Theorem mulgcddvds 12816

Description: One half of rpmulgcd2 12817, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.)

Assertion

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

Proof of Theorem mulgcddvds

Step	Hyp	Ref	Expression
1		simp1 1024	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
2		simp2 1025	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
3		simp3 1026	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
4	2, 3	zmulcld 9724	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
5	1, 4	gcdcld 12689	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) e. NN0 )
6	5	nn0zd 9716	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) e. ZZ )
7		dvds0 12517	. . . . 5 |- ( ( K gcd ( M x. N ) ) e. ZZ -> ( K gcd ( M x. N ) ) || 0 )
8	6, 7	syl 14	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || 0 )
9	8	adantr 276	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) = 0 ) -> ( K gcd ( M x. N ) ) || 0 )
10		oveq2 6066	. . . 4 |- ( ( K gcd N ) = 0 -> ( ( K gcd M ) x. ( K gcd N ) ) = ( ( K gcd M ) x. 0 ) )
11	1, 2	gcdcld 12689	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) e. NN0 )
12	11	nn0cnd 9572	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) e. CC )
13	12	mul01d 8683	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd M ) x. 0 ) = 0 )
14	10, 13	sylan9eqr 2289	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) = 0 ) -> ( ( K gcd M ) x. ( K gcd N ) ) = 0 )
15	9, 14	breqtrrd 4142	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) = 0 ) -> ( K gcd ( M x. N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) )
16	6	adantr 276	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd ( M x. N ) ) e. ZZ )
17	16	zcnd 9719	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd ( M x. N ) ) e. CC )
18	1, 3	gcdcld 12689	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. NN0 )
19	18	nn0zd 9716	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. ZZ )
20	19	adantr 276	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) e. ZZ )
21	20	zcnd 9719	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) e. CC )
22		0zd 9606	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
23		zapne 9669	. . . . . 6 |- ( ( ( K gcd N ) e. ZZ /\ 0 e. ZZ ) -> ( ( K gcd N ) # 0 <-> ( K gcd N ) =/= 0 ) )
24	19, 22, 23	syl2anc 411	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) # 0 <-> ( K gcd N ) =/= 0 ) )
25	24	biimpar 297	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) # 0 )
26	17, 21, 25	divcanap1d 9082	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) x. ( K gcd N ) ) = ( K gcd ( M x. N ) ) )
27		gcddvds 12684	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ ( M x. N ) e. ZZ ) -> ( ( K gcd ( M x. N ) ) || K /\ ( K gcd ( M x. N ) ) || ( M x. N ) ) )
28	1, 4, 27	syl2anc 411	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd ( M x. N ) ) || K /\ ( K gcd ( M x. N ) ) || ( M x. N ) ) )
29	28	simpld 112	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || K )
30	6, 1, 19, 29	dvdsmultr1d 12543	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( K x. ( K gcd N ) ) )
31	30	adantr 276	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd ( M x. N ) ) || ( K x. ( K gcd N ) ) )
32	26, 31	eqbrtrd 4136	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) x. ( K gcd N ) ) || ( K x. ( K gcd N ) ) )
33		gcddvds 12684	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) || K /\ ( K gcd N ) || N ) )
34	1, 3, 33	syl2anc 411	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) || K /\ ( K gcd N ) || N ) )
35	34	simpld 112	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) || K )
36	34	simprd 114	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) || N )
37		dvdsmultr2 12544	. . . . . . . . . . . 12 |- ( ( ( K gcd N ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) || N -> ( K gcd N ) || ( M x. N ) ) )
38	19, 2, 3, 37	syl3anc 1274	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) || N -> ( K gcd N ) || ( M x. N ) ) )
39	36, 38	mpd 13	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) || ( M x. N ) )
40		dvdsgcd 12733	. . . . . . . . . . 11 |- ( ( ( K gcd N ) e. ZZ /\ K e. ZZ /\ ( M x. N ) e. ZZ ) -> ( ( ( K gcd N ) || K /\ ( K gcd N ) || ( M x. N ) ) -> ( K gcd N ) || ( K gcd ( M x. N ) ) ) )
41	19, 1, 4, 40	syl3anc 1274	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K gcd N ) || K /\ ( K gcd N ) || ( M x. N ) ) -> ( K gcd N ) || ( K gcd ( M x. N ) ) ) )
42	35, 39, 41	mp2and 433	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) || ( K gcd ( M x. N ) ) )
43	42	adantr 276	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) || ( K gcd ( M x. N ) ) )
44		simpr 110	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) =/= 0 )
45		dvdsval2 12501	. . . . . . . . 9 |- ( ( ( K gcd N ) e. ZZ /\ ( K gcd N ) =/= 0 /\ ( K gcd ( M x. N ) ) e. ZZ ) -> ( ( K gcd N ) || ( K gcd ( M x. N ) ) <-> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) e. ZZ ) )
