Theorem mulgcddvds 12048

Description: One half of rpmulgcd2 12049, 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 992	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
2		simp2 993	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
3		simp3 994	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
4	2, 3	zmulcld 9340	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
5	1, 4	gcdcld 11923	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) e. NN0 )
6	5	nn0zd 9332	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) e. ZZ )
7		dvds0 11768	. . . . 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 274	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) = 0 ) -> ( K gcd ( M x. N ) ) || 0 )
10		oveq2 5861	. . . 4 |- ( ( K gcd N ) = 0 -> ( ( K gcd M ) x. ( K gcd N ) ) = ( ( K gcd M ) x. 0 ) )
11	1, 2	gcdcld 11923	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) e. NN0 )
12	11	nn0cnd 9190	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) e. CC )
13	12	mul01d 8312	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd M ) x. 0 ) = 0 )
14	10, 13	sylan9eqr 2225	. . 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 4017	. 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 274	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd ( M x. N ) ) e. ZZ )
17	16	zcnd 9335	. . . 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 11923	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. NN0 )
19	18	nn0zd 9332	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. ZZ )
20	19	adantr 274	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) e. ZZ )
21	20	zcnd 9335	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) e. CC )
22		0zd 9224	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
23		zapne 9286	. . . . . 6 |- ( ( ( K gcd N ) e. ZZ /\ 0 e. ZZ ) -> ( ( K gcd N ) # 0 <-> ( K gcd N ) =/= 0 ) )
24	19, 22, 23	syl2anc 409	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) # 0 <-> ( K gcd N ) =/= 0 ) )
25	24	biimpar 295	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) # 0 )
26	17, 21, 25	divcanap1d 8708	. . 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 11918	. . . . . . . . . . 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 409	. . . . . . . . . 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 111	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || K )
30	6, 1, 19, 29	dvdsmultr1d 11794	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( K x. ( K gcd N ) ) )
31	30	adantr 274	. . . . . . 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 4011	. . . . . 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 11918	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) || K /\ ( K gcd N ) || N ) )
34	1, 3, 33	syl2anc 409	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd N ) || K /\ ( K gcd N ) || N ) )
35	34	simpld 111	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) || K )
36	34	simprd 113	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) || N )
37		dvdsmultr2 11795	. . . . . . . . . . . 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 1233	. . . . . . . . . . 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 11967	. . . . . . . . . . 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 1233	. . . . . . . . . 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 431	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) || ( K gcd ( M x. N ) ) )
43	42	adantr 274	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) || ( K gcd ( M x. N ) ) )
44		simpr 109	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd N ) =/= 0 )
45		dvdsval2 11752	. . . . . . . . 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 1233	. . . . . . . 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 146	. . . . . . 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 274	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> K e. ZZ )
49		dvdsmulcr 11783	. . . . . . 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 1237	. . . . . 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 146	. . . . 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 11049	. . . . . . . . . . . . . . 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 9332	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( abs ` M ) e. ZZ )
55		dvdsmultr2 11795	. . . . . . . . . . . . 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 1233	. . . . . . . . . . . 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 113	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( M x. N ) )
59		iddvds 11766	. . . . . . . . . . . . . . 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 11772	. . . . . . . . . . . . . . 15 |- ( ( M e. ZZ /\ M e. ZZ ) -> ( M || M <-> M || ( abs ` M ) ) )
62	2, 2, 61	syl2anc 409	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M || M <-> M || ( abs ` M ) ) )
63	60, 62	mpbid 146	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M || ( abs ` M ) )
64		dvdsmulc 11781	. . . . . . . . . . . . . 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 1233	. . . . . . . . . . . . 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 9340	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) x. N ) e. ZZ )
68		dvdstr 11790	. . . . . . . . . . . . 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 1233	. . . . . . . . . . . 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 431	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( ( abs ` M ) x. N ) )
71	54, 1	zmulcld 9340	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) x. K ) e. ZZ )
72		dvdsgcd 11967	. . . . . . . . . . . 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 1233	. . . . . . . . . . 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 431	. . . . . . . . . 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 9189	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. RR )
76	18	nn0ge0d 9191	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 <_ ( K gcd N ) )
77	75, 76	absidd 11131	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( abs ` ( K gcd N ) ) = ( K gcd N ) )
78	77	oveq2d 5869	. . . . . . . . . . 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 9335	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
80	18	nn0cnd 9190	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd N ) e. CC )
81	79, 80	absmuld 11158	. . . . . . . . . . 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 11971	. . . . . . . . . . . 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 1233	. . . . . . . . . . 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 2213	. . . . . . . . . 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 4017	. . . . . . . . 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 9340	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M x. ( K gcd N ) ) e. ZZ )
87		dvdsabsb 11772	. . . . . . . . . 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 409	. . . . . . . . 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 166	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd ( M x. N ) ) || ( M x. ( K gcd N ) ) )
90	89	adantr 274	. . . . . . 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 4011	. . . . . 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 274	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> M e. ZZ )
93		dvdsmulcr 11783	. . . . . . 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 1237	. . . . . 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 146	. . . . 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 11967	. . . . . 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 1233	. . . . 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 431	. . . 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 9332	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) e. ZZ )
100	99	adantr 274	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd N ) =/= 0 ) -> ( K gcd M ) e. ZZ )
101		dvdsmulc 11781	. . . . 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 1233	. . . 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 4013	. 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 9287	. . . 4 |- ( ( ( K gcd N ) e. ZZ /\ 0 e. ZZ ) -> DECID ( K gcd N ) = 0 )
106	19, 22, 105	syl2anc 409	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( K gcd N ) = 0 )
107		exmiddc 831	. . . 4 |- (DECID ( K gcd N ) = 0 -> ( ( K gcd N ) = 0 \/ -. ( K gcd N ) = 0 ) )
108		df-ne 2341	. . . . 5 |- ( ( K gcd N ) =/= 0 <-> -. ( K gcd N ) = 0 )
109	108	orbi2i 757	. . . 4 |- ( ( ( K gcd N ) = 0 \/ ( K gcd N ) =/= 0 ) <-> ( ( K gcd N ) = 0 \/ -. ( K gcd N ) = 0 ) )
110	107, 109	sylibr 133	. . 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 793	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 103   <-> wb 104   \/ wo 703  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   0cc0 7774   x. cmul 7779   # cap 8500   / cdiv 8589   NN0cn0 9135   ZZcz 9212   abscabs 10961   || cdvds 11749   gcd cgcd 11897

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898

This theorem is referenced by:  rpmulgcd2  12049  rpmul  12052