Proof of Theorem aannenlem1
Step | Hyp | Ref
| Expression |
1 | | breq2 4406 |
. . . . . . 7
  deg  deg     |
2 | | breq2 4406 |
. . . . . . . 8
      coeff          coeff         |
3 | 2 | ralbidv 2827 |
. . . . . . 7
  
    coeff           coeff         |
4 | 1, 3 | 3anbi23d 1342 |
. . . . . 6
    deg 
     coeff         deg       coeff          |
5 | 4 | rabbidv 3036 |
. . . . 5
  Poly 
  deg       coeff         Poly 
  deg 
     coeff          |
6 | 5 | rexeqdv 2994 |
. . . 4
  
 Poly 
  deg       coeff              Poly 
  deg 
     coeff               |
7 | 6 | rabbidv 3036 |
. . 3
  
 Poly    deg       coeff              
 Poly 
  deg       coeff               |
8 | | aannenlem.a |
. . 3
    Poly 
  deg 
     coeff               |
9 | | cnex 9620 |
. . . 4
 |
10 | 9 | rabex 4554 |
. . 3
 
 Poly    deg       coeff              |
11 | 7, 8, 10 | fvmpt 5948 |
. 2

       Poly 
  deg 
     coeff               |
12 | | iunrab 4325 |
. . 3
  Poly    deg 
     coeff                 Poly 
  deg 
     coeff              |
13 | | fzfi 12185 |
. . . . . . 7
      |
14 | | fzfi 12185 |
. . . . . . 7
     |
15 | | mapfi 7870 |
. . . . . . 7
                         |
16 | 13, 14, 15 | mp2an 678 |
. . . . . 6
     
      |
17 | 16 | a1i 11 |
. . . . 5

     
       |
18 | | ovex 6318 |
. . . . . 6
     
      |
19 | | neeq1 2686 |
. . . . . . . . . . 11
       |
20 | | fveq2 5865 |
. . . . . . . . . . . 12
 deg  deg    |
21 | 20 | breq1d 4412 |
. . . . . . . . . . 11
  deg  deg     |
22 | | fveq2 5865 |
. . . . . . . . . . . . . . 15
 coeff  coeff    |
23 | 22 | fveq1d 5867 |
. . . . . . . . . . . . . 14
  coeff      coeff       |
24 | 23 | fveq2d 5869 |
. . . . . . . . . . . . 13
     coeff          coeff        |
25 | 24 | breq1d 4412 |
. . . . . . . . . . . 12
      coeff          coeff         |
26 | 25 | ralbidv 2827 |
. . . . . . . . . . 11
  
    coeff           coeff         |
27 | 19, 21, 26 | 3anbi123d 1339 |
. . . . . . . . . 10
    deg 
     coeff         deg       coeff          |
28 | 27 | elrab 3196 |
. . . . . . . . 9
  Poly    deg       coeff       
 Poly    deg 
     coeff          |
29 | | simp3 1010 |
. . . . . . . . . 10
   deg 
     coeff      
     coeff        |
30 | 29 | anim2i 573 |
. . . . . . . . 9
  Poly 
  deg 
     coeff         Poly 
     coeff         |
31 | 28, 30 | sylbi 199 |
. . . . . . . 8
  Poly    deg       coeff         Poly 
     coeff         |
32 | | 0z 10948 |
. . . . . . . . . . . . . . 15
 |
33 | | eqid 2451 |
. . . . . . . . . . . . . . . 16
coeff  coeff   |
34 | 33 | coef2 23185 |
. . . . . . . . . . . . . . 15
  Poly 
 coeff        |
35 | 32, 34 | mpan2 677 |
. . . . . . . . . . . . . 14
 Poly  coeff        |
36 | 35 | ad2antrl 734 |
. . . . . . . . . . . . 13
   Poly  
    coeff     
 
coeff        |
37 | | ffn 5728 |
. . . . . . . . . . . . 13
 coeff      coeff    |
38 | 36, 37 | syl 17 |
. . . . . . . . . . . 12
   Poly  
    coeff     
 
