Theorem gcdaddm 12761

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 12740	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
2	1	3adant1 1046	. . . . . . . 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 1028	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. ZZ )
5		1zzd 9671	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 1 e. ZZ )
6		gcdcl 12743	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
7	6	3adant1 1046	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
8	7	nn0zd 9766	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
9		simp2 1029	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
10		simp3 1030	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
11		dvds2ln 12591	. . . . . . . . . 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 1285	. . . . . . . . 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 9769	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
15	14	mullidd 8344	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 1 x. N ) = N )
16	15	oveq2d 6101	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + ( 1 x. N ) ) = ( ( K x. M ) + N ) )
17	13, 16	breqtrd 4156	. . . . . . 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 9774	. . . . . . . 8 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. ZZ )
20	19, 10	zaddcld 9772	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) e. ZZ )
21		dvdslegcd 12741	. . . . . . . 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 1278	. . . . . 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 12740	. . . . . . . . 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 415	. . . . . . . 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 9770	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> -u K e. ZZ )
29	9, 20	gcdcld 12745	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. NN0 )
30	29	nn0zd 9766	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. ZZ )
31		dvds2ln 12591	. . . . . . . . . 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 1285	. . . . . . . . 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 9769	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> K e. CC )
35	9	zcnd 9769	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
36	34, 35	mulneg1d 8738	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u K x. M ) = -u ( K x. M ) )
37	20	zcnd 9769	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) e. CC )
38	37	mullidd 8344	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 1 x. ( ( K x. M ) + N ) ) = ( ( K x. M ) + N ) )
39	36, 38	oveq12d 6103	. . . . . . . . 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 8346	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K x. M ) e. CC )
41	40	negcld 8624	. . . . . . . . . . . . 13 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> -u ( K x. M ) e. CC )
42	40, 41	addcomd 8477	. . . . . . . . . . . 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 8627	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + -u ( K x. M ) ) = 0 )
44	42, 43	eqtr3d 2273	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u ( K x. M ) + ( K x. M ) ) = 0 )
45	44	oveq1d 6100	. . . . . . . . . 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 8348	. . . . . . . . . 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 8476	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( 0 + N ) = N )
48	45, 46, 47	3eqtr3d 2279	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -u ( K x. M ) + ( ( K x. M ) + N ) ) = N )
49	39, 48	eqtrd 2271	. . . . . . . 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 4156	. . . . . . 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 12741	. . . . . . . 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 1324	. . . . . 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 9621	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. RR )
58	29	nn0red 9621	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd ( ( K x. M ) + N ) ) e. RR )
59	57, 58	letri3d 8441	. . . 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 9656	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> 0 e. ZZ )
62		zdceq 9720	. . . . . . 7 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
63	9, 61, 62	syl2anc 415	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID M = 0 )
64		zdceq 9720	. . . . . . 7 |- ( ( ( ( K x. M ) + N ) e. ZZ /\ 0 e. ZZ ) -> DECID ( ( K x. M ) + N ) = 0 )
65	20, 61, 64	syl2anc 415	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( ( K x. M ) + N ) = 0 )
66	63, 65	dcand 945	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) )
67		zdceq 9720	. . . . . . 7 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
68	10, 61, 67	syl2anc 415	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID N = 0 )
69	63, 68	dcand 945	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
70		orandc 952	. . . . 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 415	. . . 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 6101	. . . . . . . . . . 11 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. M ) = ( K x. 0 ) )
74	34	mul01d 8720	. . . . . . . . . . . 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 2271	. . . . . . . . . 10 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( K x. M ) = 0 )
77	76	oveq1d 6100	. . . . . . . . 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 2271	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( K x. M ) + N ) = N )
80	79	eqeq1d 2247	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( ( K x. M ) + N ) = 0 <-> N = 0 ) )
81	80	pm5.32da 456	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) <-> ( M = 0 /\ N = 0 ) ) )
82		oveq12 6094	. . . . . . . . 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 6094	. . . . . . . . . 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 2274	. . . . . . 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 729	. . . 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 854	. . . . . 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 854	. . . . . 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 945	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID ( -. ( M = 0 /\ ( ( K x. M ) + N ) = 0 ) /\ -. ( M = 0 /\ N = 0 ) ) )
97		exmiddc 848	. . . 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 730	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( ( K x. M ) + N ) ) )
100	40, 14	addcomd 8477	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) + N ) = ( N + ( K x. M ) ) )
101	100	oveq2d 6101	. 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 2271	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd ( N + ( K x. M ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   <_ cle 8361   -ucneg 8498   NN0cn0 9563   ZZcz 9644   || cdvds 12554   gcd cgcd 12730

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731

This theorem is used by:  gcdadd  12762  gcdid  12763  modgcd  12768  gcdmultipled  12770  gcdmultiple  12797  pythagtriplem4  13047  gcdi  13199