Proof of Theorem 2sqpwodd
Step | Hyp | Ref
| Expression |
1 | | oddpwdc.j |
. . . . . . . . 9
   |
2 | | oddpwdc.f |
. . . . . . . . 9
 
        |
3 | 1, 2 | oddpwdc 12157 |
. . . . . . . 8
       |
4 | | f1ocnv 5470 |
. . . . . . . 8
      
         |
5 | | f1of 5457 |
. . . . . . . 8
                 |
6 | 3, 4, 5 | mp2b 8 |
. . . . . . 7
        |
7 | 6 | ffvelcdmi 5646 |
. . . . . 6
          |
8 | | xp2nd 6161 |
. . . . . 6
       
           |
9 | 7, 8 | syl 14 |
. . . . 5
            |
10 | 9 | nn0zd 9362 |
. . . 4
            |
11 | | 2nn 9069 |
. . . . . 6
 |
12 | 11 | a1i 9 |
. . . . 5
   |
13 | 12 | nnzd 9363 |
. . . 4
   |
14 | 10, 13 | zmulcld 9370 |
. . 3
              |
15 | | dvdsmul2 11805 |
. . . 4
          
              |
16 | 10, 13, 15 | syl2anc 411 |
. . 3
              |
17 | | oddp1even 11864 |
. . . . 5
            
                           |
18 | 17 | biimprd 158 |
. . . 4
            
            
              |
19 | 18 | con2d 624 |
. . 3
            
                           |
20 | 14, 16, 19 | sylc 62 |
. 2
                |
21 | | xp1st 6160 |
. . . . . . . . . . 11
       
           |
22 | 7, 21 | syl 14 |
. . . . . . . . . 10
            |
23 | | breq2 4004 |
. . . . . . . . . . . . 13
         

            |
24 | 23 | notbid 667 |
. . . . . . . . . . . 12
         

            |
25 | 24, 1 | elrab2 2896 |
. . . . . . . . . . 11
         
                      |
26 | 25 | simplbi 274 |
. . . . . . . . . 10
                     |
27 | 22, 26 | syl 14 |
. . . . . . . . 9
            |
28 | 27 | nnsqcld 10660 |
. . . . . . . 8
                |
29 | 25 | simprbi 275 |
. . . . . . . . . . 11
         
           |
30 | 22, 29 | syl 14 |
. . . . . . . . . 10
            |
31 | | 2prm 12110 |
. . . . . . . . . . 11
 |
32 | 27 | nnzd 9363 |
. . . . . . . . . . 11
            |
33 | | euclemma 12129 |
. . . . . . . . . . . 12
                     
                   
                       |
34 | | oridm 757 |
. . . . . . . . . . . 12
                                |
35 | 33, 34 | bitrdi 196 |
. . . . . . . . . . 11
                     
                   
            |
36 | 31, 32, 32, 35 | mp3an2i 1342 |
. . . . . . . . . 10
 
                   
            |
37 | 30, 36 | mtbird 673 |
. . . . . . . . 9
                       |
38 | 27 | nncnd 8922 |
. . . . . . . . . . 11
            |
39 | 38 | sqvald 10636 |
. . . . . . . . . 10
                                    |
40 | 39 | breq2d 4012 |
. . . . . . . . 9
 
            
                       |
41 | 37, 40 | mtbird 673 |
. . . . . . . 8
                |
42 | | breq2 4004 |
. . . . . . . . . 10
              
                |
43 | 42 | notbid 667 |
. . . . . . . . 9
              
                |
44 | 43, 1 | elrab2 2896 |
. . . . . . . 8
             
                              |
45 | 28, 41, 44 | sylanbrc 417 |
. . . . . . 7
                |
46 | 12 | nnnn0d 9218 |
. . . . . . . . 9
   |
47 | 9, 46 | nn0mulcld 9223 |
. . . . . . . 8
              |
48 | | peano2nn0 9205 |
. . . . . . . 8
           
