Step | Hyp | Ref
| Expression |
1 | | eqid 2471 |
. . . . 5
               
                         
           |
2 | 1 | grp1 16836 |
. . . 4
                
            |
3 | | fvex 5889 |
. . . . . . 7
         
                     |
4 | | lmod1.m |
. . . . . . . . 9
                
         
 Scalar                 
       |
5 | | snex 4641 |
. . . . . . . . . . . . 13
 
 |
6 | 1 | grpbase 15315 |
. . . . . . . . . . . . 13
                       
             |
7 | 5, 6 | ax-mp 5 |
. . . . . . . . . . . 12
 
         
                     |
8 | 7 | opeq2i 4162 |
. . . . . . . . . . 11
              
         
                      |
9 | | tpeq1 4051 |
. . . . . . . . . . 11
               
         
                    
               
         
 Scalar                     
                                      Scalar       |
10 | 8, 9 | ax-mp 5 |
. . . . . . . . . 10
               
         
 Scalar                     
                                      Scalar      |
11 | 10 | uneq1i 3575 |
. . . . . . . . 9
       
                    Scalar           
                             
                                      Scalar           
             |
12 | 4, 11 | eqtri 2493 |
. . . . . . . 8
                 
                                      Scalar           
             |
13 | 12 | lmodbase 15340 |
. . . . . . 7
                   
                    
                          |
14 | 3, 13 | ax-mp 5 |
. . . . . 6
         
                         |
15 | 14 | eqcomi 2480 |
. . . . 5
                      
            |
16 | | fvex 5889 |
. . . . . . 7
        
                     |
17 | | snex 4641 |
. . . . . . . . . . . . 13
         |
18 | 1 | grpplusg 15316 |
. . . . . . . . . . . . 13
                         
                      |
19 | 17, 18 | ax-mp 5 |
. . . . . . . . . . . 12
                         
            |
20 | 19 | opeq2i 4162 |
. . . . . . . . . . 11
                  
        
                      |
21 | | tpeq2 4052 |
. . . . . . . . . . 11
  
                
        
                    
               
         
 Scalar                    
        
                       Scalar       |
22 | 20, 21 | ax-mp 5 |
. . . . . . . . . 10
               
         
 Scalar                    
        
                       Scalar      |
23 | 22 | uneq1i 3575 |
. . . . . . . . 9
       
                    Scalar           
                            
        
                       Scalar                         |
24 | 4, 23 | eqtri 2493 |
. . . . . . . 8
                
        
                       Scalar                         |
25 | 24 | lmodplusg 15341 |
. . . . . . 7
                  
                   
                         |
26 | 16, 25 | ax-mp 5 |
. . . . . 6
        
                        |
27 | 26 | eqcomi 2480 |
. . . . 5
                    
            |
28 | 15, 27 | grpprop 16763 |
. . . 4

               
            |
29 | 2, 28 | sylibr 217 |
. . 3
   |
30 | 29 | adantr 472 |
. 2
 

  |
31 | 4 | lmodsca 15342 |
. . . . 5

Scalar    |
32 | 31 | eqcomd 2477 |
. . . 4

Scalar    |
33 | 32 | adantl 473 |
. . 3
 
 Scalar    |
34 | | simpr 468 |
. . 3
 

  |
35 | 33, 34 | eqeltrd 2549 |
. 2
 
 Scalar    |
36 | 33 | fveq2d 5883 |
. . . . . . 7
 
    Scalar         |
37 | 36 | eleq2d 2534 |
. . . . . 6
 
     Scalar  
       |
38 | 36 | eleq2d 2534 |
. . . . . 6
 
     Scalar  
       |
39 | 37, 38 | anbi12d 725 |
. . . . 5
 
      Scalar  
   Scalar   
    
        |
40 | | simpll 768 |
. . . . . . . . . 10
  
     
        |
41 | | simplr 770 |
. . . . . . . . . 10
  
     
        |
42 | | simprr 774 |
. . . . . . . . . 10
  
     
            |
43 | 40, 41, 42 | 3jca 1210 |
. . . . . . . . 9
  
     
      
       |
44 | 4 | lmod1lem1 40788 |
. . . . . . . . 9
 
                 |
45 | 43, 44 | syl 17 |
. . . . . . . 8
  
     
                  |
46 | 4 | lmod1lem2 40789 |
. . . . . . . . 9
 
                                             |
47 | 43, 46 | syl 17 |
. . . . . . . 8
  
     
                                              |
48 | 4 | lmod1lem3 40790 |
. . . . . . . 8
  
     
           Scalar                                     |
49 | 45, 47, 48 | 3jca 1210 |
. . . . . . 7
  
     
                                                            Scalar                                      |
50 | 4 | lmod1lem4 40791 |
. . . . . . 7
  
     
            Scalar                              |
51 | 4 | lmod1lem5 40792 |
. . . . . . . 8
 
     Scalar            |
52 | 51 | adantr 472 |
. . . . . . 7
  
     
          Scalar            |
53 | 49, 50, 52 | jca32 544 |
. . . . . 6
  
     
                                                             Scalar                                           Scalar                                Scalar              |
54 | 53 | ex 441 |
. . . . 5
 
      
    
                                                       Scalar                                           Scalar                                Scalar               |
55 | 39, 54 | sylbid 223 |
. . . 4
 
