Theorem gcdaddm 12380

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 12359	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
2	1	3adant1 1018	. . . . . . . 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 1000	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
5		1zzd 9419	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 1 e. ZZ )
6		gcdcl 12362	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
7	6	3adant1 1018	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
8	7	nn0zd 9513	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
9		simp2 1001	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
10		simp3 1002	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
11		dvds2ln 12210	. . . . . . . . . 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 1257	. . . . . . . . 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 9516	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
15	14	mulid2d 8111	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 1 x. N ) = N )
16	15	oveq2d 5973	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + ( 1 x. N ) ) = ( ( K x. M ) + N ) )
17	13, 16	breqtrd 4077	. . . . . . 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 9521	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
20	19, 10	zaddcld 9519	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) e. ZZ )
21		dvdslegcd 12360	. . . . . . . 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 1250	. . . . . 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 12359	. . . . . . . . 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 9517	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> -u K e. ZZ )
29	9, 20	gcdcld 12364	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. NN0 )
30	29	nn0zd 9513	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. ZZ )
31		dvds2ln 12210	. . . . . . . . . 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 1257	. . . . . . . . 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 9516	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. CC )
35	9	zcnd 9516	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
36	34, 35	mulneg1d 8503	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u K x. M ) = -u ( K x. M ) )
37	20	zcnd 9516	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) e. CC )
38	37	mulid2d 8111	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 1 x. ( ( K x. M ) + N ) ) = ( ( K x. M ) + N ) )
39	36, 38	oveq12d 5975	. . . . . . . . 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 8113	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. CC )
41	40	negcld 8390	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> -u ( K x. M ) e. CC )
42	40, 41	addcomd 8243	. . . . . . . . . . . 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 8393	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + -u ( K x. M ) ) = 0 )
44	42, 43	eqtr3d 2241	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u ( K x. M ) + ( K x. M ) ) = 0 )
45	44	oveq1d 5972	. . . . . . . . . 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 8115	. . . . . . . . . 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	addlidd 8242	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 0 + N ) = N )
48	45, 46, 47	3eqtr3d 2247	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u ( K x. M ) + ( ( K x. M ) + N ) ) = N )
49	39, 48	eqtrd 2239	. . . . . . . 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 4077	. . . . . . 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 12360	. . . . . . . 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, 53	syld3an1 1296	. . . . . 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 9369	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. RR )
58	29	nn0red 9369	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. RR )
59	57, 58	letri3d 8208	. . . 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 9404	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
62		zdceq 9468	. . . . . . 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 9468	. . . . . . 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	63, 65	dcand 935	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) )
67		zdceq 9468	. . . . . . 7 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
68	10, 61, 67	syl2anc 411	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID N = 0 )
69	63, 68	dcand 935	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
70		orandc 942	. . . . 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 ) ) ) )
71	66, 69, 70	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 ) ) ) )
72		simpr 110	. . . . . . . . . . . 12 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> M = 0 )
73	72	oveq2d 5973	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. M ) = ( K x. 0 ) )
74	34	mul01d 8485	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. 0 ) = 0 )
75	74	adantr 276	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. 0 ) = 0 )
76	73, 75	eqtrd 2239	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. M ) = 0 )
77	76	oveq1d 5972	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( K x. M ) + N ) = ( 0 + N ) )
78	47	adantr 276	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( 0 + N ) = N )
79	77, 78	eqtrd 2239	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( K x. M ) + N ) = N )
80	79	eqeq1d 2215	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( ( K x. M ) + N ) = 0 <-> N = 0 ) )
81	80	pm5.32da 452	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) <-> ( M = 0 /\ N = 0 ) ) )
82		oveq12 5966	. . . . . . . . 9 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( 0 gcd 0 ) )
83	82	adantl 277	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( 0 gcd 0 ) )
84		oveq12 5966	. . . . . . . . . 10 |- ( ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) -> ( M gcd ( ( K x. M ) + N ) ) = ( 0 gcd 0 ) )
85	81, 84	biimtrrdi 164	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) -> ( M gcd ( ( K x. M ) + N ) ) = ( 0 gcd 0 ) ) )
86	85	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 ) )
87	83, 86	eqtr4d 2242	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) )
88	87	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 ) ) ) )
89	81, 88	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 ) ) ) )
90	89, 88	jaod 719	. . . 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 ) ) ) )
91	71, 90	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 ) ) ) )
92		dcn 844	. . . . . 6 |- (DECID ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) -> DECID -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) )
93	66, 92	syl 14	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) )
94		dcn 844	. . . . . 6 |- (DECID ( M = 0 /\ N = 0 ) -> DECID -. ( M = 0 /\ N = 0 ) )
95	69, 94	syl 14	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID -. ( M = 0 /\ N = 0 ) )
96	93, 95	dcand 935	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) )
97		exmiddc 838	. . . 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 ) ) ) )
98	96, 97	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 ) ) ) )
99	60, 91, 98	mpjaod 720	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) )
100	40, 14	addcomd 8243	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) = ( N + ( K x. M ) ) )
101	100	oveq2d 5973	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) = ( M gcd ( N + ( K x. M ) ) ) )
102	99, 101	eqtrd 2239	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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2177   class class class wbr 4051  (class class class)co 5957   0cc0 7945   1c1 7946   + caddc 7948   x. cmul 7950   <_ cle 8128   -ucneg 8264   NN0cn0 9315   ZZcz 9392   || cdvds 12173   gcd cgcd 12349

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-sup 7101  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-fz 10151  df-fzo 10285  df-fl 10435  df-mod 10490  df-seqfrec 10615  df-exp 10706  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385  df-dvds 12174  df-gcd 12350

This theorem is referenced by:  gcdadd  12381  gcdid  12382  modgcd  12387  gcdmultipled  12389  gcdmultiple  12416  pythagtriplem4  12666  gcdi  12818