Step | Hyp | Ref
| Expression |
1 | | mapd0.h |
. . 3
     |
2 | | mapd0.u |
. . 3
         |
3 | | eqid 2471 |
. . 3
         |
4 | | eqid 2471 |
. . 3
LFnl  LFnl   |
5 | | eqid 2471 |
. . 3
LKer  LKer   |
6 | | eqid 2471 |
. . 3
                 |
7 | | mapd0.m |
. . 3
 mapd      |
8 | | mapd0.k |
. . 3
     |
9 | 1, 2, 8 | dvhlmod 34749 |
. . . 4
   |
10 | | mapd0.o |
. . . . 5
     |
11 | 10, 3 | lsssn0 18249 |
. . . 4

 
      |
12 | 9, 11 | syl 17 |
. . 3
         |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 12 | mapdval 35267 |
. 2
        LFnl                          LKer        LKer                 LKer     
      |
14 | | simprrr 783 |
. . . . . . . . . . 11
 
 LFnl                          LKer        LKer                 LKer                       LKer          |
15 | 9 | adantr 472 |
. . . . . . . . . . . 12
 
 LFnl                          LKer        LKer                 LKer             |
16 | 8 | adantr 472 |
. . . . . . . . . . . . 13
 
 LFnl                          LKer        LKer                 LKer               |
17 | | eqid 2471 |
. . . . . . . . . . . . . 14
         |
18 | | simprl 772 |
. . . . . . . . . . . . . 14
 
 LFnl                          LKer        LKer                 LKer           LFnl    |
19 | 17, 4, 5, 15, 18 | lkrssv 32733 |
. . . . . . . . . . . . 13
 
 LFnl                          LKer        LKer                 LKer            LKer    
      |
20 | 1, 2, 17, 3, 6 | dochlss 34993 |
. . . . . . . . . . . . 13
     LKer    
                 LKer            |
21 | 16, 19, 20 | syl2anc 673 |
. . . . . . . . . . . 12
 
 LFnl                          LKer        LKer                 LKer                       LKer            |
22 | 10, 3 | lssle0 18251 |
. . . . . . . . . . . 12
              LKer                        LKer       
            LKer           |
23 | 15, 21, 22 | syl2anc 673 |
. . . . . . . . . . 11
 
 LFnl                          LKer        LKer                 LKer                        LKer     
              LKer           |
24 | 14, 23 | mpbid 215 |
. . . . . . . . . 10
 
 LFnl                          LKer        LKer                 LKer                       LKer          |
25 | 24 | fveq2d 5883 |
. . . . . . . . 9
 
 LFnl                          LKer        LKer                 LKer                                  LKer                       |
26 | | simprrl 782 |
. . . . . . . . 9
 
 LFnl                          LKer        LKer                 LKer                                  LKer        LKer       |
27 | 1, 2, 6, 17, 10 | doch0 34997 |
. . . . . . . . . . 11
 
                     |
28 | 8, 27 | syl 17 |
. . . . . . . . . 10
                     |
29 | 28 | adantr 472 |
. . . . . . . . 9
 
 LFnl                          LKer        LKer                 LKer                               |
30 | 25, 26, 29 | 3eqtr3d 2513 |
. . . . . . . 8
 
 LFnl                          LKer        LKer                 LKer            LKer           |
31 | | eqid 2471 |
. . . . . . . . . 10
Scalar  Scalar   |
32 | | eqid 2471 |
. . . . . . . . . 10
   Scalar      Scalar    |
33 | 31, 32, 17, 4, 5 | lkr0f 32731 |
. . . . . . . . 9
  LFnl  
  LKer                  Scalar        |
34 | 15, 18, 33 | syl2anc 673 |
. . . . . . . 8
 
 LFnl                          LKer        LKer                 LKer             LKer        
         Scalar        |
35 | 30, 34 | mpbid 215 |
. . . . . . 7
 
 LFnl                          LKer        LKer                 LKer                    Scalar       |
36 | | mapd0.c |
. . . . . . . . 9
 LCDual      |
37 | | mapd0.z |
. . . . . . . . 9
     |
38 | 1, 2, 17, 31, 32, 36, 37, 8 | lcd0v 35250 |
. . . . . . . 8
          Scalar       |
39 | 38 | adantr 472 |
. . . . . . 7
 
 LFnl                          LKer        LKer                 LKer          
         Scalar       |
40 | 35, 39 | eqtr4d 2508 |
. . . . . 6
 
