Proof of Theorem lgsdir2
Step | Hyp | Ref
| Expression |
1 | | 0cnd 7941 |
. . . . . . 7
    
  |
2 | | 1cnd 7964 |
. . . . . . . . 9
 
   |
3 | | neg1cn 9013 |
. . . . . . . . . 10
  |
4 | 3 | a1i 9 |
. . . . . . . . 9
 
    |
5 | | simpr 110 |
. . . . . . . . . . . . . 14
 
   |
6 | | 8nn 9075 |
. . . . . . . . . . . . . . 15
 |
7 | 6 | a1i 9 |
. . . . . . . . . . . . . 14
 
   |
8 | 5, 7 | zmodcld 10331 |
. . . . . . . . . . . . 13
 
     |
9 | 8 | nn0zd 9362 |
. . . . . . . . . . . 12
 
     |
10 | | 1zzd 9269 |
. . . . . . . . . . . 12
 
   |
11 | | zdceq 9317 |
. . . . . . . . . . . 12
   
 DECID     |
12 | 9, 10, 11 | syl2anc 411 |
. . . . . . . . . . 11
 
 DECID     |
13 | | 7nn 9074 |
. . . . . . . . . . . . . 14
 |
14 | 13 | nnzi 9263 |
. . . . . . . . . . . . 13
 |
15 | 14 | a1i 9 |
. . . . . . . . . . . 12
 
   |
16 | | zdceq 9317 |
. . . . . . . . . . . 12
   
 DECID     |
17 | 9, 15, 16 | syl2anc 411 |
. . . . . . . . . . 11
 
 DECID     |
18 | | dcor 935 |
. . . . . . . . . . 11
DECID  
DECID
  DECID          |
19 | 12, 17, 18 | sylc 62 |
. . . . . . . . . 10
 
 DECID         |
20 | | elprg 3611 |
. . . . . . . . . . . 12
  
     
         |
21 | 8, 20 | syl 14 |
. . . . . . . . . . 11
 
                |
22 | 21 | dcbid 838 |
. . . . . . . . . 10
 
 DECID     
DECID          |
23 | 19, 22 | mpbird 167 |
. . . . . . . . 9
 
 DECID        |
24 | 2, 4, 23 | ifcldcd 3569 |
. . . . . . . 8
 
              |
25 | 24 | adantr 276 |
. . . . . . 7
    
             |
26 | | 2nn 9069 |
. . . . . . . . 9
 |
27 | 26 | a1i 9 |
. . . . . . . 8
    
  |
28 | | simplr 528 |
. . . . . . . 8
    
  |
29 | | dvdsdc 11789 |
. . . . . . . 8
 
 DECID   |
30 | 27, 28, 29 | syl2anc 411 |
. . . . . . 7
    
DECID
  |
31 | 1, 25, 30 | ifcldcd 3569 |
. . . . . 6
    
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                |
32 | 31 | mul02d 8339 |
. . . . 5
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33 | | iftrue 3539 |
. . . . . . 7
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1"         |
34 | 33 | adantl 277 |
. . . . . 6
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0"               |
35 | 34 | oveq1d 5884 |
. . . . 5
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1"    
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1"           
                |
36 | | 2z 9270 |
. . . . . . . 8
 |
37 | | dvdsmultr1 11822 |
. . . . . . . 8
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38 | 36, 37 | mp3an1 1324 |
. . . . . . 7
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39 | 38 | imp 124 |
. . . . . 6
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)"  |
40 | 39 | iftrued 3541 |
. . . . 5
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41 | 32, 35, 40 | 3eqtr4d 2220 |
. . . 4
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1"                  
           |
42 | | 0cnd 7941 |
. . . . . . . 8
 
   |
43 | | simpl 109 |
. . . . . . . . . . . . . 14
 
   |
44 | 43, 7 | zmodcld 10331 |
. . . . . . . . . . . . 13
 
     |
45 | 44 | nn0zd 9362 |
. . . . . . . . . . . 12
 
     |
46 | | zdceq 9317 |
. . . . . . . . . . . 12
   
 DECID     |
47 | 45, 10, 46 | syl2anc 411 |
. . . . . . . . . . 11
 
 DECID     |
48 | | zdceq 9317 |
. . . . . . . . . . . 12
   
 DECID     |
49 | 45, 15, 48 | syl2anc 411 |
. . . . . . . . . . 11
 
 DECID     |
50 | | dcor 935 |
. . . . . . . . . . 11
DECID  
DECID
  DECID          |
51 | 47, 49, 50 | sylc 62 |
. . . . . . . . . 10
 
