Step | Hyp | Ref
| Expression |
1 | | ply1rem.g |
. . . 4
       |
2 | | ply1rem.1 |
. . . . . . . 8

NzRing |
3 | | nzrring 18485 |
. . . . . . . 8
 NzRing   |
4 | 2, 3 | syl 17 |
. . . . . . 7
   |
5 | | ply1rem.p |
. . . . . . . 8
Poly1   |
6 | 5 | ply1ring 18841 |
. . . . . . 7

  |
7 | 4, 6 | syl 17 |
. . . . . 6
   |
8 | | ringgrp 17785 |
. . . . . 6

  |
9 | 7, 8 | syl 17 |
. . . . 5
   |
10 | | ply1rem.x |
. . . . . . 7
var1   |
11 | | ply1rem.b |
. . . . . . 7
     |
12 | 10, 5, 11 | vr1cl 18810 |
. . . . . 6

  |
13 | 4, 12 | syl 17 |
. . . . 5
   |
14 | | ply1rem.a |
. . . . . . . 8
algSc   |
15 | | ply1rem.k |
. . . . . . . 8
     |
16 | 5, 14, 15, 11 | ply1sclf 18878 |
. . . . . . 7

      |
17 | 4, 16 | syl 17 |
. . . . . 6
       |
18 | | ply1rem.3 |
. . . . . 6
   |
19 | 17, 18 | ffvelrnd 6023 |
. . . . 5
       |
20 | | ply1rem.m |
. . . . . 6
     |
21 | 11, 20 | grpsubcl 16734 |
. . . . 5
 
             |
22 | 9, 13, 19, 21 | syl3anc 1268 |
. . . 4
         |
23 | 1, 22 | syl5eqel 2533 |
. . 3
   |
24 | 1 | fveq2i 5868 |
. . . . . . 7
               |
25 | | ply1rem.d |
. . . . . . . 8
deg1    |
26 | 25, 5, 11 | deg1xrcl 23031 |
. . . . . . . . . . 11
               |
27 | 19, 26 | syl 17 |
. . . . . . . . . 10
           |
28 | | 0xr 9687 |
. . . . . . . . . . 11
 |
29 | 28 | a1i 11 |
. . . . . . . . . 10
   |
30 | | 1re 9642 |
. . . . . . . . . . 11
 |
31 | | rexr 9686 |
. . . . . . . . . . 11
   |
32 | 30, 31 | mp1i 13 |
. . . . . . . . . 10
   |
33 | 25, 5, 15, 14 | deg1sclle 23061 |
. . . . . . . . . . 11
             |
34 | 4, 18, 33 | syl2anc 667 |
. . . . . . . . . 10
        
  |
35 | | 0lt1 10136 |
. . . . . . . . . . 11
 |
36 | 35 | a1i 11 |
. . . . . . . . . 10
   |
37 | 27, 29, 32, 34, 36 | xrlelttrd 11457 |
. . . . . . . . 9
        
  |
38 | | eqid 2451 |
. . . . . . . . . . . . . 14
mulGrp  mulGrp   |
39 | 38, 11 | mgpbas 17729 |
. . . . . . . . . . . . 13
   mulGrp    |
40 | | eqid 2451 |
. . . . . . . . . . . . 13
.g mulGrp   .g mulGrp    |
41 | 39, 40 | mulg1 16765 |
. . . . . . . . . . . 12
   .g mulGrp       |
42 | 13, 41 | syl 17 |
. . . . . . . . . . 11
   .g mulGrp       |
43 | 42 | fveq2d 5869 |
. . . . . . . . . 10
      .g mulGrp            |
44 | | 1nn0 10885 |
. . . . . . . . . . 11
 |
45 | 25, 5, 10, 38, 40 | deg1pw 23069 |
. . . . . . . . . . 11
  NzRing
      .g mulGrp        |
46 | 2, 44, 45 | sylancl 668 |
. . . . . . . . . 10
      .g mulGrp        |
47 | 43, 46 | eqtr3d 2487 |
. . . . . . . . 9
       |
48 | 37, 47 | breqtrrd 4429 |
. . . . . . . 8
        
      |
49 | 5, 25, 4, 11, 20, 13, 19, 48 | deg1sub 23057 |
. . . . . . 7
                 |
50 | 24, 49 | syl5eq 2497 |
. . . . . 6
           |
51 | 50, 47 | eqtrd 2485 |
. . . . 5
       |
52 | 51, 44 | syl6eqel 2537 |
. . . 4
       |
53 | | eqid 2451 |
. . . . . 6
         |
54 | 25, 5, 53, 11 | deg1nn0clb 23039 |
. . . . 5
       
