. . . . . . . . . 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 1216	. . . . . . . . 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 429	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || N )
33		dvdsgcd 11689	. . . . . . . . 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 1216	. . . . . . . 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 429	. . . . . . 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 109	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( M gcd N ) = 1 )
37	35, 36	breqtrd 3949	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || 1 )
38		dvds1 11540	. . . . . . 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 14	. . . . . 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 146	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) = 1 )
41		coprmdvds2 11763	. . . . 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 1219	. . . 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 429	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. ( K gcd N ) ) || K )
44		dvdscmul 11509	. . . . . 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 1216	. . . . 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 11510	. . . . . 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 1216	. . . . 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 9172	. . . . . 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 9172	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. N ) e. ZZ )
50		dvdstr 11519	. . . . . 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 1216	. . . . 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 293	. . . 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 429	. . 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 11689	. . . 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 1216	. . 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 429	. 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 11535	. 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 1217	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   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3924  (class class class)co 5767   1c1 7614   x. cmul 7618   NN0cn0 8970   ZZcz 9047   || cdvds 11482   gcd cgcd 11624

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-sup 6864  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-fl 10036  df-mod 10089  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-gcd 11625

This theorem is referenced by: (None)