Proof of Theorem mulgcd
Step | Hyp | Ref
| Expression |
1 | | elnn0 9167 |
. . 3
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2 | | simp1 997 |
. . . . . . . . 9
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3 | 2 | nnzd 9363 |
. . . . . . . 8
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4 | | simp2 998 |
. . . . . . . 8
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5 | 3, 4 | zmulcld 9370 |
. . . . . . 7
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6 | | simp3 999 |
. . . . . . . 8
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7 | 3, 6 | zmulcld 9370 |
. . . . . . 7
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8 | 5, 7 | gcdcld 11952 |
. . . . . 6
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9 | 2 | nnnn0d 9218 |
. . . . . . 7
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10 | | gcdcl 11950 |
. . . . . . . 8
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11 | 10 | 3adant1 1015 |
. . . . . . 7
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12 | 9, 11 | nn0mulcld 9223 |
. . . . . 6
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13 | 8 | nn0cnd 9220 |
. . . . . . . 8
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14 | 2 | nncnd 8922 |
. . . . . . . 8
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15 | 2 | nnap0d 8954 |
. . . . . . . 8
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16 | 13, 14, 15 | divcanap2d 8738 |
. . . . . . 7
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17 | | gcddvds 11947 |
. . . . . . . . . . . . 13
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)" 
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)"  |
18 | 5, 7, 17 | syl2anc 411 |
. . . . . . . . . . . 12
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("   
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(" 
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)"   |
19 | 18 | simpld 112 |
. . . . . . . . . . 11
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)"  |
20 | 16, 19 | eqbrtrd 4022 |
. . . . . . . . . 10
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)"  |
21 | | dvdsmul1 11804 |
. . . . . . . . . . . . . 14
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22 | 3, 4, 21 | syl2anc 411 |
. . . . . . . . . . . . 13
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)"  |
23 | | dvdsmul1 11804 |
. . . . . . . . . . . . . 14
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24 | 3, 6, 23 | syl2anc 411 |
. . . . . . . . . . . . 13
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)"  |
25 | | dvdsgcd 11996 |
. . . . . . . . . . . . . 14
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26 | 3, 5, 7, 25 | syl3anc 1238 |
. . . . . . . . . . . . 13
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27 | 22, 24, 26 | mp2and 433 |
. . . . . . . . . . . 12
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28 | 2 | nnne0d 8953 |
. . . . . . . . . . . . 13
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29 | 8 | nn0zd 9362 |
. . . . . . . . . . . . 13
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30 | | dvdsval2 11781 |
. . . . . . . . . . . . 13
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31 | 3, 28, 29, 30 | syl3anc 1238 |
. . . . . . . . . . . 12
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32 | 27, 31 | mpbid 147 |
. . . . . . . . . . 11
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33 | | dvdscmulr 11811 |
. . . . . . . . . . 11
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34 | 32, 4, 3, 28, 33 | syl112anc 1242 |
. . . . . . . . . 10
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("   
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)"  |
35 | 20, 34 | mpbid 147 |
. . . . . . . . 9
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36 | 18 | simprd 114 |
. . . . . . . . . . 11
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)"  |
37 | 16, 36 | eqbrtrd 4022 |
. . . . . . . . . 10
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)"  |
38 | | dvdscmulr 11811 |
. . . . . . . . . . 11
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39 | 32, 6, 3, 28, 38 | syl112anc 1242 |
. . . . . . . . . 10
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)"  |
40 | 37, 39 | mpbid 147 |
. . . . . . . . 9
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41 | | dvdsgcd 11996 |
. . . . . . . . . 10
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42 | 32, 4, 6, 41 | syl3anc 1238 |
. . . . . . . . 9
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)" 
     |
43 | 35, 40, 42 | mp2and 433 |
. . . . . . . 8
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44 | 11 | nn0zd 9362 |
. . . . . . . . 9
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45 | | dvdscmul 11809 |
. . . . . . . . 9
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)"  
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)" 
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(" 
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      |
46 | 32, 44, 3, 45 | syl3anc 1238 |
. . . . . . . 8
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("       
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(" 
     
      |
47 | 43, 46 | mpd 13 |
. . . . . . 7
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(" 
     
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48 | 16, 47 | eqbrtrrd 4024 |
. . . . . 6
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(" 
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49 | | gcddvds 11947 |
. . . . . . . . . 10
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    |
50 | 49 | 3adant1 1015 |
. . . . . . . . 9
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(" 
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51 | 50 | simpld 112 |
. . . . . . . 8
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52 | | dvdscmul 11809 |
. . . . . . . . 9
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(" 
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53 | 44, 4, 3, 52 | syl3anc 1238 |
. . . . . . . 8
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(" 
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54 | 51, 53 | mpd 13 |
. . . . . . 7
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)"  |
55 | 50 | simprd 114 |
. . . . . . . 8
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56 | | dvdscmul 11809 |
. . . . . . . . 9
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57 | 44, 6, 3, 56 | syl3anc 1238 |
. . . . . . . 8
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58 | 55, 57 | mpd 13 |
. . . . . . 7
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)"  |
59 | 12 | nn0zd 9362 |
. . . . . . . 8
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60 | | dvdsgcd 11996 |
. . . . . . . 8
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)"    |
61 | 59, 5, 7, 60 | syl3anc 1238 |
. . . . . . 7
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)"    |
62 | 54, 58, 61 | mp2and 433 |
. . . . . 6
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)"  |
63 | | dvdseq 11837 |
. . . . . 6
                     
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                       |
64 | 8, 12, 48, 62, 63 | syl22anc 1239 |
. . . . 5
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(" 
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65 | 64 | 3expib 1206 |
. . . 4
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("               |
66 | | gcd0val 11944 |
. . . . . . 7
   |
67 | 10 | 3adant1 1015 |
. . . . . . . . 9
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68 | 67 | nn0cnd 9220 |
. . . . . . . 8
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69 | 68 | mul02d 8339 |
. . . . . . 7
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70 | 66, 69 | eqtr4id 2229 |
. . . . . 6
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         |
71 | | simp1 997 |
. . . . . . . . 9
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   |
72 | 71 | oveq1d 5884 |
. . . . . . . 8
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       |
73 | | zcn 9247 |
. . . . . . . . . 10
   |
74 | 73 | 3ad2ant2 1019 |
. . . . . . . . 9
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   |
75 | 74 | mul02d 8339 |
. . . . . . . 8
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     |
76 | 72, 75 | eqtrd 2210 |
. . . . . . 7
 
     |
77 | 71 | oveq1d 5884 |
. . . . . . . 8
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       |
78 | | zcn 9247 |
. . . . . . . . . 10
   |
79 | 78 | 3ad2ant3 1020 |
. . . . . . . . 9
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   |
80 | 79 | mul02d 8339 |
. . . . . . . 8
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     |
81 | 77, 80 | eqtrd 2210 |
. . . . . . 7
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     |
82 | 76, 81 | oveq12d 5887 |
. . . . . 6
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      |
83 | 71 | oveq1d 5884 |
. . . . . 6
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           |
84 | 70, 82, 83 | 3eqtr4d 2220 |
. . . . 5
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        |
85 | 84 | 3expib 1206 |
. . . 4
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86 | 65, 85 | jaoi 716 |
. . 3
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87 | 1, 86 | sylbi 121 |
. 2
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88 | 87 | 3impib 1201 |
1
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