Step | Hyp | Ref
| Expression |
1 | | decpmatid.p |
. . . . . 6
Poly1   |
2 | | decpmatid.c |
. . . . . 6
 Mat   |
3 | 1, 2 | pmatring 19710 |
. . . . 5
 

  |
4 | 3 | 3adant3 1027 |
. . . 4
 
   |
5 | | eqid 2450 |
. . . . 5
         |
6 | | decpmatid.i |
. . . . 5
     |
7 | 5, 6 | ringidcl 17794 |
. . . 4

      |
8 | 4, 7 | syl 17 |
. . 3
 
       |
9 | | simp3 1009 |
. . 3
 
   |
10 | 2, 5 | decpmatval 19782 |
. . 3
        decompPMat     coe1            |
11 | 8, 9, 10 | syl2anc 666 |
. 2
 
  decompPMat     coe1            |
12 | | eqid 2450 |
. . . . . . 7
         |
13 | | eqid 2450 |
. . . . . . 7
         |
14 | | simp11 1037 |
. . . . . . 7
   
   |
15 | | simp12 1038 |
. . . . . . 7
   
   |
16 | | simp2 1008 |
. . . . . . 7
   
   |
17 | | simp3 1009 |
. . . . . . 7
   
   |
18 | 1, 2, 12, 13, 14, 15, 16, 17, 6 | pmat1ovd 19714 |
. . . . . 6
   
                    |
19 | 18 | fveq2d 5867 |
. . . . 5
   
 coe1      coe1                 |
20 | 19 | fveq1d 5865 |
. . . 4
   
  coe1          coe1                    |
21 | | fvif 5874 |
. . . . . . 7
coe1                  coe1       coe1        |
22 | 21 | fveq1i 5864 |
. . . . . 6
 coe1                      coe1       coe1           |
23 | | iffv 5875 |
. . . . . 6
    coe1       coe1              coe1           coe1           |
24 | 22, 23 | eqtri 2472 |
. . . . 5
 coe1                      coe1           coe1           |
25 | | eqid 2450 |
. . . . . . . . . . . . 13
var1  var1   |
26 | | eqid 2450 |
. . . . . . . . . . . . 13
mulGrp  mulGrp   |
27 | | eqid 2450 |
. . . . . . . . . . . . 13
.g mulGrp   .g mulGrp    |
28 | 1, 25, 26, 27 | ply1idvr1 18879 |
. . . . . . . . . . . 12

  .g mulGrp    var1         |
29 | 28 | 3ad2ant2 1029 |
. . . . . . . . . . 11
 
   .g mulGrp    var1         |
30 | 29 | eqcomd 2456 |
. . . . . . . . . 10
 
       .g mulGrp    var1     |
31 | 30 | fveq2d 5867 |
. . . . . . . . 9
 
 coe1      coe1   .g mulGrp    var1      |
32 | 31 | fveq1d 5865 |
. . . . . . . 8
 
  coe1          coe1   .g mulGrp    var1         |
33 | 1 | ply1lmod 18838 |
. . . . . . . . . . . . 13

  |
34 | 33 | 3ad2ant2 1029 |
. . . . . . . . . . . 12
 
   |
35 | | 0nn0 10881 |
. . . . . . . . . . . . . 14
 |
36 | | eqid 2450 |
. . . . . . . . . . . . . . 15
         |
37 | 1, 25, 26, 27, 36 | ply1moncl 18857 |
. . . . . . . . . . . . . 14
     .g mulGrp    var1         |
38 | 35, 37 | mpan2 676 |
. . . . . . . . . . . . 13

  .g mulGrp    var1         |
39 | 38 | 3ad2ant2 1029 |
. . . . . . . . . . . 12
 
   .g mulGrp    var1         |
40 | | eqid 2450 |
. . . . . . . . . . . . 13
Scalar  Scalar   |
41 | | eqid 2450 |
. . . . . . . . . . . . 13
         |
42 | | eqid 2450 |
. . . . . . . . . . . . 13
   Scalar      Scalar    |
43 | 36, 40, 41, 42 | lmodvs1 18112 |
. . . . . . . . . . . 12
    .g mulGrp    var1       
    Scalar           .g mulGrp    var1      .g mulGrp    var1     |
44 | 34, 39, 43 | syl2anc 666 |
. . . . . . . . . . 11
 
