Theorem rpmulgcd2 11776

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 9179	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( M x. N ) e. ZZ )
5	1, 4	gcdcld 11657	. 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 11657	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) e. NN0 )
7	1, 3	gcdcld 11657	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) e. NN0 )
8	6, 7	nn0mulcld 9035	. 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 11775	. . 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 11652	. . . . . 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 11652	. . . . . 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 9171	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) e. ZZ )
18	7	nn0zd 9171	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) e. ZZ )
19		gcddvds 11652	. . . . . . . . . . 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 11657	. . . . . . . . . . 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 9171	. . . . . . . . . 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 11530	. . . . . . . . . 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 11530	. . . . . . . . . 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 11700	. . . . . . . . 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 3954	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || 1 )
38		dvds1 11551	. . . . . . 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 11774	. . . . 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 11520	. . . . . 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 11521	. . . . . 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 9179	. . . . . 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 9179	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. N ) e. ZZ )
50		dvdstr 11530	. . . . . 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 11700	. . . 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 11546	. 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 3929  (class class class)co 5774   1c1 7621   x. cmul 7625   NN0cn0 8977   ZZcz 9054   || cdvds 11493   gcd cgcd 11635

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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494  df-gcd 11636

This theorem is referenced by: (None)