Theorem gcdaddm 11988

Description: Adding a multiple of one operand of the gcd operator to the other does not alter the result. (Contributed by Paul Chapman, 31-Mar-2011.)

Assertion

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

Proof of Theorem gcdaddm

Step	Hyp	Ref	Expression
1		gcddvds 11967	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
2	1	3adant1 1015	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
3	2	simpld 112	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || M )
4		simp1 997	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
5		1zzd 9283	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 1 e. ZZ )
6		gcdcl 11970	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
7	6	3adant1 1015	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
8	7	nn0zd 9376	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
9		simp2 998	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
10		simp3 999	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
11		dvds2ln 11834	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ 1 e. ZZ ) /\ ( ( M gcd N ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || N ) -> ( M gcd N ) || ( ( K x. M ) + ( 1 x. N ) ) ) )
12	4, 5, 8, 9, 10, 11	syl23anc 1245	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || N ) -> ( M gcd N ) || ( ( K x. M ) + ( 1 x. N ) ) ) )
13	2, 12	mpd 13	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || ( ( K x. M ) + ( 1 x. N ) ) )
14	10	zcnd 9379	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
15	14	mulid2d 7979	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 1 x. N ) = N )
16	15	oveq2d 5894	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + ( 1 x. N ) ) = ( ( K x. M ) + N ) )
17	13, 16	breqtrd 4031	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || ( ( K x. M ) + N ) )
18	3, 17	jca 306	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || ( ( K x. M ) + N ) ) )
19	4, 9	zmulcld 9384	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
20	19, 10	zaddcld 9382	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) e. ZZ )
21		dvdslegcd 11968	. . . . . . . 8 |- ( ( ( ( M gcd N ) e. ZZ /\ M e. ZZ /\ ( ( K x. M ) + N ) e. ZZ ) /\ -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || ( ( K x. M ) + N ) ) -> ( M gcd N ) <_ ( M gcd ( ( K x. M ) + N ) ) ) )
22	21	ex 115	. . . . . . 7 |- ( ( ( M gcd N ) e. ZZ /\ M e. ZZ /\ ( ( K x. M ) + N ) e. ZZ ) -> ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || ( ( K x. M ) + N ) ) -> ( M gcd N ) <_ ( M gcd ( ( K x. M ) + N ) ) ) ) )
23	8, 9, 20, 22	syl3anc 1238	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || ( ( K x. M ) + N ) ) -> ( M gcd N ) <_ ( M gcd ( ( K x. M ) + N ) ) ) ) )
24	18, 23	mpid 42	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) -> ( M gcd N ) <_ ( M gcd ( ( K x. M ) + N ) ) ) )
25		gcddvds 11967	. . . . . . . . 9 |- ( ( M e. ZZ /\ ( ( K x. M ) + N ) e. ZZ ) -> ( ( M gcd ( ( K x. M ) + N ) ) || M /\ ( M gcd ( ( K x. M ) + N ) ) || ( ( K x. M ) + N ) ) )
26	9, 20, 25	syl2anc 411	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd ( ( K x. M ) + N ) ) || M /\ ( M gcd ( ( K x. M ) + N ) ) || ( ( K x. M ) + N ) ) )
27	26	simpld 112	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) || M )
28	4	znegcld 9380	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> -u K e. ZZ )
29	9, 20	gcdcld 11972	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. NN0 )
30	29	nn0zd 9376	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. ZZ )
31		dvds2ln 11834	. . . . . . . . . 10 |- ( ( ( -u K e. ZZ /\ 1 e. ZZ ) /\ ( ( M gcd ( ( K x. M ) + N ) ) e. ZZ /\ M e. ZZ /\ ( ( K x. M ) + N ) e. ZZ ) ) -> ( ( ( M gcd ( ( K x. M ) + N ) ) || M /\ ( M gcd ( ( K x. M ) + N ) ) || ( ( K x. M ) + N ) ) -> ( M gcd ( ( K x. M ) + N ) ) || ( ( -u K x. M ) + ( 1 x. ( ( K x. M ) + N ) ) ) ) )
32	28, 5, 30, 9, 20, 31	syl23anc 1245	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( M gcd ( ( K x. M ) + N ) ) || M /\ ( M gcd ( ( K x. M ) + N ) ) || ( ( K x. M ) + N ) ) -> ( M gcd ( ( K x. M ) + N ) ) || ( ( -u K x. M ) + ( 1 x. ( ( K x. M ) + N ) ) ) ) )
33	26, 32	mpd 13	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) || ( ( -u K x. M ) + ( 1 x. ( ( K x. M ) + N ) ) ) )
34	4	zcnd 9379	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. CC )
35	9	zcnd 9379	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