     Scalar           .g mulGrp    var1      .g mulGrp    var1     |
45 | 44 | eqcomd 2456 |
. . . . . . . . . 10
 
   .g mulGrp    var1       Scalar           .g mulGrp    var1      |
46 | 45 | fveq2d 5867 |
. . . . . . . . 9
 
 coe1   .g mulGrp    var1    coe1     Scalar           .g mulGrp    var1       |
47 | 46 | fveq1d 5865 |
. . . . . . . 8
 
  coe1   .g mulGrp    var1        coe1     Scalar           .g mulGrp    var1          |
48 | | simp2 1008 |
. . . . . . . . . 10
 
   |
49 | 1 | ply1sca 18839 |
. . . . . . . . . . . . . 14

Scalar    |
50 | 49 | 3ad2ant2 1029 |
. . . . . . . . . . . . 13
 
 Scalar    |
51 | 50 | eqcomd 2456 |
. . . . . . . . . . . 12
 
 Scalar    |
52 | 51 | fveq2d 5867 |
. . . . . . . . . . 11
 
    Scalar         |
53 | | eqid 2450 |
. . . . . . . . . . . . 13
         |
54 | | eqid 2450 |
. . . . . . . . . . . . 13
         |
55 | 53, 54 | ringidcl 17794 |
. . . . . . . . . . . 12

          |
56 | 55 | 3ad2ant2 1029 |
. . . . . . . . . . 11
 
           |
57 | 52, 56 | eqeltrd 2528 |
. . . . . . . . . 10
 
    Scalar         |
58 | 35 | a1i 11 |
. . . . . . . . . 10
 
   |
59 | | eqid 2450 |
. . . . . . . . . . 11
         |
60 | 59, 53, 1, 25, 41, 26, 27 | coe1tm 18859 |
. . . . . . . . . 10
     Scalar      
 coe1     Scalar           .g mulGrp    var1            Scalar            |
61 | 48, 57, 58, 60 | syl3anc 1267 |
. . . . . . . . 9
 
 coe1     Scalar           .g mulGrp    var1            Scalar            |
62 | | eqeq1 2454 |
. . . . . . . . . . 11
     |
63 | 62 | ifbid 3902 |
. . . . . . . . . 10
       Scalar               Scalar           |
64 | 63 | adantl 468 |
. . . . . . . . 9
           Scalar               Scalar           |
65 | | fvex 5873 |
. . . . . . . . . . 11
   Scalar    |
66 | | fvex 5873 |
. . . . . . . . . . 11
     |
67 | 65, 66 | ifex 3948 |
. . . . . . . . . 10
      Scalar          |
68 | 67 | a1i 11 |
. . . . . . . . 9
 
  
    Scalar           |
69 | 61, 64, 9, 68 | fvmptd 5952 |
. . . . . . . 8
 
  coe1     Scalar           .g mulGrp    var1              Scalar           |
70 | 32, 47, 69 | 3eqtrd 2488 |
. . . . . . 7
 
  coe1               Scalar           |
71 | 1, 12, 59 | coe1z 18849 |
. . . . . . . . . 10

coe1                |
72 | 71 | 3ad2ant2 1029 |
. . . . . . . . 9
 
 coe1                |
73 | 72 | fveq1d 5865 |
. . . . . . . 8
 
  coe1                       |
74 | 66 | a1i 11 |
. . . . . . . . 9
 
       |
75 | | fvconst2g 6116 |
. . . . . . . . 9
     
                   |
76 | 74, 9, 75 | syl2anc 666 |
. . . . . . . 8
 
  
                |
77 | 73, 76 | eqtrd 2484 |
. . . . . . 7
 
  coe1               |
78 | 70, 77 | ifeq12d 3900 |
. . . . . 6
 
     coe1           coe1                   Scalar                 |
79 | 78 | 3ad2ant1 1028 |
. . . . 5
   