 DECID         |
52 | | elprg 3611 |
. . . . . . . . . . . 12
  
     
         |
53 | 44, 52 | syl 14 |
. . . . . . . . . . 11
 
                |
54 | 53 | dcbid 838 |
. . . . . . . . . 10
 
 DECID     
DECID          |
55 | 51, 54 | mpbird 167 |
. . . . . . . . 9
 
 DECID        |
56 | 2, 4, 55 | ifcldcd 3569 |
. . . . . . . 8
 
              |
57 | 26 | a1i 9 |
. . . . . . . . 9
 
   |
58 | | dvdsdc 11789 |
. . . . . . . . 9
 
 DECID   |
59 | 57, 43, 58 | syl2anc 411 |
. . . . . . . 8
 
 DECID   |
60 | 42, 56, 59 | ifcldcd 3569 |
. . . . . . 7
 
   
               |
61 | 60 | mul01d 8340 |
. . . . . 6
 
                     |
62 | 61 | adantr 276 |
. . . . 5
    
                    |
63 | | iftrue 3539 |
. . . . . . 7
                   |
64 | 63 | adantl 277 |
. . . . . 6
    
 
                |
65 | 64 | oveq2d 5885 |
. . . . 5
    
                   
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1"                       |
66 | | dvdsmultr2 11824 |
. . . . . . . 8
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67 | 36, 66 | mp3an1 1324 |
. . . . . . 7
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68 | 67 | imp 124 |
. . . . . 6
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)"  |
69 | 68 | iftrued 3541 |
. . . . 5
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70 | 62, 65, 69 | 3eqtr4d 2220 |
. . . 4
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71 | 41, 70 | jaodan 797 |
. . 3
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        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        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1"      |
72 | | ioran 752 |
. . . 4
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("    |
73 | 24 | ad2antrr 488 |
. . . . . . . . 9
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{"                 |
74 | 73 | mulid2d 7966 |
. . . . . . . 8
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{"                           |
75 | | iftrue 3539 |
. . . . . . . . . 10
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1"        |
76 | 75 | adantl 277 |
. . . . . . . . 9
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{"                               |
78 | | lgsdir2lem4 14099 |
. . . . . . . . . 10
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1"  
   
   
        |
79 | 78 | adantlr 477 |
. . . . . . . . 9
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               
        |
80 | 79 | ifbid 3555 |
. . . . . . . 8
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                                  |
81 | 74, 77, 80 | 3eqtr4d 2220 |
. . . . . . 7
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1"                 |
82 | 56 | mulid1d 7965 |
. . . . . . . . 9
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1"        |
83 | 82 | ad2antrr 488 |
. . . . . . . 8
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                   |
84 | | iftrue 3539 |
. . . . . . . . . 10
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1"        |
85 | 84 | adantl 277 |
. . . . . . . . 9
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{"                 |
86 | 85 | oveq2d 5885 |
. . . . . . . 8
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1"                            |
87 | | zcn 9247 |
. . . . . . . . . . . . . 14
   |
88 | | zcn 9247 |
. . . . . . . . . . . . . 14
   |
89 | | mulcom 7931 |
. . . . . . . . . . . . . 14
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       |
90 | 87, 88, 89 | syl2an 289 |
. . . . . . . . . . . . 13
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       |
91 | 90 | ad2antrr 488 |
. . . . . . . . . . . 12
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        
     |
92 | 91 | oveq1d 5884 |
. . . . . . . . . . 11
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{"      
       |
93 | 92 | eleq1d 2246 |
. . . . . . . . . 10
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               
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        |
94 | | lgsdir2lem4 14099 |
. . . . . . . . . . . 12
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1"  
   
   
        |
95 | 94 | ancom1s 569 |
. . . . . . . . . . 11
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1"  
   
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        |
96 | 95 | adantlr 477 |
. . . . . . . . . 10
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("      
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1"     |
97 | 93, 96 | bitrd 188 |
. . . . . . . . 9
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("      
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1"     |
98 | 97 | ifbid 3555 |
. . . . . . . 8
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1"        |
99 | 83, 86, 98 | 3eqtr4d 2220 |
. . . . . . 7
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1"                      |
100 | 81, 99 | jaodan 797 |
. . . . . 6
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{"            
          |
101 | | ioran 752 |
. . . . . . 7
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1"                      |
102 | | neg1mulneg1e1 9120 |
. . . . . . . 8
     |
103 | | iffalse 3542 |
. . . . . . . . . 10
                    |
104 | | iffalse 3542 |
. . . . . . . . . 10
                    |
105 | 103, 104 | oveqan12d 5888 |
. . . . . . . . 9
                                           |
106 | 105 | adantl 277 |
. . . . . . . 8
     