36	34, 35	mulneg1d 8371	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u K x. M ) = -u ( K x. M ) )
37	20	zcnd 9379	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) e. CC )
38	37	mulid2d 7979	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 1 x. ( ( K x. M ) + N ) ) = ( ( K x. M ) + N ) )
39	36, 38	oveq12d 5896	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( -u K x. M ) + ( 1 x. ( ( K x. M ) + N ) ) ) = ( -u ( K x. M ) + ( ( K x. M ) + N ) ) )
40	34, 35	mulcld 7981	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. CC )
41	40	negcld 8258	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> -u ( K x. M ) e. CC )
42	40, 41	addcomd 8111	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + -u ( K x. M ) ) = ( -u ( K x. M ) + ( K x. M ) ) )
43	40	negidd 8261	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + -u ( K x. M ) ) = 0 )
44	42, 43	eqtr3d 2212	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u ( K x. M ) + ( K x. M ) ) = 0 )
45	44	oveq1d 5893	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( -u ( K x. M ) + ( K x. M ) ) + N ) = ( 0 + N ) )
46	41, 40, 14	addassd 7983	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( -u ( K x. M ) + ( K x. M ) ) + N ) = ( -u ( K x. M ) + ( ( K x. M ) + N ) ) )
47	14	addid2d 8110	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 0 + N ) = N )
48	45, 46, 47	3eqtr3d 2218	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u ( K x. M ) + ( ( K x. M ) + N ) ) = N )
49	39, 48	eqtrd 2210	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( -u K x. M ) + ( 1 x. ( ( K x. M ) + N ) ) ) = N )
50	33, 49	breqtrd 4031	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) || N )
51	27, 50	jca 306	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd ( ( K x. M ) + N ) ) || M /\ ( M gcd ( ( K x. M ) + N ) ) || N ) )
52		dvdslegcd 11968	. . . . . . . 8 |- ( ( ( ( M gcd ( ( K x. M ) + N ) ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( ( M gcd ( ( K x. M ) + N ) ) || M /\ ( M gcd ( ( K x. M ) + N ) ) || N ) -> ( M gcd ( ( K x. M ) + N ) ) <_ ( M gcd N ) ) )
53	52	ex 115	. . . . . . 7 |- ( ( ( M gcd ( ( K x. M ) + N ) ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) -> ( ( ( M gcd ( ( K x. M ) + N ) ) || M /\ ( M gcd ( ( K x. M ) + N ) ) || N ) -> ( M gcd ( ( K x. M ) + N ) ) <_ ( M gcd N ) ) ) )
54	30, 9, 10, 53	syl3anc 1238	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) -> ( ( ( M gcd ( ( K x. M ) + N ) ) || M /\ ( M gcd ( ( K x. M ) + N ) ) || N ) -> ( M gcd ( ( K x. M ) + N ) ) <_ ( M gcd N ) ) ) )
55	51, 54	mpid 42	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) -> ( M gcd ( ( K x. M ) + N ) ) <_ ( M gcd N ) ) )
56	24, 55	anim12d 335	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) <_ ( M gcd ( ( K x. M ) + N ) ) /\ ( M gcd ( ( K x. M ) + N ) ) <_ ( M gcd N ) ) ) )
57	7	nn0red 9233	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. RR )
58	29	nn0red 9233	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. RR )
59	57, 58	letri3d 8076	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) <-> ( ( M gcd N ) <_ ( M gcd ( ( K x. M ) + N ) ) /\ ( M gcd ( ( K x. M ) + N ) ) <_ ( M gcd N ) ) ) )
60	56, 59	sylibrd 169	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) ) )
61		0zd 9268	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
62		zdceq 9331	. . . . . . 7 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
63	9, 61, 62	syl2anc 411	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID M = 0 )
64		zdceq 9331	. . . . . . 7 |- ( ( ( ( K x. M ) + N ) e. ZZ /\ 0 e. ZZ ) -> DECID ( ( K x. M ) + N ) = 0 )
65	20, 61, 64	syl2anc 411	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( ( K x. M ) + N ) = 0 )
66		dcan2 934	. . . . . 6 |- (DECID M = 0 -> (DECID ( ( K x. M ) + N ) = 0 -> DECID ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) ) )
67	63, 65, 66	sylc 62	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) )
68		zdceq 9331	. . . . . . 7 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
69	10, 61, 68	syl2anc 411	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID N = 0 )
70		dcan2 934	. . . . . 6 |- (DECID M = 0 -> (DECID N = 0 -> DECID ( M = 0 /\ N = 0 ) ) )
71	63, 69, 70	sylc 62	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