      Scalar  
   Scalar                                                           Scalar                                           Scalar                                Scalar               |
56 | 55 | ralrimivv 2813 |
. . 3
 
 
   Scalar        Scalar                                                           Scalar                                           Scalar                                Scalar              |
57 | | oveq2 6316 |
. . . . . . . . . . . 12
                 |
58 | 57 | oveq2d 6324 |
. . . . . . . . . . 11
                                 |
59 | | oveq2 6316 |
. . . . . . . . . . . 12
                   |
60 | 59 | oveq2d 6324 |
. . . . . . . . . . 11
                                                 |
61 | 58, 60 | eqeq12d 2486 |
. . . . . . . . . 10
                                       
                                         |
62 | 61 | 3anbi2d 1370 |
. . . . . . . . 9
                                                        Scalar                                                                                          Scalar                                       |
63 | 62 | anbi1d 719 |
. . . . . . . 8
                                                         Scalar                                           Scalar                                Scalar           
                                                       Scalar                                           Scalar                                Scalar               |
64 | 63 | ralbidv 2829 |
. . . . . . 7
  
                                                         Scalar                                           Scalar                                Scalar           
                                                          Scalar                                           Scalar                                Scalar               |
65 | 64 | ralsng 3997 |
. . . . . 6
  
                                                             Scalar                                           Scalar                                Scalar           
                                                          Scalar                                           Scalar                                Scalar               |
66 | 65 | adantr 472 |
. . . . 5
 
      
                                                         Scalar                                           Scalar                                Scalar           
                                                          Scalar                                           Scalar                                Scalar               |
67 | | oveq2 6316 |
. . . . . . . . . 10
                   |
68 | 67 | eleq1d 2533 |
. . . . . . . . 9
                         |
69 | | oveq1 6315 |
. . . . . . . . . . 11
                 |
70 | 69 | oveq2d 6324 |
. . . . . . . . . 10
                                 |
71 | 67 | oveq1d 6323 |
. . . . . . . . . 10
                                                 |
72 | 70, 71 | eqeq12d 2486 |
. . . . . . . . 9
                                       
                                         |
73 | | oveq2 6316 |
. . . . . . . . . 10
      Scalar                 Scalar              |
74 | | oveq2 6316 |
. . . . . . . . . . 11
                   |
75 | 74, 67 | oveq12d 6326 |
. . . . . . . . . 10
                                                 |
76 | 73, 75 | eqeq12d 2486 |
. . . . . . . . 9
       Scalar                                  
     Scalar                                      |
77 | 68, 72, 76 | 3anbi123d 1365 |
. . . . . . . 8
                                                        Scalar                                                                                          Scalar                                       |
78 | | oveq2 6316 |
. . . . . . . . . 10
       Scalar                  Scalar              |
79 | 67 | oveq2d 6324 |
. . . . . . . . . 10
                                   |
80 | 78, 79 | eqeq12d 2486 |
. . . . . . . . 9
        Scalar                           
      Scalar                               |
81 | | oveq2 6316 |
. . . . . . . . . 10
     Scalar              Scalar            |
82 | | id 22 |
. . . . . . . . . 10
   |
83 | 81, 82 | eqeq12d 2486 |
. . . . . . . . 9
      Scalar              Scalar             |
84 | 80, 83 | anbi12d 725 |
. . . . . . . 8
         Scalar                                Scalar                  Scalar                                Scalar              |
85 | 77, 84 | anbi12d 725 |
. . . . . . 7
                                                         Scalar                                           Scalar                                Scalar           
                                                       Scalar                                           Scalar                                Scalar               |
86 | 85 | ralsng 3997 |
. . . . . 6
  
                                                         Scalar                                           Scalar                                Scalar           
                                                       Scalar                                           Scalar                                Scalar               |
87 | 86 | adantr 472 |
. . . . 5
 
                                                            Scalar                                           Scalar                                Scalar           
                                                       Scalar                                           Scalar                                Scalar               |
88 | 66, 87 | bitrd 261 |
. . . 4
 
      
                                                         Scalar                                           Scalar                                Scalar           
                                                       Scalar                                           Scalar                                Scalar               |
89 | 88 | 2ralbidv 2832 |
. . 3
 
      Scalar    
   Scalar        
                                                         Scalar                                           Scalar                                Scalar           
    Scalar    
   Scalar                                                           Scalar                                           Scalar                                Scalar               |
90 | 56, 89 | mpbird 240 |
. 2
 
 
   Scalar        Scalar    
                                                             Scalar                                           Scalar                                Scalar              |
91 | 4 | lmodbase 15340 |
. . . 4
           |
92 | 5, 91 | ax-mp 5 |
. . 3
 
     |
93 | | eqid 2471 |
. . 3
       |
94 | | eqid 2471 |
. . 3
         |
95 | | eqid 2471 |
. . 3
Scalar  Scalar   |
96 | | eqid 2471 |
. . 3
   Scalar      Scalar    |
97 | | eqid 2471 |
. . 3
  Scalar     Scalar    |
98 | | eqid 2471 |
. . 3
   Scalar      Scalar    |
99 | | eqid 2471 |
. . 3
   Scalar      Scalar    |
100 | 92, 93, 94, 95, 96, 97, 98, 99 | islmod 18173 |
. 2


Scalar  
   Scalar        Scalar    
                                                             Scalar                                           Scalar                                Scalar               |
101 | 30, 35, 90, 100 | syl3anbrc 1214 |
1
 

  |