Proof of Theorem lgsdir2lem5
Step | Hyp | Ref
| Expression |
1 | | 8nn 9075 |
. . . . . . . . 9
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2 | | zmodcl 10330 |
. . . . . . . . 9
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     |
3 | 1, 2 | mpan2 425 |
. . . . . . . 8
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NN0"  |
4 | 3 | adantr 276 |
. . . . . . 7
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5 | | elprg 3611 |
. . . . . . 7
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6 | 4, 5 | syl 14 |
. . . . . 6
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3"           |
7 | | zmodcl 10330 |
. . . . . . . . 9
 
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8 | 1, 7 | mpan2 425 |
. . . . . . . 8
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NN0"  |
9 | 8 | adantl 277 |
. . . . . . 7
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10 | | elprg 3611 |
. . . . . . 7
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11 | 9, 10 | syl 14 |
. . . . . 6
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12 | 6, 11 | anbi12d 473 |
. . . . 5
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{"                    |
13 | | simpll 527 |
. . . . . . . . 9
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14 | | 3z 9271 |
. . . . . . . . . 10
 |
15 | 14 | a1i 9 |
. . . . . . . . 9
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16 | | simplr 528 |
. . . . . . . . 9
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17 | | nnq 9622 |
. . . . . . . . . . 11
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18 | 1, 17 | ax-mp 5 |
. . . . . . . . . 10
 |
19 | 18 | a1i 9 |
. . . . . . . . 9
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20 | | 8pos 9011 |
. . . . . . . . . 10
 |
21 | 20 | a1i 9 |
. . . . . . . . 9
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  |
22 | | simprl 529 |
. . . . . . . . . 10
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("    |
23 | | lgsdir2lem1 14096 |
. . . . . . . . . . . 12
                 |
24 | 23 | simpri 113 |
. . . . . . . . . . 11
        |
25 | 24 | simpli 111 |
. . . . . . . . . 10
   |
26 | 22, 25 | eqtr4di 2228 |
. . . . . . . . 9
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("      |
27 | | simprr 531 |
. . . . . . . . . 10
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("    |
28 | 27, 25 | eqtr4di 2228 |
. . . . . . . . 9
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("      |
29 | 13, 15, 16, 15, 19, 21, 26, 28 | modqmul12d 10364 |
. . . . . . . 8
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("          |
30 | 29 | orcd 733 |
. . . . . . 7
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("                     |
31 | 30 | ex 115 |
. . . . . 6
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("                     |
32 | | simpll 527 |
. . . . . . . . . 10
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33 | | znegcl 9273 |
. . . . . . . . . . 11
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34 | 14, 33 | mp1i 10 |
. . . . . . . . . 10
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35 | | simplr 528 |
. . . . . . . . . 10
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36 | 14 | a1i 9 |
. . . . . . . . . 10
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37 | 18 | a1i 9 |
. . . . . . . . . 10
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38 | 20 | a1i 9 |
. . . . . . . . . 10
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39 | | simprl 529 |
. . . . . . . . . . 11
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("    |
40 | 24 | simpri 113 |
. . . . . . . . . . 11
 
  |
41 | 39, 40 | eqtr4di 2228 |
. . . . . . . . . 10
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("   
   |
42 | | simprr 531 |
. . . . . . . . . . 11
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("    |
43 | 42, 25 | eqtr4di 2228 |
. . . . . . . . . 10
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("      |
44 | 32, 34, 35, 36, 37, 38, 41, 43 | modqmul12d 10364 |
. . . . . . . . 9
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("           |
45 | | 3cn 8983 |
. . . . . . . . . . 11
 |
46 | 45, 45 | mulneg1i 8351 |
. . . . . . . . . 10
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47 | 46 | oveq1i 5879 |
. . . . . . . . 9
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-u"         |
48 | 44, 47 | eqtrdi 2226 |
. . . . . . . 8
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("           |
49 | 48 | olcd 734 |
. . . . . . 7
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("                     |
50 | 49 | ex 115 |
. . . . . 6
 
                              |
51 | | simpll 527 |
. . . . . . . . . 10
          
  |
52 | 14 | a1i 9 |
. . . . . . . . . 10
             |
53 | | simplr 528 |
. . . . . . . . . 10
          
  |
54 | 14, 33 | mp1i 10 |
. . . . . . . . . 10
              |
55 | 18 | a1i 9 |
. . . . . . . . . 10
             |
56 | 20 | a1i 9 |
. . . . . . . . . 10
          
  |
57 | | simprl 529 |
. . . . . . . . . . 11
               |
58 | 57, 25 | eqtr4di 2228 |
. . . . . . . . . 10
                 |
59 | | simprr 531 |
. . . . . . . . . . 11
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("    |
60 | 59, 40 | eqtr4di 2228 |
. . . . . . . . . 10
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("   
   |
61 | 51, 52, 53, 54, 55, 56, 58, 60 | modqmul12d 10364 |
. . . . . . . . 9
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("       
   |
62 | 45, 45 | mulneg2i 8352 |
. . . . . . . . . 10
       |
63 | 62 | oveq1i 5879 |
. . . . . . . . 9
   
       |
64 | 61, 63 | eqtrdi 2226 |
. . . . . . . 8
                      |
65 | 64 | olcd 734 |
. . . . . . 7
                                |
66 | 65 | ex 115 |
. . . . . 6
 
