Theorem gcdaddm 12573

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 12552	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
2	1	3adant1 1041	. . . . . . . 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 1023	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
5		1zzd 9506	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 1 e. ZZ )
6		gcdcl 12555	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
7	6	3adant1 1041	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
8	7	nn0zd 9600	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
9		simp2 1024	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
10		simp3 1025	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
11		dvds2ln 12403	. . . . . . . . . 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 1280	. . . . . . . . 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 9603	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
15	14	mulid2d 8198	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 1 x. N ) = N )
16	15	oveq2d 6034	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + ( 1 x. N ) ) = ( ( K x. M ) + N ) )
17	13, 16	breqtrd 4114	. . . . . . 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 9608	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
20	19, 10	zaddcld 9606	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) e. ZZ )
21		dvdslegcd 12553	. . . . . . . 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 1273	. . . . . 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 12552	. . . . . . . . 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 9604	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> -u K e. ZZ )
29	9, 20	gcdcld 12557	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. NN0 )
30	29	nn0zd 9600	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. ZZ )
31		dvds2ln 12403	. . . . . . . . . 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 1280	. . . . . . . . 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 9603	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. CC )
35	9	zcnd 9603	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
36	34, 35	mulneg1d 8590	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u K x. M ) = -u ( K x. M ) )
37	20	zcnd 9603	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) e. CC )
38	37	mulid2d 8198	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 1 x. ( ( K x. M ) + N ) ) = ( ( K x. M ) + N ) )
39	36, 38	oveq12d 6036	. . . . . . . . 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 8200	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. CC )
41	40	negcld 8477	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> -u ( K x. M ) e. CC )
42	40, 41	addcomd 8330	. . . . . . . . . . . 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 8480	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + -u ( K x. M ) ) = 0 )
44	42, 43	eqtr3d 2266	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u ( K x. M ) + ( K x. M ) ) = 0 )
45	44	oveq1d 6033	. . . . . . . . . 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 8202	. . . . . . . . . 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 8329	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 0 + N ) = N )
48	45, 46, 47	3eqtr3d 2272	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u ( K x. M ) + ( ( K x. M ) + N ) ) = N )
49	39, 48	eqtrd 2264	. . . . . . . 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 4114	. . . . . . 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 12553	. . . . . . . 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 1319	. . . . . 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 9456	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. RR )
58	29	nn0red 9456	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. RR )
59	57, 58	letri3d 8295	. . . 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 9491	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
62		zdceq 9555	. . . . . . 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 9555	. . . . . . 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 940	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) )
67		zdceq 9555	. . . . . . 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 940	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
70		orandc 947	. . . . 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 6034	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. M ) = ( K x. 0 ) )
74	34	mul01d 8572	. . . . . . . . . . . 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 2264	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. M ) = 0 )
77	76	oveq1d 6033	. . . . . . . . 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 2264	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( K x. M ) + N ) = N )
80	79	eqeq1d 2240	. . . . . . 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 6027	. . . . . . . . 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 6027	. . . . . . . . . 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 2267	. . . . . . 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 724	. . . 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 849	. . . . . 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 849	. . . . . 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 940	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) )
97		exmiddc 843	. . . 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 725	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) )
100	40, 14	addcomd 8330	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) = ( N + ( K x. M ) ) )
101	100	oveq2d 6034	. 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 2264	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 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6018   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   <_ cle 8215   -ucneg 8351   NN0cn0 9402   ZZcz 9479   || cdvds 12366   gcd cgcd 12542

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-dvds 12367  df-gcd 12543

This theorem is referenced by:  gcdadd  12574  gcdid  12575  modgcd  12580  gcdmultipled  12582  gcdmultiple  12609  pythagtriplem4  12859  gcdi  13011