 LFnl                          LKer        LKer                 LKer            |
41 | 40 | ex 441 |
. . . . 5
   LFnl                          LKer        LKer                 LKer            |
42 | | eqid 2471 |
. . . . . . . 8
         |
43 | 1, 36, 42, 37, 8 | lcd0vcl 35253 |
. . . . . . . 8
       |
44 | 1, 36, 42, 2, 4, 8,
43 | lcdvbaselfl 35234 |
. . . . . . 7
 LFnl    |
45 | 31, 32, 17, 4, 5 | lkr0f 32731 |
. . . . . . . . . . . . 13
  LFnl  
  LKer                Scalar        |
46 | 9, 44, 45 | syl2anc 673 |
. . . . . . . . . . . 12
   LKer  
   
         Scalar        |
47 | 38, 46 | mpbird 240 |
. . . . . . . . . . 11
  LKer  
      |
48 | 47 | fveq2d 5883 |
. . . . . . . . . 10
             LKer                      |
49 | 48 | fveq2d 5883 |
. . . . . . . . 9
                        LKer  
                                |
50 | 1, 2, 6, 17, 8 | dochoc1 35000 |
. . . . . . . . 9
                                   |
51 | 49, 50 | eqtrd 2505 |
. . . . . . . 8
                        LKer  
        |
52 | 51, 47 | eqtr4d 2508 |
. . . . . . 7
                        LKer  
   LKer     |
53 | 1, 2, 6, 17, 10 | doch1 34998 |
. . . . . . . . . 10
 
                     |
54 | 8, 53 | syl 17 |
. . . . . . . . 9
                     |
55 | 48, 54 | eqtrd 2505 |
. . . . . . . 8
             LKer        |
56 | | eqimss 3470 |
. . . . . . . 8
             LKer  
               LKer        |
57 | 55, 56 | syl 17 |
. . . . . . 7
             LKer        |
58 | 44, 52, 57 | jca32 544 |
. . . . . 6
 LFnl                          LKer  
   LKer               LKer  

      |
59 | | eleq1 2537 |
. . . . . . 7
  LFnl 
LFnl     |
60 | | fveq2 5879 |
. . . . . . . . . . 11
  LKer      LKer     |
61 | 60 | fveq2d 5883 |
. . . . . . . . . 10
             LKer                  LKer  
   |
62 | 61 | fveq2d 5883 |
. . . . . . . . 9
                        LKer                              LKer  
    |
63 | 62, 60 | eqeq12d 2486 |
. . . . . . . 8
                         LKer        LKer    
                       LKer      LKer  
   |
64 | 61 | sseq1d 3445 |
. . . . . . . 8
              LKer                    LKer         |
65 | 63, 64 | anbi12d 725 |
. . . . . . 7
                          LKer        LKer                 LKer        
                        LKer      LKer  
            LKer          |
66 | 59, 65 | anbi12d 725 |
. . . . . 6
   LFnl                          LKer        LKer                 LKer     
   
LFnl                          LKer      LKer  
            LKer           |
67 | 58, 66 | syl5ibrcom 230 |
. . . . 5
   LFnl                          LKer        LKer                 LKer     
       |
68 | 41, 67 | impbid 195 |
. . . 4
   LFnl                          LKer        LKer                 LKer         
  |
69 | | fveq2 5879 |
. . . . . . . . 9
  LKer      LKer       |
70 | 69 | fveq2d 5883 |
. . . . . . . 8
             LKer                  LKer        |
71 | 70 | fveq2d 5883 |
. . . . . . 7
                        LKer                              LKer         |
72 | 71, 69 | eqeq12d 2486 |
. . . . . 6
                         LKer        LKer    
                       LKer        LKer        |
73 | 70 | sseq1d 3445 |
. . . . . 6
              LKer                    LKer           |
74 | 72, 73 | anbi12d 725 |
. . . . 5
                          LKer        LKer                 LKer        
                        LKer        LKer                 LKer            |
75 | 74 | elrab 3184 |
. . . 4
  LFnl                          LKer        LKer                 LKer         
 LFnl                          LKer        LKer                 LKer            |
76 | | elsn 3973 |
. . . 4
  |
77 | 68, 75, 76 | 3bitr4g 296 |
. . 3
   LFnl                          LKer        LKer                 LKer     
   
   |
78 | 77 | eqrdv 2469 |
. 2
  LFnl                          LKer        LKer                 LKer     
      |
79 | 13, 78 | eqtrd 2505 |
1
         |