     coe1           coe1                   Scalar                 |
80 | 24, 79 | syl5eq 2496 |
. . . 4
   
  coe1                           Scalar                 |
81 | 20, 80 | eqtrd 2484 |
. . 3
   
  coe1                  Scalar                 |
82 | 81 | mpt2eq3dva 6352 |
. 2
 
    coe1           
         Scalar                  |
83 | 50 | adantl 468 |
. . . . . . . . 9
    
Scalar    |
84 | 83 | eqcomd 2456 |
. . . . . . . 8
     Scalar    |
85 | 84 | fveq2d 5867 |
. . . . . . 7
        Scalar         |
86 | 85 | ifeq1d 3898 |
. . . . . 6
           Scalar                        |
87 | 86 | mpt2eq3dv 6354 |
. . . . 5
      
      Scalar                            |
88 | | iftrue 3886 |
. . . . . . . 8
  
    Scalar            Scalar     |
89 | 88 | ifeq1d 3898 |
. . . . . . 7
          Scalar                     Scalar           |
90 | 89 | adantr 467 |
. . . . . 6
              Scalar                     Scalar           |
91 | 90 | mpt2eq3dv 6354 |
. . . . 5
      
         Scalar                        Scalar            |
92 | | decpmatid.1 |
. . . . . . . 8
     |
93 | | decpmatid.a |
. . . . . . . . 9
 Mat   |
94 | 93, 54, 59 | mat1 19465 |
. . . . . . . 8
 
                       |
95 | 92, 94 | syl5eq 2496 |
. . . . . . 7
 
                   |
96 | 95 | 3adant3 1027 |
. . . . . 6
 
                   |
97 | 96 | adantl 468 |
. . . . 5
                       |
98 | 87, 91, 97 | 3eqtr4d 2494 |
. . . 4
      
         Scalar                 |
99 | | iftrue 3886 |
. . . . . 6
  
  |
100 | 99 | eqcomd 2456 |
. . . . 5
      |
101 | 100 | adantr 467 |
. . . 4
          |
102 | 98, 101 | eqtrd 2484 |
. . 3
      
         Scalar                     |
103 | | ifid 3917 |
. . . . . . 7
                  |
104 | 103 | a1i 11 |
. . . . . 6
  
                     |
105 | 104 | mpt2eq3dv 6354 |
. . . . 5
  
                           |
106 | | iffalse 3889 |
. . . . . . . 8
       Scalar               |
107 | 106 | adantr 467 |
. . . . . . 7
  
   
    Scalar               |
108 | 107 | ifeq1d 3898 |
. . . . . 6
  
           Scalar                              |
109 | 108 | mpt2eq3dv 6354 |
. . . . 5
  
             Scalar                                  |
110 | | 3simpa 1004 |
. . . . . . 7
 
 
   |
111 | 110 | adantl 468 |
. . . . . 6
  
  
   |
112 | | decpmatid.0 |
. . . . . . 7
     |
113 | 93, 59 | mat0op 19437 |
. . . . . . 7
 
              |
114 | 112, 113 | syl5eq 2496 |
. . . . . 6
 
          |
115 | 111, 114 | syl 17 |
. . . . 5
  
           |
116 | 105, 109,
115 | 3eqtr4d 2494 |
. . . 4
  
             Scalar                 |
117 | | iffalse 3889 |
. . . . . 6
   
 |
118 | 117 | eqcomd 2456 |
. . . . 5
      |
119 | 118 | adantr 467 |
. . . 4
  
       |
120 | 116, 119 | eqtrd 2484 |
. . 3
  
             Scalar                     |
121 | 102, 120 | pm2.61ian 798 |
. 2
 
            Scalar                     |
122 | 11, 82, 121 | 3eqtrd 2488 |
1
 
  decompPMat       |