Proof of Theorem sincos6thpi
Step | Hyp | Ref
| Expression |
1 | | 2cn 8979 |
. . . . 5
 |
2 | 1 | a1i 9 |
. . . 4
  |
3 | | pire 13874 |
. . . . . . . 8
 |
4 | | 6re 8989 |
. . . . . . . 8
 |
5 | | 6pos 9009 |
. . . . . . . . 9
 |
6 | 4, 5 | gt0ap0ii 8575 |
. . . . . . . 8
#  |
7 | 3, 4, 6 | redivclapi 8725 |
. . . . . . 7
   |
8 | 7 | recni 7960 |
. . . . . 6
   |
9 | | sincl 11698 |
. . . . . 6
           |
10 | 8, 9 | ax-mp 5 |
. . . . 5
       |
11 | 10 | a1i 9 |
. . . 4
   
    |
12 | | 2ap0 9001 |
. . . . 5
#  |
13 | 12 | a1i 9 |
. . . 4
#
  |
14 | | recoscl 11713 |
. . . . . . . . . . . 12
           |
15 | 7, 14 | ax-mp 5 |
. . . . . . . . . . 11
       |
16 | 15 | recni 7960 |
. . . . . . . . . 10
       |
17 | 1, 10, 16 | mulassi 7957 |
. . . . . . . . 9
            
              
     |
18 | | sin2t 11741 |
. . . . . . . . . 10
                
            |
19 | 8, 18 | ax-mp 5 |
. . . . . . . . 9
    
              
     |
20 | 17, 19 | eqtr4i 2201 |
. . . . . . . 8
            
            |
21 | | 3cn 8983 |
. . . . . . . . . . . 12
 |
22 | | 3ap0 9004 |
. . . . . . . . . . . 12
#  |
23 | 1, 21, 22 | divclapi 8700 |
. . . . . . . . . . 11
   |
24 | 21, 22 | recclapi 8688 |
. . . . . . . . . . 11
   |
25 | | df-3 8968 |
. . . . . . . . . . . . 13
   |
26 | 25 | oveq1i 5879 |
. . . . . . . . . . . 12
       |
27 | 21, 22 | dividapi 8691 |
. . . . . . . . . . . 12
   |
28 | | ax-1cn 7895 |
. . . . . . . . . . . . 13
 |
29 | 1, 28, 21, 22 | divdirapi 8715 |
. . . . . . . . . . . 12
           |
30 | 26, 27, 29 | 3eqtr3ri 2207 |
. . . . . . . . . . 11
       |
31 | | sincosq1eq 13927 |
. . . . . . . . . . 11
                             
     |
32 | 23, 24, 30, 31 | mp3an 1337 |
. . . . . . . . . 10
      
              |
33 | | picn 13875 |
. . . . . . . . . . . . 13
 |
34 | 1, 21, 33, 1, 22, 12 | divmuldivapi 8718 |
. . . . . . . . . . . 12
             |
35 | | 3t2e6 9064 |
. . . . . . . . . . . . 13
   |
36 | 35 | oveq2i 5880 |
. . . . . . . . . . . 12
           |
37 | | 6cn 8990 |
. . . . . . . . . . . . 13
 |
38 | 1, 33, 37, 6 | divassapi 8714 |
. . . . . . . . . . . 12
     
   |
39 | 34, 36, 38 | 3eqtri 2202 |
. . . . . . . . . . 11
       
   |
40 | 39 | fveq2i 5514 |
. . . . . . . . . 10
      
            |
41 | 32, 40 | eqtr3i 2200 |
. . . . . . . . 9
      
            |
42 | 28, 21, 33, 1, 22, 12 | divmuldivapi 8718 |
. . . . . . . . . . 11
             |
43 | 33 | mulid2i 7951 |
. . . . . . . . . . . 12
   |
44 | 43, 35 | oveq12i 5881 |
. . . . . . . . . . 11
         |
45 | 42, 44 | eqtri 2198 |
. . . . . . . . . 10
         |
46 | 45 | fveq2i 5514 |
. . . . . . . . 9
      
          |
47 | 41, 46 | eqtr3i 2200 |
. . . . . . . 8
    
          |
48 | 20, 47 | eqtri 2198 |
. . . . . . 7
            
          |
49 | 16 | mulid2i 7951 |
. . . . . . 7
    
          |
50 | 48, 49 | eqtr4i 2201 |
. . . . . 6
            
            |
51 | 1, 10 | mulcli 7953 |
. . . . . . 7
    
    |
52 | | pipos 13876 |
. . . . . . . . . . . . 13
 |
53 | 3, 4, 52, 5 | divgt0ii 8865 |
. . . . . . . . . . . 12
   |
54 | | 2lt6 9090 |
. . . . . . . . . . . . 13
 |
55 | | 2re 8978 |
. . . . . . . . . . . . . . 15
 |
56 | | 2pos 8999 |
. . . . . . . . . . . . . . 15
 |
57 | 55, 56 | pm3.2i 272 |
. . . . . . . . . . . . . 14
   |
58 | 4, 5 | pm3.2i 272 |
. . . . . . . . . . . . . 14
   |
59 | 3, 52 | pm3.2i 272 |
. . . . . . . . . . . . . 14
   |
60 | | ltdiv2 8833 |
. . . . . . . . . . . . . 14
            
    |
61 | 57, 58, 59, 60 | mp3an 1337 |
. . . . . . . . . . . . 13

  
   |
62 | 54, 61 | mpbi 145 |
. . . . . . . . . . . 12
     |
63 | | 0re 7948 |
. . . . . . . . . . . . 13
 |
64 | | halfpire 13880 |
. . . . . . . . . . . . 13
   |
65 | | rexr 7993 |
. . . . . . . . . . . . . 14
   |
66 | | rexr 7993 |
. . . . . . . . . . . . . 14
   
   |
67 | | elioo2 9908 |
. . . . . . . . . . . . . 14
  
          
 
