Proof of Theorem trlval4
Step | Hyp | Ref
| Expression |
1 | | simp1 988 |
. 2
       
               |
2 | | simp21 1021 |
. 2
       
          
  |
3 | | simp22 1022 |
. 2
       
               |
4 | | simp23 1023 |
. 2
       
               |
5 | | simp3r 1017 |
. . 3
       
                   |
6 | | simpl1l 1039 |
. . . . . . . 8
     

    
                       |
7 | | simp23l 1109 |
. . . . . . . . 9
       
          
  |
8 | 7 | adantr 465 |
. . . . . . . 8
     

    
                       |
9 | | simpl1 991 |
. . . . . . . . 9
     

    
                     
   |
10 | | simpl21 1066 |
. . . . . . . . 9
     

    
                       |
11 | | trlval3.l |
. . . . . . . . . 10
     |
12 | | trlval3.a |
. . . . . . . . . 10
     |
13 | | trlval3.h |
. . . . . . . . . 10
     |
14 | | trlval3.t |
. . . . . . . . . 10
         |
15 | 11, 12, 13, 14 | ltrnat 34142 |
. . . . . . . . 9
   

      |
16 | 9, 10, 8, 15 | syl3anc 1219 |
. . . . . . . 8
     

    
                           |
17 | | trlval3.j |
. . . . . . . . 9
     |
18 | 11, 17, 12 | hlatlej1 33377 |
. . . . . . . 8
 
             |
19 | 6, 8, 16, 18 | syl3anc 1219 |
. . . . . . 7
     

    
                             |
20 | | simpl22 1067 |
. . . . . . . . 9
     

    
                     
   |
21 | | trlval3.r |
. . . . . . . . . 10
         |
22 | 11, 17, 12, 13, 14, 21 | trljat1 34168 |
. . . . . . . . 9
      
              |
23 | 9, 10, 20, 22 | syl3anc 1219 |
. . . . . . . 8
     

    
                                   |
24 | | simpr 461 |
. . . . . . . 8
     

    
                                   |
25 | 23, 24 | eqtrd 2495 |
. . . . . . 7
     

    
                                   |
26 | 19, 25 | breqtrrd 4429 |
. . . . . 6
     

    
                             |
27 | | simpl3r 1044 |
. . . . . . . . 9
     

    
                             |
28 | | simpll1 1027 |
. . . . . . . . . . . . 13
      

    
                          
   |
29 | 20 | adantr 465 |
. . . . . . . . . . . . 13
      

    
                          
   |
30 | 10 | adantr 465 |
. . . . . . . . . . . . 13
      

    
                            |
31 | | simpr 461 |
. . . . . . . . . . . . 13
      

    
                                |
32 | | eqid 2454 |
. . . . . . . . . . . . . 14
         |
33 | 11, 32, 12, 13, 14, 21 | trl0 34172 |
. . . . . . . . . . . . 13
      
                |
34 | 28, 29, 30, 31, 33 | syl112anc 1223 |
. . . . . . . . . . . 12
      

    
                                    |
35 | | hlatl 33363 |
. . . . . . . . . . . . . . 15
   |
36 | 6, 35 | syl 16 |
. . . . . . . . . . . . . 14
     

    
                       |
37 | | simp22l 1107 |
. . . . . . . . . . . . . . . 16
       
          
  |
38 | 37 | adantr 465 |
. . . . . . . . . . . . . . 15
     

    
                       |
39 | | eqid 2454 |
. . . . . . . . . . . . . . . 16
         |
40 | 39, 17, 12 | hlatjcl 33369 |
. . . . . . . . . . . . . . 15
 
         |
41 | 6, 38, 8, 40 | syl3anc 1219 |
. . . . . . . . . . . . . 14
     

    
                             |
42 | 39, 11, 32 | atl0le 33307 |
. . . . . . . . . . . . . 14
             
   |
43 | 36, 41, 42 | syl2anc 661 |
. . . . . . . . . . . . 13
     

    
                         
   |
44 | 43 | adantr 465 |
. . . . . . . . . . . 12
      

    
                              
   |
45 | 34, 44 | eqbrtrd 4423 |
. . . . . . . . . . 11
      

    
                                  |
46 | 45 | ex 434 |
. . . . . . . . . 10
     

    
                                   |
47 | 46 | necon3bd 2664 |
. . . . . . . . 9
     

    
                           
       |
48 | 27, 47 | mpd 15 |
. . . . . . . 8
     

    
                           |
49 | 11, 12, 13, 14, 21 | trlat 34171 |
. . . . . . . 8
      
            |
50 | 9, 20, 10, 48, 49 | syl112anc 1223 |
. . . . . . 7
     

    
                           |
51 | | simpl3l 1043 |
. . . . . . . 8
     

    
                       |
52 | 51 | necomd 2723 |
. . . . . . 7
     

    
                       |
53 | 11, 17, 12 | hlatexch1 33397 |
. . . . . . 7
  
   


      
         |
54 | 6, 8, 50, 38, 52, 53 | syl131anc 1232 |
. . . . . 6
     

    
                           
         |
55 | 26, 54 | mpd 15 |
. . . . 5
     

    
                             |
56 | 55 | ex 434 |
. . . 4
       
                                 |
57 | 56 | necon3bd 2664 |
. . 3
       
                
                |
58 | 5, 57 | mpd 15 |
. 2
       
                         |
59 | | trlval3.m |
. . 3
     |
60 | 11, 17, 59, 12, 13, 14, 21 | trlval3 34189 |
. 2
       
              
          
         |
61 | 1, 2, 3, 4, 58, 60 | syl113anc 1231 |
1
       
                               |