72		orandc 939	. . . . 5 |- ( (DECID ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ DECID ( M = 0 /\ N = 0 ) ) -> ( ( ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) \/ ( M = 0 /\ N = 0 ) ) <-> -. ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) ) )
73	67, 71, 72	syl2anc 411	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) \/ ( M = 0 /\ N = 0 ) ) <-> -. ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) ) )
74		simpr 110	. . . . . . . . . . . 12 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> M = 0 )
75	74	oveq2d 5894	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. M ) = ( K x. 0 ) )
76	34	mul01d 8353	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. 0 ) = 0 )
77	76	adantr 276	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. 0 ) = 0 )
78	75, 77	eqtrd 2210	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. M ) = 0 )
79	78	oveq1d 5893	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( K x. M ) + N ) = ( 0 + N ) )
80	47	adantr 276	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( 0 + N ) = N )
81	79, 80	eqtrd 2210	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( K x. M ) + N ) = N )
82	81	eqeq1d 2186	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( ( K x. M ) + N ) = 0 <-> N = 0 ) )
83	82	pm5.32da 452	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) <-> ( M = 0 /\ N = 0 ) ) )
84		oveq12 5887	. . . . . . . . 9 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( 0 gcd 0 ) )
85	84	adantl 277	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( 0 gcd 0 ) )
86		oveq12 5887	. . . . . . . . . 10 |- ( ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) -> ( M gcd ( ( K x. M ) + N ) ) = ( 0 gcd 0 ) )
87	83, 86	syl6bir 164	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) -> ( M gcd ( ( K x. M ) + N ) ) = ( 0 gcd 0 ) ) )
88	87	imp 124	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd ( ( K x. M ) + N ) ) = ( 0 gcd 0 ) )
89	85, 88	eqtr4d 2213	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) )
90	89	ex 115	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) ) )
91	83, 90	sylbid 150	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) ) )
92	91, 90	jaod 717	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) \/ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) ) )
93	73, 92	sylbird 170	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -. ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) ) )
94		dcn 842	. . . . . 6 |- (DECID ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) -> DECID -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) )
95	67, 94	syl 14	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) )
96		dcn 842	. . . . . 6 |- (DECID ( M = 0 /\ N = 0 ) -> DECID -. ( M = 0 /\ N = 0 ) )
97	71, 96	syl 14	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID -. ( M = 0 /\ N = 0 ) )
98		dcan2 934	. . . . 5 |- (DECID -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) -> (DECID -. ( M = 0 /\ N = 0 ) -> DECID ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) ) )
99	95, 97, 98	sylc 62	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) )
100		exmiddc 836	. . . 4 |- (DECID ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) \/ -. ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) ) )
101	99, 100	syl 14	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) \/ -. ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) ) )
102	60, 93, 101	mpjaod 718	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) )
103	40, 14	addcomd 8111	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) = ( N + ( K x. M ) ) )
104	103	oveq2d 5894	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) = ( M gcd ( N + ( K x. M ) ) ) )
105	102, 104	eqtrd 2210	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( N + ( K x. M ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5878   0cc0 7814   1c1 7815   + caddc 7817   x. cmul 7819   <_ cle 7996   -ucneg 8132   NN0cn0 9179   ZZcz 9256   || cdvds 11797   gcd cgcd 11946

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-sup 6986  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-fz 10012  df-fzo 10146  df-fl 10273  df-mod 10326  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-dvds 11798  df-gcd 11947

This theorem is referenced by:  gcdadd  11989  gcdid  11990  modgcd  11995  gcdmultipled  11997  gcdmultiple  12024  pythagtriplem4  12271