              
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1"                   |
107 | | lgsdir2lem3 14098 |
. . . . . . . . . . . . . 14
 
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108 | 107 | ad2ant2r 509 |
. . . . . . . . . . . . 13
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("            |
109 | | elun 3276 |
. . . . . . . . . . . . 13
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{"           |
110 | 108, 109 | sylib 122 |
. . . . . . . . . . . 12
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("              |
111 | 110 | orcanai 928 |
. . . . . . . . . . 11
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{"           |
112 | | lgsdir2lem3 14098 |
. . . . . . . . . . . . . 14
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             |
113 | 112 | ad2ant2l 508 |
. . . . . . . . . . . . 13
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("            |
114 | | elun 3276 |
. . . . . . . . . . . . 13
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{"           |
115 | 113, 114 | sylib 122 |
. . . . . . . . . . . 12
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("              |
116 | 115 | orcanai 928 |
. . . . . . . . . . 11
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{"     |
117 | 111, 116 | anim12dan 600 |
. . . . . . . . . 10
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{"           |
118 | | lgsdir2lem5 14100 |
. . . . . . . . . . 11
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{"                   |
119 | 118 | adantlr 477 |
. . . . . . . . . 10
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                        |
120 | 117, 119 | syldan 282 |
. . . . . . . . 9
     
              
 
       |
121 | 120 | iftrued 3541 |
. . . . . . . 8
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              
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("             |
122 | 102, 106,
121 | 3eqtr4a 2236 |
. . . . . . 7
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{"                      |
124 | | dcor 935 |
. . . . . . . . 9
DECID     
DECID
     DECID                |
125 | 55, 23, 124 | sylc 62 |
. . . . . . . 8
 
 DECID       
       |
126 | 125 | adantr 276 |
. . . . . . 7
    
  DECID       
       |
127 | | exmiddc 836 |
. . . . . . 7
DECID                                
        |
128 | 126, 127 | syl 14 |
. . . . . 6
    
                              |
129 | 100, 123,
128 | mpjaodan 798 |
. . . . 5
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                              
          |
130 | | iffalse 3542 |
. . . . . . 7
                              |
131 | | iffalse 3542 |
. . . . . . 7
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1"               |
132 | 130, 131 | oveqan12d 5888 |
. . . . . 6
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133 | 132 | adantl 277 |
. . . . 5
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1"               |
134 | | 2prm 12110 |
. . . . . . . . . 10
 |
135 | | euclemma 12129 |
. . . . . . . . . 10
   
 
     |
136 | 134, 135 | mp3an1 1324 |
. . . . . . . . 9
 
  
      |
137 | 136 | notbid 667 |
. . . . . . . 8
 
  

     |
138 | 137 | biimpar 297 |
. . . . . . 7
      

   |
139 | 72, 138 | sylan2br 288 |
. . . . . 6
    
 
    |
140 | | iffalse 3542 |
. . . . . 6
     
                                |
141 | 139, 140 | syl 14 |
. . . . 5
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1"                 |
142 | 129, 133,
141 | 3eqtr4d 2220 |
. . . 4
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               |
143 | 72, 142 | sylan2b 287 |
. . 3
                                          
           |
144 | 57, 5, 29 | syl2anc 411 |
. . . . 5
 
 DECID   |
145 | | dcor 935 |
. . . . 5
DECID DECID DECID 
    |
146 | 59, 144, 145 | sylc 62 |
. . . 4
 
 DECID     |
147 | | exmiddc 836 |
. . . 4
DECID    
 
    |
148 | 146, 147 | syl 14 |
. . 3
 
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 
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149 | 71, 143, 148 | mpjaodan 798 |
. 2
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150 | | lgs2 14085 |
. . 3
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151 | | lgs2 14085 |
. . 3
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152 | 150, 151 | oveqan12d 5888 |
. 2
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153 | | zmulcl 9295 |
. . 3
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154 | | lgs2 14085 |
. . 3
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("                           |
155 | 153, 154 | syl 14 |
. 2
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1"    |
156 | 149, 152,
155 | 3eqtr4rd 2221 |
1
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