Theorem gcdaddm 12121

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 12100	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
2	1	3adant1 1017	. . . . . . . 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 999	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
5		1zzd 9344	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 1 e. ZZ )
6		gcdcl 12103	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
7	6	3adant1 1017	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
8	7	nn0zd 9437	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
9		simp2 1000	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
10		simp3 1001	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
11		dvds2ln 11967	. . . . . . . . . 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 1256	. . . . . . . . 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 9440	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
15	14	mulid2d 8038	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 1 x. N ) = N )
16	15	oveq2d 5934	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + ( 1 x. N ) ) = ( ( K x. M ) + N ) )
17	13, 16	breqtrd 4055	. . . . . . 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 9445	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
20	19, 10	zaddcld 9443	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) e. ZZ )
21		dvdslegcd 12101	. . . . . . . 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 1249	. . . . . 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 12100	. . . . . . . . 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 9441	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> -u K e. ZZ )
29	9, 20	gcdcld 12105	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. NN0 )
30	29	nn0zd 9437	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. ZZ )
31		dvds2ln 11967	. . . . . . . . . 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 1256	. . . . . . . . 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 9440	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. CC )
35	9	zcnd 9440	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
36	34, 35	mulneg1d 8430	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u K x. M ) = -u ( K x. M ) )
37	20	zcnd 9440	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) e. CC )
38	37	mulid2d 8038	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 1 x. ( ( K x. M ) + N ) ) = ( ( K x. M ) + N ) )
39	36, 38	oveq12d 5936	. . . . . . . . 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 8040	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. CC )
41	40	negcld 8317	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> -u ( K x. M ) e. CC )
42	40, 41	addcomd 8170	. . . . . . . . . . . 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 8320	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + -u ( K x. M ) ) = 0 )
44	42, 43	eqtr3d 2228	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u ( K x. M ) + ( K x. M ) ) = 0 )
45	44	oveq1d 5933	. . . . . . . . . 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 8042	. . . . . . . . . 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 8169	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 0 + N ) = N )
48	45, 46, 47	3eqtr3d 2234	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u ( K x. M ) + ( ( K x. M ) + N ) ) = N )
49	39, 48	eqtrd 2226	. . . . . . . 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 4055	. . . . . . 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 12101	. . . . . . . 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 1295	. . . . . 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 9294	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. RR )
58	29	nn0red 9294	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. RR )
59	57, 58	letri3d 8135	. . . 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 9329	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
62		zdceq 9392	. . . . . . 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 9392	. . . . . . 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 934	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) )
67		zdceq 9392	. . . . . . 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 934	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
70		orandc 941	. . . . 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 5934	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. M ) = ( K x. 0 ) )
74	34	mul01d 8412	. . . . . . . . . . . 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 2226	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. M ) = 0 )
77	76	oveq1d 5933	. . . . . . . . 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 2226	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( K x. M ) + N ) = N )
80	79	eqeq1d 2202	. . . . . . 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 5927	. . . . . . . . 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 5927	. . . . . . . . . 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 2229	. . . . . . 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 718	. . . 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 843	. . . . . 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 843	. . . . . 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 934	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) )
97		exmiddc 837	. . . 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 719	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) )
100	40, 14	addcomd 8170	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) = ( N + ( K x. M ) ) )
101	100	oveq2d 5934	. 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 2226	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 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   <_ cle 8055   -ucneg 8191   NN0cn0 9240   ZZcz 9317   || 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-stab 832  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:  gcdadd  12122  gcdid  12123  modgcd  12128  gcdmultipled  12130  gcdmultiple  12157  pythagtriplem4  12406