coeff    |
39 | 35 | adantl 468 |
. . . . . . . . . . . . . . . . . . . 20
 
Poly   coeff        |
40 | 39 | ffvelrnda 6022 |
. . . . . . . . . . . . . . . . . . 19
   Poly     coeff       |
41 | 40 | zred 11040 |
. . . . . . . . . . . . . . . . . 18
   Poly     coeff       |
42 | | nn0re 10878 |
. . . . . . . . . . . . . . . . . . 19

  |
43 | 42 | ad2antrr 732 |
. . . . . . . . . . . . . . . . . 18
   Poly      |
44 | 41, 43 | absled 13492 |
. . . . . . . . . . . . . . . . 17
   Poly         coeff         coeff      coeff         |
45 | | nn0z 10960 |
. . . . . . . . . . . . . . . . . . . 20

  |
46 | 45 | ad2antrr 732 |
. . . . . . . . . . . . . . . . . . 19
   Poly      |
47 | 46 | znegcld 11042 |
. . . . . . . . . . . . . . . . . 18
   Poly       |
48 | | elfz 11790 |
. . . . . . . . . . . . . . . . . 18
   coeff         coeff         
   coeff      coeff         |
49 | 40, 47, 46, 48 | syl3anc 1268 |
. . . . . . . . . . . . . . . . 17
   Poly      coeff         
   coeff      coeff         |
50 | 44, 49 | bitr4d 260 |
. . . . . . . . . . . . . . . 16
   Poly         coeff       coeff             |
51 | 50 | biimpd 211 |
. . . . . . . . . . . . . . 15
   Poly         coeff       coeff             |
52 | 51 | ralimdva 2796 |
. . . . . . . . . . . . . 14
 
Poly    
    coeff        coeff             |
53 | 52 | impr 625 |
. . . . . . . . . . . . 13
   Poly  
    coeff     
 
  coeff            |
54 | | fnfvrnss 6051 |
. . . . . . . . . . . . 13
  coeff    coeff          
coeff 
       |
55 | 38, 53, 54 | syl2anc 667 |
. . . . . . . . . . . 12
   Poly  
    coeff     
 
coeff         |
56 | | df-f 5586 |
. . . . . . . . . . . 12
 coeff            coeff  coeff          |
57 | 38, 55, 56 | sylanbrc 670 |
. . . . . . . . . . 11
   Poly  
    coeff     
 
coeff             |
58 | | elfznn0 11887 |
. . . . . . . . . . . 12
       |
59 | 58 | ssriv 3436 |
. . . . . . . . . . 11
     |
60 | | fssres 5749 |
. . . . . . . . . . 11
  coeff               
 coeff                      |
61 | 57, 59, 60 | sylancl 668 |
. . . . . . . . . 10
   Poly  
    coeff     
 
 coeff                      |
62 | | ovex 6318 |
. . . . . . . . . . 11
      |
63 | | ovex 6318 |
. . . . . . . . . . 11
     |
64 | 62, 63 | elmap 7500 |
. . . . . . . . . 10
  coeff                   coeff                      |
65 | 61, 64 | sylibr 216 |
. . . . . . . . 9
   Poly  
    coeff     
 
 coeff                    |
66 | 65 | ex 436 |
. . . . . . . 8

  Poly 
     coeff        coeff                     |
67 | 31, 66 | syl5 33 |
. . . . . . 7

  Poly    deg       coeff         coeff                     |
68 | | simp2 1009 |
. . . . . . . . . 10
   deg 
     coeff      
deg 
  |
69 | 68 | anim2i 573 |
. . . . . . . . 9
  Poly 
  deg 
     coeff         Poly 
deg     |
70 | 28, 69 | sylbi 199 |
. . . . . . . 8
  Poly    deg       coeff         Poly 
deg     |
71 | | neeq1 2686 |
. . . . . . . . . . 11
       |
72 | | fveq2 5865 |
. . . . . . . . . . . 12
 deg  deg    |
73 | 72 | breq1d 4412 |
. . . . . . . . . . 11
  deg  deg     |
74 | | fveq2 5865 |
. . . . . . . . . . . . . . 15
 coeff  coeff    |
75 | 74 | fveq1d 5867 |
. . . . . . . . . . . . . 14
  coeff      coeff       |
76 | 75 | fveq2d 5869 |
. . . . . . . . . . . . 13
     coeff          coeff        |
77 | 76 | breq1d 4412 |
. . . . . . . . . . . 12
      coeff          coeff         |
78 | 77 | ralbidv 2827 |
. . . . . . . . . . 11
  