  
  
     |
68 | 65, 66, 67 | syl2an 289 |
. . . . . . . . . . . . 13
  
        
      
  
     |
69 | 63, 64, 68 | mp2an 426 |
. . . . . . . . . . . 12
        
 
  
  
    |
70 | 7, 53, 62, 69 | mpbir3an 1179 |
. . . . . . . . . . 11
     
   |
71 | | sincosq1sgn 13914 |
. . . . . . . . . . 11
         
         
     |
72 | 70, 71 | ax-mp 5 |
. . . . . . . . . 10
    
     
    |
73 | 72 | simpri 113 |
. . . . . . . . 9
   
   |
74 | 15, 73 | gt0ap0ii 8575 |
. . . . . . . 8
      #  |
75 | 16, 74 | pm3.2i 272 |
. . . . . . 7
             #   |
76 | | mulcanap2 8612 |
. . . . . . 7
                       #         
              
  
    
      |
77 | 51, 28, 75, 76 | mp3an 1337 |
. . . . . 6
             
          
    
     |
78 | 50, 77 | mpbi 145 |
. . . . 5
    
    |
79 | 78 | a1i 9 |
. . . 4
          |
80 | 2, 11, 13, 79 | mvllmulapd 8788 |
. . 3
   
      |
81 | 80 | mptru 1362 |
. 2
         |
82 | | 3re 8982 |
. . . . . . . 8
 |
83 | | 3pos 9002 |
. . . . . . . 8
 |
84 | 82, 83 | sqrtpclii 11123 |
. . . . . . 7
     |
85 | 84 | recni 7960 |
. . . . . 6
     |
86 | 85, 1, 12 | sqdivapi 10589 |
. . . . 5
                         |
87 | 63, 82, 83 | ltleii 8050 |
. . . . . . 7
 |
88 | 82 | sqsqrti 11117 |
. . . . . . 7
           |
89 | 87, 88 | ax-mp 5 |
. . . . . 6
         |
90 | | sq2 10601 |
. . . . . 6
     |
91 | 89, 90 | oveq12i 5881 |
. . . . 5
                 |
92 | 86, 91 | eqtri 2198 |
. . . 4
             |
93 | 92 | fveq2i 5514 |
. . 3
                     |
94 | 82 | sqrtge0i 11118 |
. . . . . 6
       |
95 | 87, 94 | ax-mp 5 |
. . . . 5
     |
96 | 84, 55 | divge0i 8857 |
. . . . 5
 
             |
97 | 95, 56, 96 | mp2an 426 |
. . . 4
       |
98 | 84, 55, 12 | redivclapi 8725 |
. . . . 5
       |
99 | 98 | sqrtsqi 11116 |
. . . 4
                             |
100 | 97, 99 | ax-mp 5 |
. . 3
                     |
101 | | 4cn 8986 |
. . . . . . . 8
 |
102 | | 4ap0 9007 |
. . . . . . . 8
#  |
103 | 101, 102 | dividapi 8691 |
. . . . . . 7
   |
104 | 103 | oveq1i 5879 |
. . . . . 6
           |
105 | 101, 102 | pm3.2i 272 |
. . . . . . . 8
 #   |
106 | | divsubdirap 8654 |
. . . . . . . 8
 
 #               |
107 | 101, 28, 105, 106 | mp3an 1337 |
. . . . . . 7
           |
108 | | 4m1e3 9029 |
. . . . . . . 8
   |
109 | 108 | oveq1i 5879 |
. . . . . . 7
       |
110 | 107, 109 | eqtr3i 2200 |
. . . . . 6
         |
111 | 101, 102 | recclapi 8688 |
. . . . . . 7
   |
112 | 16 | sqcli 10586 |
. . . . . . 7
           |
113 | 81 | oveq1i 5879 |
. . . . . . . . . 10
                 |
114 | | 2z 9270 |
. . . . . . . . . . 11
 |
115 | | exprecap 10547 |
. . . . . . . . . . 11
  #
               |
116 | 1, 12, 114, 115 | mp3an 1337 |
. . . . . . . . . 10
             |
117 | 90 | oveq2i 5880 |
. . . . . . . . . 10
         |
118 | 113, 116,
117 | 3eqtri 2202 |
. . . . . . . . 9
             |
119 | 118 | oveq1i 5879 |
. . . . . . . 8
     
                               |
120 | | sincossq 11740 |
. . . . . . . . 9
        
                  |
121 | 8, 120 | ax-mp 5 |
. . . . . . . 8
     
                 |
122 | 119, 121 | eqtr3i 2200 |
. . . . . . 7
               |
123 | 28, 111, 112, 122 | subaddrii 8236 |
. . . . . 6
        
      |
124 | 104, 110,
123 | 3eqtr3ri 2207 |
. . . . 5
             |
125 | 124 | fveq2i 5514 |
. . . 4
       
             |
126 | 63, 15, 73 | ltleii 8050 |
. . . . 5
   
   |
127 | 15 | sqrtsqi 11116 |
. . . . 5
    
 
       
              |
128 | 126, 127 | ax-mp 5 |
. . . 4
       
             |
129 | 125, 128 | eqtr3i 2200 |
. . 3
             |
130 | 93, 100, 129 | 3eqtr3ri 2207 |
. 2
             |
131 | 81, 130 | pm3.2i 272 |
1
                       |