Step | Hyp | Ref
| Expression |
1 | | eqid 2451 |
. . . . . . 7
         |
2 | | eqid 2451 |
. . . . . . 7
         |
3 | | eqid 2451 |
. . . . . . 7
         |
4 | | eqid 2451 |
. . . . . . 7
         |
5 | 1, 2, 3, 4 | rngohomf 28912 |
. . . . . 6
 

 
       
      |
6 | 5 | 3expa 1188 |
. . . . 5
  

                 |
7 | 6 | 3adantl1 1144 |
. . . 4
  
   
       
      |
8 | 7 | adantrl 715 |
. . 3
  
 
             
      |
9 | | eqid 2451 |
. . . . . . 7
         |
10 | | eqid 2451 |
. . . . . . 7
         |
11 | 9, 10, 1, 2 | rngohomf 28912 |
. . . . . 6
 

 
       
      |
12 | 11 | 3expa 1188 |
. . . . 5
  

                 |
13 | 12 | 3adantl3 1146 |
. . . 4
  
   
       
      |
14 | 13 | adantrr 716 |
. . 3
  
 
             
      |
15 | | fco 5668 |
. . 3
             
       
                     |
16 | 8, 14, 15 | syl2anc 661 |
. 2
  
 
      
        
      |
17 | | eqid 2451 |
. . . . . . 7
         |
18 | | eqid 2451 |
. . . . . . 7
GId      GId       |
19 | 10, 17, 18 | rngo1cl 24053 |
. . . . . 6
 GId            |
20 | 19 | 3ad2ant1 1009 |
. . . . 5
 
 GId            |
21 | 20 | adantr 465 |
. . . 4
  
 
      GId            |
22 | | fvco3 5869 |
. . . 4
             
GId                GId             GId          |
23 | 14, 21, 22 | syl2anc 661 |
. . 3
  
 
           GId             GId          |
24 | | eqid 2451 |
. . . . . . . . 9
         |
25 | | eqid 2451 |
. . . . . . . . 9
GId      GId       |
26 | 17, 18, 24, 25 | rngohom1 28914 |
. . . . . . . 8
 

 
   GId       GId        |
27 | 26 | 3expa 1188 |
. . . . . . 7
  

      GId       GId        |
28 | 27 | 3adantl3 1146 |
. . . . . 6
  
   
   GId       GId        |
29 | 28 | adantrr 716 |
. . . . 5
  
 
         GId       GId        |
30 | 29 | fveq2d 5795 |
. . . 4
  
 
            GId           GId         |
31 | | eqid 2451 |
. . . . . . . 8
         |
32 | | eqid 2451 |
. . . . . . . 8
GId      GId       |
33 | 24, 25, 31, 32 | rngohom1 28914 |
. . . . . . 7
 

 
   GId       GId        |
34 | 33 | 3expa 1188 |
. . . . . 6
  

      GId       GId        |
35 | 34 | 3adantl1 1144 |
. . . . 5
  
   
   GId       GId        |
36 | 35 | adantrl 715 |
. . . 4
  
 
         GId       GId        |
37 | 30, 36 | eqtrd 2492 |
. . 3
  
 
            GId        GId        |
38 | 23, 37 | eqtrd 2492 |
. 2
  
 
           GId       GId        |
39 | 9, 10, 1 | rngohomadd 28915 |
. . . . . . . . . . . 12
  
                                            |
40 | 39 | ex 434 |
. . . . . . . . . . 11
 

 
     
                                    |
41 | 40 | 3expa 1188 |
. . . . . . . . . 10
  

        
                                    |
42 | 41 | 3adantl3 1146 |
. . . . . . . . 9
  
   
     
                                    |
43 | 42 | imp 429 |
. . . . . . . 8
   

   
                                        |
44 | 43 | adantlrr 720 |
. . . . . . 7
   
 
 
        
                                    |
45 | 44 | fveq2d 5795 |
. . . . . 6
   
 
 
        
                                            |
46 | 9, 10, 1, 2 | rngohomcl 28913 |
. . . . . . . . . . . . 13
  
                  |
47 | 9, 10, 1, 2 | rngohomcl 28913 |
. . . . . . . . . . . . 13
  
                  |
48 | 46, 47 | anim12da 28744 |
. . . . . . . . . . . 12
  
                                  |
49 | 48 | ex 434 |
. . . . . . . . . . 11
 

 
     
                          |
50 | 49 | 3expa 1188 |
. . . . . . . . . 10
  

        
                          |
51 | 50 | 3adantl3 1146 |
. . . . . . . . 9
  
   
     
                          |
52 | 51 | imp 429 |
. . . . . . . 8
   

   
                              |
53 | 52 | adantlrr 720 |
. . . . . . 7
   
 
 
        
                          |
54 | 1, 2, 3 | rngohomadd 28915 |
. . . . . . . . . . . 12
  
                                                                    |
55 | 54 | ex 434 |
. . . . . . . . . . 11
 

 
                                                                  |
56 | 55 | 3expa 1188 |
. . . . . . . . . 10
  

            
        
