Theorem rpmulgcd2 12233

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 1002	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> K e. ZZ )
2		simpl2 1003	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> M e. ZZ )
3		simpl3 1004	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> N e. ZZ )
4	2, 3	zmulcld 9445	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( M x. N ) e. ZZ )
5	1, 4	gcdcld 12105	. 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 12105	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) e. NN0 )
7	1, 3	gcdcld 12105	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) e. NN0 )
8	6, 7	nn0mulcld 9298	. 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 12232	. . 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 276	. 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 12100	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( ( K gcd M ) || K /\ ( K gcd M ) || M ) )
12	1, 2, 11	syl2anc 411	. . . . 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 112	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) || K )
14		gcddvds 12100	. . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) || K /\ ( K gcd N ) || N ) )
15	1, 3, 14	syl2anc 411	. . . . 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 112	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) || K )
17	6	nn0zd 9437	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) e. ZZ )
18	7	nn0zd 9437	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) e. ZZ )
19		gcddvds 12100	. . . . . . . . . . 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 411	. . . . . . . . . 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 112	. . . . . . . . 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 114	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) || M )
23	17, 18	gcdcld 12105	. . . . . . . . . . 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 9437	. . . . . . . . . 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 11971	. . . . . . . . . 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 1249	. . . . . . . . 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 433	. . . . . . . 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 114	. . . . . . . . 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 114	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) || N )
30		dvdstr 11971	. . . . . . . . . 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 1249	. . . . . . . . 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 433	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || N )
33		dvdsgcd 12149	. . . . . . . . 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 1249	. . . . . . . 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 433	. . . . . . 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 110	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( M gcd N ) = 1 )
37	35, 36	breqtrd 4055	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || 1 )
38		dvds1 11995	. . . . . . 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 147	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) = 1 )
41		coprmdvds2 12231	. . . . 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 1252	. . . 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 433	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. ( K gcd N ) ) || K )
44		dvdscmul 11961	. . . . . 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 1249	. . . . 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 11962	. . . . . 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 1249	. . . . 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 9445	. . . . . 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 9445	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. N ) e. ZZ )
50		dvdstr 11971	. . . . . 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 1249	. . . . 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 295	. . . 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 433	. . 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 12149	. . . 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 1249	. . 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 433	. 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 11990	. 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 1250	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   1c1 7873   x. cmul 7877   NN0cn0 9240   ZZcz 9317   || cdvds 11930   gcd cgcd 12079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080

This theorem is referenced by: (None)