               |
49 | 47, 48 | syl 14 |
. . . . . . 7
                |
50 | | opelxp 4653 |
. . . . . . 7
                                                              |
51 | 45, 49, 50 | sylanbrc 417 |
. . . . . 6
                                  |
52 | 12 | nncnd 8922 |
. . . . . . . . . . 11
   |
53 | 52, 47 | expp1d 10640 |
. . . . . . . . . 10
                                     |
54 | 52, 47 | expcld 10639 |
. . . . . . . . . . 11
                  |
55 | 54, 52 | mulcomd 7969 |
. . . . . . . . . 10
                                     |
56 | 52, 46, 9 | expmuld 10642 |
. . . . . . . . . . 11
                                   |
57 | 56 | oveq2d 5885 |
. . . . . . . . . 10
                                       |
58 | 53, 55, 57 | 3eqtrd 2214 |
. . . . . . . . 9
                                       |
59 | 58 | oveq1d 5884 |
. . . . . . . 8
                                                                     |
60 | 12, 49 | nnexpcld 10661 |
. . . . . . . . . 10
                    |
61 | 60, 28 | nnmulcld 8957 |
. . . . . . . . 9
                                   |
62 | | oveq2 5877 |
. . . . . . . . . 10
                                         |
63 | | oveq2 5877 |
. . . . . . . . . . 11
                                     |
64 | 63 | oveq1d 5884 |
. . . . . . . . . 10
                                                                   |
65 | 62, 64, 2 | ovmpog 6003 |
. . . . . . . . 9
                           
                                                                                                 |
66 | 45, 49, 61, 65 | syl3anc 1238 |
. . . . . . . 8
                                                                 |
67 | | f1ocnvfv2 5773 |
. . . . . . . . . . . . . . . 16
       
            |
68 | 3, 67 | mpan 424 |
. . . . . . . . . . . . . . 15
            |
69 | | 1st2nd2 6170 |
. . . . . . . . . . . . . . . . 17
       
                            |
70 | 7, 69 | syl 14 |
. . . . . . . . . . . . . . . 16
                             |
71 | 70 | fveq2d 5515 |
. . . . . . . . . . . . . . 15
                                     |
72 | 68, 71 | eqtr3d 2212 |
. . . . . . . . . . . . . 14
                            |
73 | | df-ov 5872 |
. . . . . . . . . . . . . 14
                                                |
74 | 72, 73 | eqtr4di 2228 |
. . . . . . . . . . . . 13
                         |
75 | 12, 9 | nnexpcld 10661 |
. . . . . . . . . . . . . . 15
                |
76 | 75, 27 | nnmulcld 8957 |
. . . . . . . . . . . . . 14
                           |
77 | | oveq2 5877 |
. . . . . . . . . . . . . . 15
         
                       |
78 | | oveq2 5877 |
. . . . . . . . . . . . . . . 16
         
                   |
79 | 78 | oveq1d 5884 |
. . . . . . . . . . . . . . 15
         
                                         |
80 | 77, 79, 2 | ovmpog 6003 |
. . . . . . . . . . . . . 14
                                                                                             |
81 | 22, 9, 76, 80 | syl3anc 1238 |
. . . . . . . . . . . . 13
                                                 |
82 | 74, 81 | eqtrd 2210 |
. . . . . . . . . . . 12
                           |
83 | 82 | oveq1d 5884 |
. . . . . . . . . . 11
                                   |
84 | 75 | nncnd 8922 |
. . . . . . . . . . . 12
                |
85 | 84, 38 | sqmuld 10651 |
. . . . . . . . . . 11
                                                               |
86 | 83, 85 | eqtrd 2210 |
. . . . . . . . . 10
                                       |
87 | 86 | oveq2d 5885 |
. . . . . . . . 9
                                           |
88 | 56, 54 | eqeltrrd 2255 |
. . . . . . . . . 10
                    |
89 | 28 | nncnd 8922 |
. . . . . . . . . 10
                |
90 | 52, 88, 89 | mulassd 7971 |
. . . . . . . . 9
                                                                       |
91 | 87, 90 | eqtr4d 2213 |
. . . . . . . 8
                                           |
92 | 59, 66, 91 | 3eqtr4rd 2221 |
. . . . . . 7
                                       |
93 | | df-ov 5872 |
. . . . . . 7
                                                                |
94 | 92, 93 | eqtr2di 2227 |
. . . . . 6
                                          |
95 | | f1ocnvfv 5774 |
. . . . . . 7
                                                                                                                           |
96 | 3, 95 | mpan 424 |
. . . . . 6
                                                                                                                   |
97 | 51, 94, 96 | sylc 62 |
. . . . 5
                                           |
98 | 97 | fveq2d 5515 |
. . . 4
                                                   |
99 | | op2ndg 6146 |
. . . . 5
                                                                             |
100 | 45, 49, 99 | syl2anc 411 |
. . . 4
                                                 |
101 | 98, 100 | eqtrd 2210 |
. . 3
                               |
102 | 101 | breq2d 4012 |
. 2
 
              
                |
103 | 20, 102 | mtbird 673 |
1
                  |