                                               |
57 | 56 | 3adantl1 1144 |
. . . . . . . . 9
  
   
                                                                  |
58 | 57 | imp 429 |
. . . . . . . 8
   

                                                                    |
59 | 58 | adantlrl 719 |
. . . . . . 7
   
 
 
            
                                                        |
60 | 53, 59 | syldan 470 |
. . . . . 6
   
 
 
        
                                                    |
61 | 45, 60 | eqtrd 2492 |
. . . . 5
   
 
 
        
                                                |
62 | 9, 10 | rngogcl 24015 |
. . . . . . . . 9
     
                   |
63 | 62 | 3expb 1189 |
. . . . . . . 8
      
                    |
64 | 63 | 3ad2antl1 1150 |
. . . . . . 7
  
     
                    |
65 | 64 | adantlr 714 |
. . . . . 6
   
 
 
        
                    |
66 | | fvco3 5869 |
. . . . . . 7
             
                                             |
67 | 14, 66 | sylan 471 |
. . . . . 6
   
 
 
                
 
                              |
68 | 65, 67 | syldan 470 |
. . . . 5
   
 
 
        
                                      |
69 | | fvco3 5869 |
. . . . . . . 8
             
                     |
70 | 14, 69 | sylan 471 |
. . . . . . 7
   
 
 
        
 
              |
71 | | fvco3 5869 |
. . . . . . . 8
             
                     |
72 | 14, 71 | sylan 471 |
. . . . . . 7
   
 
 
        
 
              |
73 | 70, 72 | anim12da 28744 |
. . . . . 6
   
 
 
        
                                      |
74 | | oveq12 6201 |
. . . . . 6
                                                                             |
75 | 73, 74 | syl 16 |
. . . . 5
   
 
 
        
                                                    |
76 | 61, 68, 75 | 3eqtr4d 2502 |
. . . 4
   
 
 
        
                                          |
77 | 9, 10, 17, 24 | rngohommul 28916 |
. . . . . . . . . . . 12
  
                                            |
78 | 77 | ex 434 |
. . . . . . . . . . 11
 

 
     
                                    |
79 | 78 | 3expa 1188 |
. . . . . . . . . 10
  

        
                                    |
80 | 79 | 3adantl3 1146 |
. . . . . . . . 9
  
   
     
                                    |
81 | 80 | imp 429 |
. . . . . . . 8
   

   
                                        |
82 | 81 | adantlrr 720 |
. . . . . . 7
   
 
 
        
                                    |
83 | 82 | fveq2d 5795 |
. . . . . 6
   
 
 
        
                                            |
84 | 1, 2, 24, 31 | rngohommul 28916 |
. . . . . . . . . . . 12
  
                                                                    |
85 | 84 | ex 434 |
. . . . . . . . . . 11
 

 
                                                                  |
86 | 85 | 3expa 1188 |
. . . . . . . . . 10
  

            
        
                                               |
87 | 86 | 3adantl1 1144 |
. . . . . . . . 9
  
   
                                                                  |
88 | 87 | imp 429 |
. . . . . . . 8
   

                                                                    |
89 | 88 | adantlrl 719 |
. . . . . . 7
   
 
 
            
                                                        |
90 | 53, 89 | syldan 470 |
. . . . . 6
   
 
 
        
                                                    |
91 | 83, 90 | eqtrd 2492 |
. . . . 5
   
 
 
        
                                                |
92 | 9, 17, 10 | rngocl 24006 |
. . . . . . . . 9
     
                   |
93 | 92 | 3expb 1189 |
. . . . . . . 8
      
                    |
94 | 93 | 3ad2antl1 1150 |
. . . . . . 7
  
     
                    |
95 | 94 | adantlr 714 |
. . . . . 6
   
 
 
        
                    |
96 | | fvco3 5869 |
. . . . . . 7
             
                                             |
97 | 14, 96 | sylan 471 |
. . . . . 6
   
 
 
                
 
                              |
98 | 95, 97 | syldan 470 |
. . . . 5
   
 
 
        
                                      |
99 | | oveq12 6201 |
. . . . . 6
                                                                             |
100 | 73, 99 | syl 16 |
. . . . 5
   
 
 
        
                                                    |
101 | 91, 98, 100 | 3eqtr4d 2502 |
. . . 4
   
 
 
        
                                          |
102 | 76, 101 | jca 532 |
. . 3
   
 
 
        
                                                                              |
103 | 102 | ralrimivva 2906 |
. 2
  
 
                                                                                          |
104 | 9, 17, 10, 18, 3, 31, 4, 32 | isrngohom 28911 |
. . . 4
       
 
        
         GId       GId                                                                                            |
105 | 104 | 3adant2 1007 |
. . 3
 
                    
 
   GId       GId                                                                                            |
106 | 105 | adantr 465 |
. 2
  
 
          
 
        
         GId       GId                                                                                            |
107 | 16, 38, 103, 106 | mpbir3and 1171 |
1
  
 
      
     |