Proof of Theorem mulgcd
Step | Hyp | Ref
| Expression |
1 | | elnn0 11294 |
. . 3
|
2 | | simp1 1061 |
. . . . . . . . 9
|
3 | 2 | nnzd 11481 |
. . . . . . . 8
|
4 | | simp2 1062 |
. . . . . . . 8
|
5 | 3, 4 | zmulcld 11488 |
. . . . . . 7
|
6 | | simp3 1063 |
. . . . . . . 8
|
7 | 3, 6 | zmulcld 11488 |
. . . . . . 7
|
8 | 5, 7 | gcdcld 15230 |
. . . . . 6
|
9 | 2 | nnnn0d 11351 |
. . . . . . 7
|
10 | | gcdcl 15228 |
. . . . . . . 8
|
11 | 10 | 3adant1 1079 |
. . . . . . 7
|
12 | 9, 11 | nn0mulcld 11356 |
. . . . . 6
|
13 | 8 | nn0cnd 11353 |
. . . . . . . 8
|
14 | 2 | nncnd 11036 |
. . . . . . . 8
|
15 | 2 | nnne0d 11065 |
. . . . . . . 8
|
16 | 13, 14, 15 | divcan2d 10803 |
. . . . . . 7
|
17 | | gcddvds 15225 |
. . . . . . . . . . . . 13
|
18 | 5, 7, 17 | syl2anc 693 |
. . . . . . . . . . . 12
|
19 | 18 | simpld 475 |
. . . . . . . . . . 11
|
20 | 16, 19 | eqbrtrd 4675 |
. . . . . . . . . 10
|
21 | | dvdsmul1 15003 |
. . . . . . . . . . . . . 14
|
22 | 3, 4, 21 | syl2anc 693 |
. . . . . . . . . . . . 13
|
23 | | dvdsmul1 15003 |
. . . . . . . . . . . . . 14
|
24 | 3, 6, 23 | syl2anc 693 |
. . . . . . . . . . . . 13
|
25 | | dvdsgcd 15261 |
. . . . . . . . . . . . . 14
|
26 | 3, 5, 7, 25 | syl3anc 1326 |
. . . . . . . . . . . . 13
|
27 | 22, 24, 26 | mp2and 715 |
. . . . . . . . . . . 12
|
28 | 8 | nn0zd 11480 |
. . . . . . . . . . . . 13
|
29 | | dvdsval2 14986 |
. . . . . . . . . . . . 13
|
30 | 3, 15, 28, 29 | syl3anc 1326 |
. . . . . . . . . . . 12
|
31 | 27, 30 | mpbid 222 |
. . . . . . . . . . 11
|
32 | | dvdscmulr 15010 |
. . . . . . . . . . 11
|
33 | 31, 4, 3, 15, 32 | syl112anc 1330 |
. . . . . . . . . 10
|
34 | 20, 33 | mpbid 222 |
. . . . . . . . 9
|
35 | 18 | simprd 479 |
. . . . . . . . . . 11
|
36 | 16, 35 | eqbrtrd 4675 |
. . . . . . . . . 10
|
37 | | dvdscmulr 15010 |
. . . . . . . . . . 11
|
38 | 31, 6, 3, 15, 37 | syl112anc 1330 |
. . . . . . . . . 10
|
39 | 36, 38 | mpbid 222 |
. . . . . . . . 9
|
40 | | dvdsgcd 15261 |
. . . . . . . . . 10
|
41 | 31, 4, 6, 40 | syl3anc 1326 |
. . . . . . . . 9
|
42 | 34, 39, 41 | mp2and 715 |
. . . . . . . 8
|
43 | 11 | nn0zd 11480 |
. . . . . . . . 9
|
44 | | dvdscmul 15008 |
. . . . . . . . 9
|
45 | 31, 43, 3, 44 | syl3anc 1326 |
. . . . . . . 8
|
46 | 42, 45 | mpd 15 |
. . . . . . 7
|
47 | 16, 46 | eqbrtrrd 4677 |
. . . . . 6
|
48 | | gcddvds 15225 |
. . . . . . . . . 10
|
49 | 48 | 3adant1 1079 |
. . . . . . . . 9
|
50 | 49 | simpld 475 |
. . . . . . . 8
|
51 | | dvdscmul 15008 |
. . . . . . . . 9
|
52 | 43, 4, 3, 51 | syl3anc 1326 |
. . . . . . . 8
|
53 | 50, 52 | mpd 15 |
. . . . . . 7
|
54 | 49 | simprd 479 |
. . . . . . . 8
|
55 | | dvdscmul 15008 |
. . . . . . . . 9
|
56 | 43, 6, 3, 55 | syl3anc 1326 |
. . . . . . . 8
|
57 | 54, 56 | mpd 15 |
. . . . . . 7
|
58 | 12 | nn0zd 11480 |
. . . . . . . 8
|
59 | | dvdsgcd 15261 |
. . . . . . . 8
|
60 | 58, 5, 7, 59 | syl3anc 1326 |
. . . . . . 7
|
61 | 53, 57, 60 | mp2and 715 |
. . . . . 6
|
62 | | dvdseq 15036 |
. . . . . 6
|
63 | 8, 12, 47, 61, 62 | syl22anc 1327 |
. . . . 5
|
64 | 63 | 3expib 1268 |
. . . 4
|
65 | 10 | 3adant1 1079 |
. . . . . . . . 9
|
66 | 65 | nn0cnd 11353 |
. . . . . . . 8
|
67 | 66 | mul02d 10234 |
. . . . . . 7
|
68 | | gcd0val 15219 |
. . . . . . 7
|
69 | 67, 68 | syl6reqr 2675 |
. . . . . 6
|
70 | | simp1 1061 |
. . . . . . . . 9
|
71 | 70 | oveq1d 6665 |
. . . . . . . 8
|
72 | | zcn 11382 |
. . . . . . . . . 10
|
73 | 72 | 3ad2ant2 1083 |
. . . . . . . . 9
|
74 | 73 | mul02d 10234 |
. . . . . . . 8
|
75 | 71, 74 | eqtrd 2656 |
. . . . . . 7
|
76 | 70 | oveq1d 6665 |
. . . . . . . 8
|
77 | | zcn 11382 |
. . . . . . . . . 10
|
78 | 77 | 3ad2ant3 1084 |
. . . . . . . . 9
|
79 | 78 | mul02d 10234 |
. . . . . . . 8
|
80 | 76, 79 | eqtrd 2656 |
. . . . . . 7
|
81 | 75, 80 | oveq12d 6668 |
. . . . . 6
|
82 | 70 | oveq1d 6665 |
. . . . . 6
|
83 | 69, 81, 82 | 3eqtr4d 2666 |
. . . . 5
|
84 | 83 | 3expib 1268 |
. . . 4
|
85 | 64, 84 | jaoi 394 |
. . 3
|
86 | 1, 85 | sylbi 207 |
. 2
|
87 | 86 | 3impib 1262 |
1
|