    coeff           coeff         |
79 | 71, 73, 78 | 3anbi123d 1339 |
. . . . . . . . . 10
    deg 
     coeff         deg       coeff          |
80 | 79 | elrab 3196 |
. . . . . . . . 9
  Poly    deg       coeff       
 Poly    deg 
     coeff          |
81 | | simp2 1009 |
. . . . . . . . . 10
   deg 
     coeff      
deg 
  |
82 | 81 | anim2i 573 |
. . . . . . . . 9
  Poly 
  deg 
     coeff         Poly 
deg     |
83 | 80, 82 | sylbi 199 |
. . . . . . . 8
  Poly    deg       coeff         Poly 
deg     |
84 | | simplll 768 |
. . . . . . . . . . . . 13
    Poly  deg    Poly 
deg    
 coeff        coeff         Poly    |
85 | | plyf 23152 |
. . . . . . . . . . . . 13
 Poly        |
86 | | ffn 5728 |
. . . . . . . . . . . . 13
       |
87 | 84, 85, 86 | 3syl 18 |
. . . . . . . . . . . 12
    Poly  deg    Poly 
deg    
 coeff        coeff           |
88 | | simplrl 770 |
. . . . . . . . . . . . 13
    Poly  deg    Poly 
deg    
 coeff        coeff         Poly    |
89 | | plyf 23152 |
. . . . . . . . . . . . 13
 Poly        |
90 | | ffn 5728 |
. . . . . . . . . . . . 13
       |
91 | 88, 89, 90 | 3syl 18 |
. . . . . . . . . . . 12
    Poly  deg    Poly 
deg    
 coeff        coeff           |
92 | | simplrr 771 |
. . . . . . . . . . . . . . . . . 18
     Poly  deg 
  Poly  deg    
 coeff        coeff           coeff        coeff         |
93 | 92 | adantr 467 |
. . . . . . . . . . . . . . . . 17
      Poly 
deg  
 Poly  deg    
 coeff        coeff                coeff        coeff         |
94 | 93 | fveq1d 5867 |
. . . . . . . . . . . . . . . 16
      Poly 
deg  
 Poly  deg    
 coeff        coeff                 coeff            coeff            |
95 | | fvres 5879 |
. . . . . . . . . . . . . . . . 17
       coeff           coeff       |
96 | 95 | adantl 468 |
. . . . . . . . . . . . . . . 16
      Poly 
deg  
 Poly  deg    
 coeff        coeff                 coeff           coeff       |
97 | | fvres 5879 |
. . . . . . . . . . . . . . . . 17
       coeff           coeff       |
98 | 97 | adantl 468 |
. . . . . . . . . . . . . . . 16
      Poly 
deg  
 Poly  deg    
 coeff        coeff                 coeff           coeff       |
99 | 94, 96, 98 | 3eqtr3d 2493 |
. . . . . . . . . . . . . . 15
      Poly 
deg  
 Poly  deg    
 coeff        coeff                coeff      coeff       |
100 | 99 | oveq1d 6305 |
. . . . . . . . . . . . . 14
      Poly 
deg  
 Poly  deg    
 coeff        coeff                 coeff            coeff            |
101 | 100 | sumeq2dv 13769 |
. . . . . . . . . . . . 13
     Poly  deg 
  Poly  deg    
 coeff        coeff                  coeff                  coeff            |
102 | | simp-4l 776 |
. . . . . . . . . . . . . 14
     Poly  deg 
  Poly  deg    
 coeff        coeff          Poly    |
103 | | simp-4r 777 |
. . . . . . . . . . . . . . 15
     Poly  deg 
  Poly  deg    
 coeff        coeff          deg 
  |
104 | | dgrcl 23187 |
. . . . . . . . . . . . . . . . 17
 Poly  deg    |
105 | | nn0z 10960 |
. . . . . . . . . . . . . . . . 17
 deg  deg    |
106 | 102, 104,
105 | 3syl 18 |
. . . . . . . . . . . . . . . 16
     Poly  deg 
  Poly  deg    
 coeff        coeff          deg    |
107 | | simplrl 770 |
