Proof of Theorem pythagtriplem12
Step | Hyp | Ref
| Expression |
1 | | pythagtriplem11.1 |
. . 3
                 |
2 | 1 | oveq1i 5879 |
. 2
                
        |
3 | | simp3 999 |
. . . . . . . . . 10
 
   |
4 | | simp2 998 |
. . . . . . . . . 10
 
   |
5 | 3, 4 | nnaddcld 8956 |
. . . . . . . . 9
 
 
   |
6 | 5 | nnrpd 9681 |
. . . . . . . 8
 
 
   |
7 | 6 | rpsqrtcld 11151 |
. . . . . . 7
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         |
8 | 7 | rpcnd 9685 |
. . . . . 6
 
         |
9 | 8 | 3ad2ant1 1018 |
. . . . 5
  
                            |
10 | 3 | nnred 8921 |
. . . . . . . . . . 11
 
   |
11 | 10 | adantr 276 |
. . . . . . . . . 10
  
                  |
12 | 4 | nnred 8921 |
. . . . . . . . . . 11
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   |
13 | 12 | adantr 276 |
. . . . . . . . . 10
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                  |
14 | 11, 13 | resubcld 8328 |
. . . . . . . . 9
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                    |
15 | | pythagtriplem10 12252 |
. . . . . . . . 9
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                
   |
16 | 14, 15 | elrpd 9680 |
. . . . . . . 8
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                    |
17 | 16 | rpsqrtcld 11151 |
. . . . . . 7
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                   
    |
18 | 17 | 3adant3 1017 |
. . . . . 6
  
                            |
19 | 18 | rpcnd 9685 |
. . . . 5
  
                            |
20 | 9, 19 | addcld 7967 |
. . . 4
  
                        
           |
21 | | 2cn 8979 |
. . . . . 6
 |
22 | | 2ap0 9001 |
. . . . . 6
#  |
23 | | sqdivap 10570 |
. . . . . 6
      
         # 
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("    
                   
                   |
24 | 21, 22, 23 | mp3an23 1329 |
. . . . 5
     
        
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("    
                   
                   |
25 | 21 | sqvali 10585 |
. . . . . 6
       |
26 | 25 | oveq2i 5880 |
. . . . 5
      
                       
                |
27 | 24, 26 | eqtrdi 2226 |
. . . 4
     
        
      
                   
                 |
28 | 20, 27 | syl 14 |
. . 3
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                                                    
           |
29 | | binom2 10617 |
. . . . . . 7
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                                 
          
                        |
30 | 9, 19, 29 | syl2anc 411 |
. . . . . 6
  
                         
                                             
        |
31 | | nnre 8915 |
. . . . . . . . . . . 12
   |
32 | | nnre 8915 |
. . . . . . . . . . . 12
   |
33 | | readdcl 7928 |
. . . . . . . . . . . 12
 
     |
34 | 31, 32, 33 | syl2anr 290 |
. . . . . . . . . . 11
 
     |
35 | 34 | 3adant1 1015 |
. . . . . . . . . 10
 
 
   |
36 | 35 | 3ad2ant1 1018 |
. . . . . . . . 9
  
                    
   |
37 | 31 | 3ad2ant3 1020 |
. . . . . . . . . . . 12
 
   |
38 | 32 | 3ad2ant2 1019 |
. . . . . . . . . . . 12
 
   |
39 | | nngt0 8933 |
. . . . . . . . . . . . 13
   |
40 | 39 | 3ad2ant3 1020 |
. . . . . . . . . . . 12
 
   |
41 | | nngt0 8933 |
. . . . . . . . . . . . 13
   |
42 | 41 | 3ad2ant2 1019 |
. . . . . . . . . . . 12
 
   |
43 | 37, 38, 40, 42 | addgt0d 8468 |
. . . . . . . . . . 11
 
     |
44 | 43 | 3ad2ant1 1018 |
. . . . . . . . . 10
  
                        |
45 | | 0re 7948 |
. . . . . . . . . . 11
 |
46 | | ltle 8035 |
. . . . . . . . . . 11
  
    
     |
47 | 45, 46 | mpan 424 |
. . . . . . . . . 10
   
       |
48 | 36, 44, 47 | sylc 62 |
. . . . . . . . 9
  
                        |
49 | | resqrtth 11024 |
. . . . . . . . 9
           
         |
50 | 36, 48, 49 | syl2anc 411 |
. . . . . . . 8
  
                        
         |
51 | 50 | oveq1d 5884 |
. . . . . . 7
  
                         
          
                   
             |
52 | | resubcl 8211 |
. . . . . . . . . . 11
 
     |
53 | 31, 32, 52 | syl2anr 290 |
. . . . . . . . . 10
 
     |
54 | 53 | 3adant1 1015 |
. . . . . . . . 9
 
 
   |
55 | 54 | 3ad2ant1 1018 |
. . . . . . . 8
  
                    
   |
56 | 15 | 3adant3 1017 |
. . . . . . . . 9
  
                        |
57 | | ltle 8035 |
. . . . . . . . . 10
  
    
     |
58 | 45, 57 | mpan 424 |
. . . . . . . . 9
   
       |
59 | 55, 56, 58 | sylc 62 |
. . . . . . . 8
  
                        |
60 | | resqrtth 11024 |
. . . . . . . 8
           
         |
61 | 55, 59, 60 | syl2anc 411 |
. . . . . . 7
  
                        
         |
62 | 51, 61 | oveq12d 5887 |
. . . . . 6
  
                                                     
                                |
63 | | nncn 8916 |
. . . . . . . . . . . 12
   |
64 | 63 | 3ad2ant3 1020 |
. . . . . . . . . . 11
 
   |
65 | 64 | 3ad2ant1 1018 |
. . . . . . . . . 10
  
                      |
66 | | nncn 8916 |
. . . . . . . . . . . 12
   |
67 | 66 | 3ad2ant2 1019 |
. . . . . . . . . . 11
 
   |
68 | 67 | 3ad2ant1 1018 |
. . . . . . . . . 10
  
                      |
69 | 65, 68, 65 | ppncand 8298 |
. . . . . . . . 9
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                              |
70 | 65 | 2timesd 9150 |
. . . . . . . . 9
  