                              |
67 | | simpll 527 |
. . . . . . . . . 10
          
  |
68 | 14, 33 | mp1i 10 |
. . . . . . . . . 10
              |
69 | | simplr 528 |
. . . . . . . . . 10
          
  |
70 | 18 | a1i 9 |
. . . . . . . . . 10
             |
71 | 20 | a1i 9 |
. . . . . . . . . 10
          
  |
72 | | simprl 529 |
. . . . . . . . . . 11
               |
73 | 72, 40 | eqtr4di 2228 |
. . . . . . . . . 10
              
   |
74 | | simprr 531 |
. . . . . . . . . . 11
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("    |
75 | 74, 40 | eqtr4di 2228 |
. . . . . . . . . 10
              
   |
76 | 67, 68, 69, 68, 70, 71, 73, 75 | modqmul12d 10364 |
. . . . . . . . 9
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("        
   |
77 | 45, 45 | mul2negi 8353 |
. . . . . . . . . 10
       |
78 | 77 | oveq1i 5879 |
. . . . . . . . 9
    
      |
79 | 76, 78 | eqtrdi 2226 |
. . . . . . . 8
                     |
80 | 79 | orcd 733 |
. . . . . . 7
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("                     |
81 | 80 | ex 115 |
. . . . . 6
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                              |
82 | 31, 50, 66, 81 | ccased 965 |
. . . . 5
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               
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(" 
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(" 
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83 | 12, 82 | sylbid 150 |
. . . 4
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84 | 83 | imp 124 |
. . 3
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-u"                           |
93 | | df-9 8974 |
. . . . . . . 8
   |
94 | | 8cn 8994 |
. . . . . . . . 9
 |
95 | | ax-1cn 7895 |
. . . . . . . . 9
 |
96 | 94, 95 | addcomi 8091 |
. . . . . . . 8
     |
97 | 93, 96 | eqtri 2198 |
. . . . . . 7
   |
98 | | 3t3e9 9065 |
. . . . . . 7
   |
99 | 94 | mulid2i 7951 |
. . . . . . . 8
   |
100 | 99 | oveq2i 5880 |
. . . . . . 7
       |
101 | 97, 98, 100 | 3eqtr4i 2208 |
. . . . . 6
       |
102 | 101 | oveq1i 5879 |
. . . . 5
           |
103 | | 1nn 8919 |
. . . . . . 7
 |
104 | | nnq 9622 |
. . . . . . 7
   |
105 | 103, 104 | ax-mp 5 |
. . . . . 6
 |
106 | | 1z 9268 |
. . . . . 6
 |
107 | | modqcyc 10345 |
. . . . . 6
                 |
108 | 105, 106,
18, 20, 107 | mp4an 427 |
. . . . 5
         |
109 | 102, 108 | eqtri 2198 |
. . . 4
       |
110 | 23 | simpli 111 |
. . . . 5
        |
111 | 110 | simpli 111 |
. . . 4
   |
112 | 109, 111 | eqtri 2198 |
. . 3
     |
113 | | znegcl 9273 |
. . . . . . . 8
    |
114 | 106, 113 | mp1i 10 |
. . . . . . 7
   |
115 | | 3nn 9070 |
. . . . . . . . . 10
 |
116 | 115, 115 | nnmulcli 8930 |
. . . . . . . . 9
   |
117 | 116 | nnzi 9263 |
. . . . . . . 8
   |
118 | 117 | a1i 9 |
. . . . . . 7
    |
119 | 106 | a1i 9 |
. . . . . . 7
  |
120 | 18 | a1i 9 |
. . . . . . 7
  |
121 | 20 | a1i 9 |
. . . . . . 7
  |
122 | | eqidd 2178 |
. . . . . . 7
        |
123 | 109 | a1i 9 |
. . . . . . 7
        |
124 | 114, 114,
118, 119, 120, 121, 122, 123 | modqmul12d 10364 |
. . . . . 6
              |
125 | 124 | mptru 1362 |
. . . . 5
             |
126 | 45, 45 | mulcli 7953 |
. . . . . . 7
   |
127 | 126 | mulm1i 8350 |
. . . . . 6
         |
128 | 127 | oveq1i 5879 |
. . . . 5
             |
129 | 95 | mulm1i 8350 |
. . . . . 6
     |
130 | 129 | oveq1i 5879 |
. . . . 5
         |
131 | 125, 128,
130 | 3eqtr3i 2206 |
. . . 4
         |
132 | 110 | simpri 113 |
. . . 4
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  |
133 | 131, 132 | eqtri 2198 |
. . 3
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134 | 112, 133 | preq12i 3673 |
. 2
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135 | 92, 134 | eleqtrdi 2270 |
1
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