Theorem rpmulgcd2 12666

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 1026	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> K e. ZZ )
2		simpl2 1027	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> M e. ZZ )
3		simpl3 1028	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> N e. ZZ )
4	2, 3	zmulcld 9607	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( M x. N ) e. ZZ )
5	1, 4	gcdcld 12538	. 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 12538	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) e. NN0 )
7	1, 3	gcdcld 12538	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) e. NN0 )
8	6, 7	nn0mulcld 9459	. 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 12665	. . 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 12533	. . . . . 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 12533	. . . . . 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 9599	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd M ) e. ZZ )
18	7	nn0zd 9599	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( K gcd N ) e. ZZ )
19		gcddvds 12533	. . . . . . . . . . 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 12538	. . . . . . . . . . 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 9599	. . . . . . . . . 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 12388	. . . . . . . . . 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 1273	. . . . . . . . 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 12388	. . . . . . . . . 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 1273	. . . . . . . . 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 12582	. . . . . . . . 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 1273	. . . . . . . 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 4114	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) gcd ( K gcd N ) ) || 1 )
38		dvds1 12413	. . . . . . 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 12664	. . . . 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 1276	. . . 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 12378	. . . . . 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 1273	. . . . 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 12379	. . . . . 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 1273	. . . . 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 9607	. . . . . 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 9607	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M gcd N ) = 1 ) -> ( ( K gcd M ) x. N ) e. ZZ )
50		dvdstr 12388	. . . . . 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 1273	. . . . 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 12582	. . . 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 1273	. . 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 12408	. 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 1274	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 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   1c1 8032   x. cmul 8036   NN0cn0 9401   ZZcz 9478   || cdvds 12347   gcd cgcd 12523

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524

This theorem is referenced by:  mpodvdsmulf1o  15713