. . . . . . . . . . . . . . . . 17
     Poly  deg 
  Poly  deg    
 coeff        coeff            |
108 | 107 | nn0zd 11038 |
. . . . . . . . . . . . . . . 16
     Poly  deg 
  Poly  deg    
 coeff        coeff            |
109 | | eluz 11172 |
. . . . . . . . . . . . . . . 16
  deg   
   deg   deg     |
110 | 106, 108,
109 | syl2anc 667 |
. . . . . . . . . . . . . . 15
     Poly  deg 
  Poly  deg    
 coeff        coeff              deg  
deg 
   |
111 | 103, 110 | mpbird 236 |
. . . . . . . . . . . . . 14
     Poly  deg 
  Poly  deg    
 coeff        coeff             deg     |
112 | | simpr 463 |
. . . . . . . . . . . . . 14
     Poly  deg 
  Poly  deg    
 coeff        coeff            |
113 | | eqid 2451 |
. . . . . . . . . . . . . . 15
deg  deg   |
114 | 33, 113 | coeid3 23194 |
. . . . . . . . . . . . . 14
  Poly 
   deg                coeff            |
115 | 102, 111,
112, 114 | syl3anc 1268 |
. . . . . . . . . . . . 13
     Poly  deg 
  Poly  deg    
 coeff        coeff              
       coeff            |
116 | | simp1rl 1073 |
. . . . . . . . . . . . . . 15
    Poly  deg    Poly 
deg    
 coeff        coeff       
 Poly    |
117 | 116 | 3expa 1208 |
. . . . . . . . . . . . . 14
     Poly  deg 
  Poly  deg    
 coeff        coeff          Poly    |
118 | | simplrr 771 |
. . . . . . . . . . . . . . . 16
    Poly  deg    Poly 
deg    
 coeff        coeff         deg 
  |
119 | 118 | adantr 467 |
. . . . . . . . . . . . . . 15
     Poly  deg 
  Poly  deg    
 coeff        coeff          deg 
  |
120 | | dgrcl 23187 |
. . . . . . . . . . . . . . . . 17
 Poly  deg    |
121 | | nn0z 10960 |
. . . . . . . . . . . . . . . . 17
 deg  deg    |
122 | 117, 120,
121 | 3syl 18 |
. . . . . . . . . . . . . . . 16
     Poly  deg 
  Poly  deg    
 coeff        coeff          deg    |
123 | | eluz 11172 |
. . . . . . . . . . . . . . . 16
  deg   
   deg   deg     |
124 | 122, 108,
123 | syl2anc 667 |
. . . . . . . . . . . . . . 15
     Poly  deg 
  Poly  deg    
 coeff        coeff              deg  
deg 
   |
125 | 119, 124 | mpbird 236 |
. . . . . . . . . . . . . 14
     Poly  deg 
  Poly  deg    
 coeff        coeff             deg     |
126 | | eqid 2451 |
. . . . . . . . . . . . . . 15
coeff  coeff   |
127 | | eqid 2451 |
. . . . . . . . . . . . . . 15
deg  deg   |
128 | 126, 127 | coeid3 23194 |
. . . . . . . . . . . . . 14
  Poly 
   deg                coeff            |
129 | 117, 125,
112, 128 | syl3anc 1268 |
. . . . . . . . . . . . 13
     Poly  deg 
  Poly  deg    
 coeff        coeff              
       coeff            |
130 | 101, 115,
129 | 3eqtr4d 2495 |
. . . . . . . . . . . 12
     Poly  deg 
  Poly  deg    
 coeff        coeff                    |
131 | 87, 91, 130 | eqfnfvd 5979 |
. . . . . . . . . . 11
    Poly  deg    Poly 
deg    
 coeff        coeff           |
132 | 131 | expr 620 |
. . . . . . . . . 10
    Poly  deg    Poly 
deg       coeff        coeff          |
133 | | fveq2 5865 |
. . . . . . . . . . 11
 coeff  coeff    |
134 | 133 | reseq1d 5104 |
. . . . . . . . . 10
  coeff        coeff         |
135 | 132, 134 | impbid1 207 |
. . . . . . . . 9
    Poly  deg    Poly 
deg       coeff        coeff      
   |
136 | 135 | expcom 437 |
. . . . . . . 8

