Proof of Theorem pythagtriplem16
Step | Hyp | Ref
| Expression |
1 | | pythagtriplem15.1 |
. . . . 5
                 |
2 | | pythagtriplem15.2 |
. . . . 5
                 |
3 | 1, 2 | oveq12i 5881 |
. . . 4
              
         
            |
4 | | simp13 1029 |
. . . . . . . . . . 11
  
                      |
5 | | simp12 1028 |
. . . . . . . . . . 11
  
                      |
6 | 4, 5 | nnaddcld 8956 |
. . . . . . . . . 10
  
                    
   |
7 | 6 | nnrpd 9681 |
. . . . . . . . 9
  
                    
   |
8 | 7 | rpsqrtcld 11151 |
. . . . . . . 8
  
                            |
9 | 8 | rpcnd 9685 |
. . . . . . 7
  
                            |
10 | 4 | nnzd 9363 |
. . . . . . . . . . . 12
  
                      |
11 | 5 | nnzd 9363 |
. . . . . . . . . . . 12
  
                      |
12 | 10, 11 | zsubcld 9369 |
. . . . . . . . . . 11
  
                    
   |
13 | 12 | zred 9364 |
. . . . . . . . . 10
  
                    
   |
14 | | pythagtriplem10 12252 |
. . . . . . . . . . 11
  
                
   |
15 | 14 | 3adant3 1017 |
. . . . . . . . . 10
  
                        |
16 | 13, 15 | elrpd 9680 |
. . . . . . . . 9
  
                    
   |
17 | 16 | rpsqrtcld 11151 |
. . . . . . . 8
  
                            |
18 | 17 | rpcnd 9685 |
. . . . . . 7
  
                            |
19 | 9, 18 | addcld 7967 |
. . . . . 6
  
                        
           |
20 | | 2cnd 8981 |
. . . . . 6
  
                      |
21 | 9, 18 | subcld 8258 |
. . . . . 6
  
                        
           |
22 | | 2ap0 9001 |
. . . . . . 7
#  |
23 | 22 | a1i 9 |
. . . . . 6
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                    #   |
24 | 19, 20, 21, 20, 23, 23 | divmuldivapd 8778 |
. . . . 5
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                                                
                               
         |
25 | 19, 21 | mulcld 7968 |
. . . . . 6
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                         
             
            |
26 | 25, 20, 20, 23, 23 | divdivap1d 8768 |
. . . . 5
  
                                 
       
                                      
         |
27 | | nnre 8915 |
. . . . . . . . . . . . 13
   |
28 | | nnre 8915 |
. . . . . . . . . . . . 13
   |
29 | | readdcl 7928 |
. . . . . . . . . . . . 13
 
     |
30 | 27, 28, 29 | syl2anr 290 |
. . . . . . . . . . . 12
 
     |
31 | 30 | 3adant1 1015 |
. . . . . . . . . . 11
 
 
   |
32 | 31 | 3ad2ant1 1018 |
. . . . . . . . . 10
  
                    
   |
33 | 27 | adantl 277 |
. . . . . . . . . . . . . 14
 
   |
34 | 28 | adantr 276 |
. . . . . . . . . . . . . 14
 
   |
35 | | nngt0 8933 |
. . . . . . . . . . . . . . 15
   |
36 | 35 | adantl 277 |
. . . . . . . . . . . . . 14
 
   |
37 | | nngt0 8933 |
. . . . . . . . . . . . . . 15
   |
38 | 37 | adantr 276 |
. . . . . . . . . . . . . 14
 
   |
39 | 33, 34, 36, 38 | addgt0d 8468 |
. . . . . . . . . . . . 13
 
     |
40 | | 0re 7948 |
. . . . . . . . . . . . . 14
 |
41 | | ltle 8035 |
. . . . . . . . . . . . . 14
  
    
     |
42 | 40, 41 | mpan 424 |
. . . . . . . . . . . . 13
   
       |
43 | 30, 39, 42 | sylc 62 |
. . . . . . . . . . . 12
 

    |
44 | 43 | 3adant1 1015 |
. . . . . . . . . . 11
 
     |
45 | 44 | 3ad2ant1 1018 |
. . . . . . . . . 10
  
                        |
46 | | resqrtth 11024 |
. . . . . . . . . 10
           
         |
47 | 32, 45, 46 | syl2anc 411 |
. . . . . . . . 9
  
                        
         |
48 | | resubcl 8211 |
. . . . . . . . . . . . 13
 
     |
49 | 27, 28, 48 | syl2anr 290 |
. . . . . . . . . . . 12
 
     |
50 | 49 | 3adant1 1015 |
. . . . . . . . . . 11
 
 
   |
51 | 50 | 3ad2ant1 1018 |
. . . . . . . . . 10
  
                    
   |
52 | | ltle 8035 |
. . . . . . . . . . . 12
  
    
     |
53 | 40, 52 | mpan 424 |
. . . . . . . . . . 11
   
       |
54 | 51, 15, 53 | sylc 62 |
. . . . . . . . . 10
  
                        |
55 | | resqrtth 11024 |
. . . . . . . . . 10
           
         |
56 | 51, 54, 55 | syl2anc 411 |
. . . . . . . . 9
  
                        
         |
57 | 47, 56 | oveq12d 5887 |
. . . . . . . 8
  
                         
                        |
58 | 57 | oveq1d 5884 |
. . . . . . 7
  
                                                
     |
59 | | subsq 10612 |
. . . . . . . . 9
     
                                          
       
            |
60 | 9, 18, 59 | syl2anc 411 |
. . . . . . . 8
  
                         
                     
             
            |
61 | 60 | oveq1d 5884 |
. . . . . . 7
  
                                                        
       
             |
62 | | nncn 8916 |
. . . . . . . . . . . 12
   |
63 | | nncn 8916 |
. . . . . . . . . . . 12
   |
64 | | pnncan 8188 |
. . . . . . . . . . . . . 14
 
           |
65 | 64 | 3anidm23 1297 |
. . . . . . . . . . . . 13
 
    
      |
66 | | 2times 9036 |
. . . . . . . . . . . . . 14
       |
67 | 66 | adantl 277 |
. . . . . . . . . . . . 13
 
       |
68 | 65, 67 | eqtr4d 2213 |
. . . . . . . . . . . 12
 
    
      |
69 | 62, 63, 68 | syl2anr 290 |
. . . . . . . . . . 11
 
    
      |
70 | 69 | 3adant1 1015 |
. . . . . . . . . 10
 
           |
71 | 70 | 3ad2ant1 1018 |
. . . . . . . . 9
  
                              |
72 | 71 | oveq1d 5884 |
. . . . . . . 8
  
                        
         |
73 | 5 | nncnd 8922 |
. . . . . . . . 9
  
                      |
74 | 73, 20, 23 | divcanap3d 8741 |
. . . . . . . 8
  
                          |
75 | 72, 74 | eqtrd 2210 |
. . . . . . 7
  
                        
     |
76 | 58, 61, 75 | 3eqtr3d 2218 |
. . . . . 6
  
                                              
       |
77 | 76 | oveq1d 5884 |
. . . . 5
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                                 
       
                |
78 | 24, 26, 77 | 3eqtr2d 2216 |
. . . 4
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                                                
         |
79 | 3, 78 | eqtrid 2222 |
. . 3
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                    
     |
80 | 79 | oveq2d 5885 |
. 2
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                              |
81 | 73, 20, 23 | divcanap2d 8738 |
. 2
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                          |
82 | 80, 81 | eqtr2d 2211 |
1
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                          |