46	20, 44, 16, 45	syl3anc 1274	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( K gcd N ) || ( K gcd ( M x. N ) ) <-> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) e. ZZ ) )
47	43, 46	mpbid 147	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) e. ZZ )
48	1	adantr 276	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> K e. ZZ )
49		dvdsmulcr 12532	. . . . . . 7 |- ( ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) e. ZZ /\ K e. ZZ /\ ( ( K gcd N ) e. ZZ /\ ( K gcd N ) =/= 0 ) ) -> ( ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) x. ( K gcd N ) ) || ( K x. ( K gcd N ) ) <-> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || K ) )
50	47, 48, 20, 44, 49	syl112anc 1278	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) x. ( K gcd N ) ) || ( K x. ( K gcd N ) ) <-> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || K ) )
51	32, 50	mpbid 147	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || K )
52		nn0abscl 11795	. . . . . . . . . . . . . . 15 |- ( M e. ZZ -> ( abs ` M ) e. NN0 )
53	2, 52	syl 14	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( abs ` M ) e. NN0 )
54	53	nn0zd 9716	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( abs ` M ) e. ZZ )
55		dvdsmultr2 12544	. . . . . . . . . . . . 13 |- ( ( ( K gcd ( M x. N ) ) e. ZZ /\ ( abs ` M ) e. ZZ /\ K e. ZZ ) -> ( ( K gcd ( M x. N ) ) || K -> ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. K ) ) )
56	6, 54, 1, 55	syl3anc 1274	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd ( M x. N ) ) || K -> ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. K ) ) )
57	29, 56	mpd 13	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. K ) )
58	28	simprd 114	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( M x. N ) )
59		iddvds 12515	. . . . . . . . . . . . . . 15 |- ( M e. ZZ -> M || M )
60	2, 59	syl 14	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M || M )
61		dvdsabsb 12521	. . . . . . . . . . . . . . 15 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M || M <-> M || ( abs ` M ) ) )
62	2, 2, 61	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || M <-> M || ( abs ` M ) ) )
63	60, 62	mpbid 147	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M || ( abs ` M ) )
64		dvdsmulc 12530	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ ( abs ` M ) e. ZZ /\ N e. ZZ ) -> ( M || ( abs ` M ) -> ( M x. N ) || ( ( abs ` M ) x. N ) ) )
65	2, 54, 3, 64	syl3anc 1274	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || ( abs ` M ) -> ( M x. N ) || ( ( abs ` M ) x. N ) ) )
66	63, 65	mpd 13	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M x. N ) || ( ( abs ` M ) x. N ) )
67	54, 3	zmulcld 9724	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) x. N ) e. ZZ )
68		dvdstr 12539	. . . . . . . . . . . . 13 |- ( ( ( K gcd ( M x. N ) ) e. ZZ /\ ( M x. N ) e. ZZ /\ ( ( abs ` M ) x. N ) e. ZZ ) -> ( ( ( K gcd ( M x. N ) ) || ( M x. N ) /\ ( M x. N ) || ( ( abs ` M ) x. N ) ) -> ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. N ) ) )
69	6, 4, 67, 68	syl3anc 1274	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K gcd ( M x. N ) ) || ( M x. N ) /\ ( M x. N ) || ( ( abs ` M ) x. N ) ) -> ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. N ) ) )
70	58, 66, 69	mp2and 433	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. N ) )
71	54, 1	zmulcld 9724	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) x. K ) e. ZZ )
72		dvdsgcd 12733	. . . . . . . . . . . 12 |- ( ( ( K gcd ( M x. N ) ) e. ZZ /\ ( ( abs ` M ) x. K ) e. ZZ /\ ( ( abs ` M ) x. N ) e. ZZ ) -> ( ( ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. K ) /\ ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. N ) ) -> ( K gcd ( M x. N ) ) || ( ( ( abs ` M ) x. K ) gcd ( ( abs ` M ) x. N ) ) ) )
73	6, 71, 67, 72	syl3anc 1274	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. K ) /\ ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. N ) ) -> ( K gcd ( M x. N ) ) || ( ( ( abs ` M ) x. K ) gcd ( ( abs ` M ) x. N ) ) ) )
74	57, 70, 73	mp2and 433	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( ( ( abs ` M ) x. K ) gcd ( ( abs ` M ) x. N ) ) )
75	18	nn0red 9571	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. RR )
76	18	nn0ge0d 9573	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 <_ ( K gcd N ) )
77	75, 76	absidd 11877	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( abs ` ( K gcd N ) ) = ( K gcd N ) )
78	77	oveq2d 6074	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) x. ( abs ` ( K gcd N ) ) ) = ( ( abs ` M ) x. ( K gcd N ) ) )
79	2	zcnd 9719	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
80	18	nn0cnd 9572	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. CC )
81	79, 80	absmuld 11904	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M x. ( K gcd N ) ) ) = ( ( abs ` M ) x. ( abs ` ( K gcd N ) ) ) )
