Theorem mulgcddvds 12232

Description: One half of rpmulgcd2 12233, 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 999	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
2		simp2 1000	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
3		simp3 1001	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
4	2, 3	zmulcld 9445	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
5	1, 4	gcdcld 12105	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) e. NN0 )
6	5	nn0zd 9437	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) e. ZZ )
7		dvds0 11949	. . . . 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 5926	. . . 4 |- ( ( K gcd N ) = 0 -> ( ( K gcd M ) x. ( K gcd N ) ) = ( ( K gcd M ) x. 0 ) )
11	1, 2	gcdcld 12105	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) e. NN0 )
12	11	nn0cnd 9295	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) e. CC )
13	12	mul01d 8412	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd M ) x. 0 ) = 0 )
14	10, 13	sylan9eqr 2248	. . 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 4057	. 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 9440	. . . 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 12105	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. NN0 )
19	18	nn0zd 9437	. . . . . 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 9440	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) e. CC )
22		0zd 9329	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
23		zapne 9391	. . . . . 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 8810	. . 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 12100	. . . . . . . . . . 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 11975	. . . . . . . 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 4051	. . . . . 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 12100	. . . . . . . . . . . 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 11976	. . . . . . . . . . . 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 1249	. . . . . . . . . . 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 12149	. . . . . . . . . . 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 1249	. . . . . . . . . 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 11933	. . . . . . . . 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 1249	. . . . . . . 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 11964	. . . . . . 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 1253	. . . . . 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 11229	. . . . . . . . . . . . . . 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 9437	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( abs ` M ) e. ZZ )
55		dvdsmultr2 11976	. . . . . . . . . . . . 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 1249	. . . . . . . . . . . 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 11947	. . . . . . . . . . . . . . 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 11953	. . . . . . . . . . . . . . 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 11962	. . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . 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 9445	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) x. N ) e. ZZ )
68		dvdstr 11971	. . . . . . . . . . . . 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 1249	. . . . . . . . . . . 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 9445	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) x. K ) e. ZZ )
72		dvdsgcd 12149	. . . . . . . . . . . 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 1249	. . . . . . . . . . 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 9294	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. RR )
76	18	nn0ge0d 9296	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 <_ ( K gcd N ) )
77	75, 76	absidd 11311	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( abs ` ( K gcd N ) ) = ( K gcd N ) )
78	77	oveq2d 5934	. . . . . . . . . . 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 9440	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
80	18	nn0cnd 9295	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. CC )
81	79, 80	absmuld 11338	. . . . . . . . . . 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 12153	. . . . . . . . . . . 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 1249	. . . . . . . . . . 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 2236	. . . . . . . . . 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 4057	. . . . . . . . 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 9445	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M x. ( K gcd N ) ) e. ZZ )
87		dvdsabsb 11953	. . . . . . . . . 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 4051	. . . . . 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 11964	. . . . . . 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 1253	. . . . . 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 12149	. . . . . 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 1249	. . . . 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 9437	. . . . . 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 11962	. . . . 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 1249	. . . 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 4053	. 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 9392	. . . 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 837	. . . 4 |- (DECID ( K gcd N ) = 0 -> ( ( K gcd N ) = 0 \/ -. ( K gcd N ) = 0 ) )
108		df-ne 2365	. . . . 5 |- ( ( K gcd N ) =/= 0 <-> -. ( K gcd N ) = 0 )
109	108	orbi2i 763	. . . 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 799	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 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   0cc0 7872   x. cmul 7877   # cap 8600   / cdiv 8691   NN0cn0 9240   ZZcz 9317   abscabs 11141   || 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:  rpmulgcd2  12233  rpmul  12236