   Poly  deg  
 Poly  deg      coeff        coeff           |
137 | 70, 83, 136 | syl2ani 662 |
. . . . . . 7

   Poly 
  deg       coeff         Poly    deg 
     coeff           coeff        coeff           |
138 | 67, 137 | dom2d 7610 |
. . . . . 6

             Poly 
  deg       coeff             
        |
139 | 18, 138 | mpi 20 |
. . . . 5

 Poly 
  deg       coeff             
       |
140 | | domfi 7793 |
. . . . 5
              Poly 
  deg 
     coeff             
     
 Poly    deg       coeff          |
141 | 17, 139, 140 | syl2anc 667 |
. . . 4

 Poly 
  deg       coeff          |
142 | | neeq1 2686 |
. . . . . . . . 9
       |
143 | | fveq2 5865 |
. . . . . . . . . 10
 deg  deg    |
144 | 143 | breq1d 4412 |
. . . . . . . . 9
  deg  deg     |
145 | | fveq2 5865 |
. . . . . . . . . . . . 13
 coeff  coeff    |
146 | 145 | fveq1d 5867 |
. . . . . . . . . . . 12
  coeff      coeff       |
147 | 146 | fveq2d 5869 |
. . . . . . . . . . 11
     coeff          coeff        |
148 | 147 | breq1d 4412 |
. . . . . . . . . 10
      coeff          coeff         |
149 | 148 | ralbidv 2827 |
. . . . . . . . 9
  
    coeff           coeff         |
150 | 142, 144,
149 | 3anbi123d 1339 |
. . . . . . . 8
    deg 
     coeff         deg       coeff          |
151 | 150 | elrab 3196 |
. . . . . . 7
  Poly    deg       coeff       
 Poly    deg 
     coeff          |
152 | | simp1 1008 |
. . . . . . . 8
   deg 
     coeff      
   |
153 | 152 | anim2i 573 |
. . . . . . 7
  Poly 
  deg 
     coeff         Poly 
    |
154 | 151, 153 | sylbi 199 |
. . . . . 6
  Poly    deg       coeff         Poly 
    |
155 | | plyf 23152 |
. . . . . . . . . . . . 13
 Poly        |
156 | | ffn 5728 |
. . . . . . . . . . . . 13
       |
157 | 155, 156 | syl 17 |
. . . . . . . . . . . 12
 Poly    |
158 | 157 | adantr 467 |
. . . . . . . . . . 11
  Poly 
    |
159 | | fniniseg 6003 |
. . . . . . . . . . 11
        
         |
160 | 158, 159 | syl 17 |
. . . . . . . . . 10
  Poly 
         
         |
161 | | fveq2 5865 |
. . . . . . . . . . . 12
           |
162 | 161 | eqeq1d 2453 |
. . . . . . . . . . 11
             |
163 | 162 | elrab 3196 |
. . . . . . . . . 10
      
        |
164 | 160, 163 | syl6rbbr 268 |
. . . . . . . . 9
  Poly 
                   |
165 | 164 | eqrdv 2449 |
. . . . . . . 8
  Poly 
                 |
166 | | eqid 2451 |
. . . . . . . . . 10
               |
167 | 166 | fta1 23261 |
. . . . . . . . 9
  Poly 
                     deg     |
168 | 167 | simpld 461 |
. . . . . . . 8
  Poly 
           |
169 | 165, 168 | eqeltrd 2529 |
. . . . . . 7
  Poly 
          |
170 | 169 | a1i 11 |
. . . . . 6

  Poly 
           |
171 | 154, 170 | syl5 33 |
. . . . 5

  Poly    deg       coeff                 |
172 | 171 | ralrimiv 2800 |
. . . 4

  Poly    deg 
     coeff                |
173 | | iunfi 7862 |
. . . 4
   Poly    deg 
     coeff          Poly 
  deg 
     coeff              
  Poly    deg 
     coeff                |
174 | 141, 172,
173 | syl2anc 667 |
. . 3

  Poly    deg 
     coeff                |
175 | 12, 174 | syl5eqelr 2534 |
. 2

 
 Poly    deg       coeff               |
176 | 11, 175 | eqeltrd 2529 |
1

      |