       |
55 | 4, 23, 54 | syl2anc 667 |
. . . 4
     
       |
56 | 52, 55 | mpbird 236 |
. . 3
       |
57 | 51 | fveq2d 5869 |
. . . 4
  coe1          coe1       |
58 | 1 | fveq2i 5868 |
. . . . . 6
coe1  coe1         |
59 | 58 | fveq1i 5866 |
. . . . 5
 coe1      coe1            |
60 | 44 | a1i 11 |
. . . . . 6
   |
61 | | eqid 2451 |
. . . . . . 7
         |
62 | 5, 11, 20, 61 | coe1subfv 18859 |
. . . . . 6
       
  coe1             coe1            coe1            |
63 | 4, 13, 19, 60, 62 | syl31anc 1271 |
. . . . 5
  coe1             coe1            coe1            |
64 | 59, 63 | syl5eq 2497 |
. . . 4
  coe1       coe1            coe1            |
65 | 42 | oveq2d 6306 |
. . . . . . . . . 10
              .g mulGrp                    |
66 | 5 | ply1sca 18846 |
. . . . . . . . . . . . 13
 NzRing Scalar    |
67 | 2, 66 | syl 17 |
. . . . . . . . . . . 12
 Scalar    |
68 | 67 | fveq2d 5869 |
. . . . . . . . . . 11
        Scalar     |
69 | 68 | oveq1d 6305 |
. . . . . . . . . 10
                 Scalar            |
70 | 5 | ply1lmod 18845 |
. . . . . . . . . . . 12

  |
71 | 4, 70 | syl 17 |
. . . . . . . . . . 11
   |
72 | | eqid 2451 |
. . . . . . . . . . . 12
Scalar  Scalar   |
73 | | eqid 2451 |
. . . . . . . . . . . 12
         |
74 | | eqid 2451 |
. . . . . . . . . . . 12
   Scalar      Scalar    |
75 | 11, 72, 73, 74 | lmodvs1 18119 |
. . . . . . . . . . 11
       Scalar            |
76 | 71, 13, 75 | syl2anc 667 |
. . . . . . . . . 10
     Scalar            |
77 | 65, 69, 76 | 3eqtrd 2489 |
. . . . . . . . 9
              .g mulGrp        |
78 | 77 | fveq2d 5869 |
. . . . . . . 8
 coe1              .g mulGrp       coe1    |
79 | 78 | fveq1d 5867 |
. . . . . . 7
  coe1              .g mulGrp           coe1       |
80 | | eqid 2451 |
. . . . . . . . . 10
         |
81 | 15, 80 | ringidcl 17801 |
. . . . . . . . 9

      |
82 | 4, 81 | syl 17 |
. . . . . . . 8
       |
83 | | ply1rem.z |
. . . . . . . . 9
     |
84 | 83, 15, 5, 10, 73, 38, 40 | coe1tmfv1 18867 |
. . . . . . . 8
        coe1              .g mulGrp                |
85 | 4, 82, 60, 84 | syl3anc 1268 |
. . . . . . 7
  coe1              .g mulGrp                |
86 | 79, 85 | eqtr3d 2487 |
. . . . . 6
  coe1           |
87 | | eqid 2451 |
. . . . . . . . . 10
         |
88 | 5, 14, 15, 87 | coe1scl 18880 |
. . . . . . . . 9
   coe1      
            |
89 | 4, 18, 88 | syl2anc 667 |
. . . . . . . 8
 coe1                   |
90 | 89 | fveq1d 5867 |
. . . . . . 7
  coe1                          |
91 | | ax-1ne0 9608 |
. . . . . . . . . . 11
 |
92 | | neeq1 2686 |
. . . . . . . . . . 11
     |
93 | 91, 92 | mpbiri 237 |
. . . . . . . . . 10
   |
94 | | ifnefalse 3893 |
. . . . . . . . . 10
  
             |
95 | 93, 94 | syl 17 |
. . . . . . . . 9
  
             |
96 | | eqid 2451 |
. . . . . . . . 9

          
           |
97 | | fvex 5875 |
. . . . . . . . 9
     |
98 | 95, 96, 97 | fvmpt 5948 |
. . . . . . . 8