                          |
71 | 69, 70 | eqtr4d 2213 |
. . . . . . . 8
  
                              |
72 | | oveq1 5876 |
. . . . . . . . . . . . 13
                                           |
73 | 72 | 3ad2ant2 1019 |
. . . . . . . . . . . 12
  
                                                |
74 | | nncn 8916 |
. . . . . . . . . . . . . . . 16
   |
75 | 74 | 3ad2ant1 1018 |
. . . . . . . . . . . . . . 15
 
   |
76 | 75 | 3ad2ant1 1018 |
. . . . . . . . . . . . . 14
  
                      |
77 | 76 | sqcld 10637 |
. . . . . . . . . . . . 13
  
                          |
78 | 68 | sqcld 10637 |
. . . . . . . . . . . . 13
  
                          |
79 | 77, 78 | pncand 8259 |
. . . . . . . . . . . 12
  
                                          |
80 | | subsq 10612 |
. . . . . . . . . . . . 13
 
                   |
81 | 65, 68, 80 | syl2anc 411 |
. . . . . . . . . . . 12
  
                                      |
82 | 73, 79, 81 | 3eqtr3rd 2219 |
. . . . . . . . . . 11
  
                                |
83 | 82 | fveq2d 5515 |
. . . . . . . . . 10
  
                                        |
84 | 36, 48, 55, 59 | sqrtmuld 11162 |
. . . . . . . . . 10
  
                                  
           |
85 | | nnre 8915 |
. . . . . . . . . . . . 13
   |
86 | 85 | 3ad2ant1 1018 |
. . . . . . . . . . . 12
 
   |
87 | 86 | 3ad2ant1 1018 |
. . . . . . . . . . 11
  
                      |
88 | | nnnn0 9172 |
. . . . . . . . . . . . . 14
   |
89 | 88 | nn0ge0d 9221 |
. . . . . . . . . . . . 13
   |
90 | 89 | 3ad2ant1 1018 |
. . . . . . . . . . . 12
 
   |
91 | 90 | 3ad2ant1 1018 |
. . . . . . . . . . 11
  
                      |
92 | 87, 91 | sqrtsqd 11158 |
. . . . . . . . . 10
  
                              |
93 | 83, 84, 92 | 3eqtr3d 2218 |
. . . . . . . . 9
  
                        
           |
94 | 93 | oveq2d 5885 |
. . . . . . . 8
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                               
        |
95 | 71, 94 | oveq12d 5887 |
. . . . . . 7
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                                                    |
96 | | addcl 7927 |
. . . . . . . . . . 11
 
     |
97 | 63, 66, 96 | syl2anr 290 |
. . . . . . . . . 10
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     |
98 | 97 | 3adant1 1015 |
. . . . . . . . 9
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 
   |
99 | 98 | 3ad2ant1 1018 |
. . . . . . . 8
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                    
   |
100 | 9, 19 | mulcld 7968 |
. . . . . . . . 9
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                        
           |
101 | | mulcl 7929 |
. . . . . . . . 9
      
                     
      |
102 | 21, 100, 101 | sylancr 414 |
. . . . . . . 8
  
                               
      |
103 | | subcl 8146 |
. . . . . . . . . . 11
 
     |
104 | 63, 66, 103 | syl2anr 290 |
. . . . . . . . . 10
 
     |
105 | 104 | 3adant1 1015 |
. . . . . . . . 9
 
 
   |
106 | 105 | 3ad2ant1 1018 |
. . . . . . . 8
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                    
   |
107 | 99, 102, 106 | add32d 8115 |
. . . . . . 7
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                                   
                          
       |
108 | | adddi 7934 |
. . . . . . . 8
 
             |
109 | 21, 65, 76, 108 | mp3an2i 1342 |
. . . . . . 7
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                                |
110 | 95, 107, 109 | 3eqtr4d 2220 |
. . . . . 6
  
                                   
         
    |
111 | 30, 62, 110 | 3eqtrd 2214 |
. . . . 5
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                         
                  |
112 | 111 | oveq1d 5884 |
. . . 4
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                                                    |
113 | | addcl 7927 |
. . . . . . . . 9
 
     |
114 | 63, 74, 113 | syl2anr 290 |
. . . . . . . 8
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     |
115 | 114 | 3adant2 1016 |
. . . . . . 7
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 
   |
116 | 115 | 3ad2ant1 1018 |
. . . . . 6
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                    
   |
117 | | mulcl 7929 |
. . . . . 6
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        |
118 | 21, 116, 117 | sylancr 414 |
. . . . 5
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2"                     |
119 | 21 | a1i 9 |
. . . . 5
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                      |
120 | 22 | a1i 9 |
. . . . 5
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                    #   |
121 | 118, 119,
119, 120, 120 | divdivap1d 8768 |
. . . 4
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2"                                 |
122 | 112, 121 | eqtr4d 2213 |
. . 3
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                                                    |
123 | 116, 119,
120 | divcanap3d 8741 |
. . . 4
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                      
       |
124 | 123 | oveq1d 5884 |
. . 3
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                                  |
125 | 28, 122, 124 | 3eqtrd 2214 |
. 2
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                                              |
126 | 2, 125 | eqtrid 2222 |
1
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                              |