82		mulgcd 12737	. . . . . . . . . . . 12 |- ( ( ( abs ` M ) e. NN0 /\ K e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) x. K ) gcd ( ( abs ` M ) x. N ) ) = ( ( abs ` M ) x. ( K gcd N ) ) )
83	53, 1, 3, 82	syl3anc 1274	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) x. K ) gcd ( ( abs ` M ) x. N ) ) = ( ( abs ` M ) x. ( K gcd N ) ) )
84	78, 81, 83	3eqtr4d 2277	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M x. ( K gcd N ) ) ) = ( ( ( abs ` M ) x. K ) gcd ( ( abs ` M ) x. N ) ) )
85	74, 84	breqtrrd 4142	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( abs ` ( M x. ( K gcd N ) ) ) )
86	2, 19	zmulcld 9724	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M x. ( K gcd N ) ) e. ZZ )
87		dvdsabsb 12521	. . . . . . . . . 10 |- ( ( ( K gcd ( M x. N ) ) e. ZZ /\ ( M x. ( K gcd N ) ) e. ZZ ) -> ( ( K gcd ( M x. N ) ) || ( M x. ( K gcd N ) ) <-> ( K gcd ( M x. N ) ) || ( abs ` ( M x. ( K gcd N ) ) ) ) )
88	6, 86, 87	syl2anc 411	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd ( M x. N ) ) || ( M x. ( K gcd N ) ) <-> ( K gcd ( M x. N ) ) || ( abs ` ( M x. ( K gcd N ) ) ) ) )
89	85, 88	mpbird 167	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( M x. ( K gcd N ) ) )
90	89	adantr 276	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd ( M x. N ) ) || ( M x. ( K gcd N ) ) )
91	26, 90	eqbrtrd 4136	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) x. ( K gcd N ) ) || ( M x. ( K gcd N ) ) )
92	2	adantr 276	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> M e. ZZ )
93		dvdsmulcr 12532	. . . . . . 7 |- ( ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) e. ZZ /\ M e. ZZ /\ ( ( K gcd N ) e. ZZ /\ ( K gcd N ) =/= 0 ) ) -> ( ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) x. ( K gcd N ) ) || ( M x. ( K gcd N ) ) <-> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || M ) )
94	47, 92, 20, 44, 93	syl112anc 1278	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) x. ( K gcd N ) ) || ( M x. ( K gcd N ) ) <-> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || M ) )
95	91, 94	mpbid 147	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || M )
96		dvdsgcd 12733	. . . . . 6 |- ( ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) e. ZZ /\ K e. ZZ /\ M e. ZZ ) -> ( ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || K /\ ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || M ) -> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || ( K gcd M ) ) )
97	47, 48, 92, 96	syl3anc 1274	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || K /\ ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || M ) -> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || ( K gcd M ) ) )
98	51, 95, 97	mp2and 433	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || ( K gcd M ) )
99	11	nn0zd 9716	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) e. ZZ )
100	99	adantr 276	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd M ) e. ZZ )
101		dvdsmulc 12530	. . . . 5 |- ( ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) e. ZZ /\ ( K gcd M ) e. ZZ /\ ( K gcd N ) e. ZZ ) -> ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || ( K gcd M ) -> ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) x. ( K gcd N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) ) )
102	47, 100, 20, 101	syl3anc 1274	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) || ( K gcd M ) -> ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) x. ( K gcd N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) ) )
103	98, 102	mpd 13	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( ( ( K gcd ( M x. N ) ) / ( K gcd N ) ) x. ( K gcd N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) )
104	26, 103	eqbrtrrd 4138	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd ( M x. N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) )
105		zdceq 9670	. . . 4 |- ( ( ( K gcd N ) e. ZZ /\ 0 e. ZZ ) -> DECID ( K gcd N ) = 0 )
106	19, 22, 105	syl2anc 411	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( K gcd N ) = 0 )
107		exmiddc 844	. . . 4 |- (DECID ( K gcd N ) = 0 -> ( ( K gcd N ) = 0 \/ -. ( K gcd N ) = 0 ) )
108		df-ne 2415	. . . . 5 |- ( ( K gcd N ) =/= 0 <-> -. ( K gcd N ) = 0 )
109	108	orbi2i 770	. . . 4 |- ( ( ( K gcd N ) = 0 \/ ( K gcd N ) =/= 0 ) <-> ( ( K gcd N ) = 0 \/ -. ( K gcd N ) = 0 ) )
110	107, 109	sylibr 134	. . 3 |- (DECID ( K gcd N ) = 0 -> ( ( K gcd N ) = 0 \/ ( K gcd N ) =/= 0 ) )
111	106, 110	syl 14	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) = 0 \/ ( K gcd N ) =/= 0 ) )
112	15, 104, 111	mpjaodan 806	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( ( K gcd M ) x. ( K gcd N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   0cc0 8143   x. cmul 8148   # cap 8872   / cdiv 8963   NN0cn0 9513   ZZcz 9594   abscabs 11707   || cdvds 12498   gcd cgcd 12674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675

This theorem is referenced by:  rpmulgcd2  12817  rpmul  12820