                     |
99 | 44, 98 | ax-mp 5 |
. . . . . . 7
                    |
100 | 90, 99 | syl6eq 2501 |
. . . . . 6
  coe1               |
101 | 86, 100 | oveq12d 6308 |
. . . . 5
   coe1            coe1                            |
102 | | ringgrp 17785 |
. . . . . . 7

  |
103 | 4, 102 | syl 17 |
. . . . . 6
   |
104 | 15, 87, 61 | grpsubid1 16739 |
. . . . . 6
                             |
105 | 103, 82, 104 | syl2anc 667 |
. . . . 5
                       |
106 | 101, 105 | eqtrd 2485 |
. . . 4
   coe1            coe1                |
107 | 57, 64, 106 | 3eqtrd 2489 |
. . 3
  coe1               |
108 | | ply1rem.u |
. . . 4
Monic1p   |
109 | 5, 11, 53, 25, 108, 80 | ismon1p 23093 |
. . 3


   
 coe1                |
110 | 23, 56, 107, 109 | syl3anbrc 1192 |
. 2
   |
111 | 1 | fveq2i 5868 |
. . . . . . . . . 10
               |
112 | 111 | fveq1i 5866 |
. . . . . . . . 9
            
          |
113 | | ply1rem.o |
. . . . . . . . . . 11
eval1   |
114 | | ply1rem.2 |
. . . . . . . . . . . 12
   |
115 | 114 | adantr 467 |
. . . . . . . . . . 11
 
   |
116 | | simpr 463 |
. . . . . . . . . . 11
 
   |
117 | 113, 10, 15, 5, 11, 115, 116 | evl1vard 18925 |
. . . . . . . . . . 11
 
 
           |
118 | 18 | adantr 467 |
. . . . . . . . . . . 12
 
   |
119 | 113, 5, 15, 14, 11, 115, 118, 116 | evl1scad 18923 |
. . . . . . . . . . 11
 
                     |
120 | 113, 5, 15, 11, 115, 116, 117, 119, 20, 61 | evl1subd 18930 |
. . . . . . . . . 10
 
            
                    |
121 | 120 | simprd 465 |
. . . . . . . . 9
 
                         |
122 | 112, 121 | syl5eq 2497 |
. . . . . . . 8
 
                   |
123 | 122 | eqeq1d 2453 |
. . . . . . 7
 
         
          |
124 | 103 | adantr 467 |
. . . . . . . 8
 
   |
125 | 15, 83, 61 | grpsubeq0 16740 |
. . . . . . . 8
 
             |
126 | 124, 116,
118, 125 | syl3anc 1268 |
. . . . . . 7
 
         
   |
127 | 123, 126 | bitrd 257 |
. . . . . 6
 
         
   |
128 | | elsn 3982 |
. . . . . 6
     |
129 | 127, 128 | syl6bbr 267 |
. . . . 5
 
         
     |
130 | 129 | pm5.32da 647 |
. . . 4
          

      |
131 | | eqid 2451 |
. . . . . . 7
 s   s   |
132 | | eqid 2451 |
. . . . . . 7
    s       s    |
133 | | fvex 5875 |
. . . . . . . . 9
     |
134 | 15, 133 | eqeltri 2525 |
. . . . . . . 8
 |
135 | 134 | a1i 11 |
. . . . . . 7
   |
136 | 113, 5, 131, 15 | evl1rhm 18920 |
. . . . . . . . . 10


RingHom  s     |
137 | 114, 136 | syl 17 |
. . . . . . . . 9
  RingHom  s     |
138 | 11, 132 | rhmf 17954 |
. . . . . . . . 9
  RingHom  s           s     |
139 | 137, 138 | syl 17 |
. . . . . . . 8
         s     |
140 | 139, 23 | ffvelrnd 6023 |
. . . . . . 7
         s     |
141 | 131, 15, 132, 2, 135, 140 | pwselbas 15387 |
. . . . . 6
           |
142 | | ffn 5728 |
. . . . . 6
               |
143 | 141, 142 | syl 17 |
. . . . 5
       |
144 | | fniniseg 6003 |
. . . . 5
                           |
145 | 143, 144 | syl 17 |
. . . 4
          

           |
146 | 18 | snssd 4117 |
. . . . . 6
     |
147 | 146 | sseld 3431 |
. . . . 5
   
   |
148 | 147 | pm4.71rd 641 |
. . . 4
   
       |
149 | 130, 145,
148 | 3bitr4d 289 |
. . 3
          
     |
150 | 149 | eqrdv 2449 |
. 2
              |
151 | 110, 51, 150 | 3jca 1188 |
1
                    |