Step | Hyp | Ref
| Expression |
1 | | decpmatid.p |
. . . . . 6
Poly1 |
2 | | decpmatid.c |
. . . . . 6
Mat |
3 | 1, 2 | pmatring 20498 |
. . . . 5
|
4 | 3 | 3adant3 1081 |
. . . 4
|
5 | | eqid 2622 |
. . . . 5
|
6 | | decpmatid.i |
. . . . 5
|
7 | 5, 6 | ringidcl 18568 |
. . . 4
|
8 | 4, 7 | syl 17 |
. . 3
|
9 | | simp3 1063 |
. . 3
|
10 | 2, 5 | decpmatval 20570 |
. . 3
decompPMat coe1 |
11 | 8, 9, 10 | syl2anc 693 |
. 2
decompPMat coe1 |
12 | | eqid 2622 |
. . . . . . 7
|
13 | | eqid 2622 |
. . . . . . 7
|
14 | | simp11 1091 |
. . . . . . 7
|
15 | | simp12 1092 |
. . . . . . 7
|
16 | | simp2 1062 |
. . . . . . 7
|
17 | | simp3 1063 |
. . . . . . 7
|
18 | 1, 2, 12, 13, 14, 15, 16, 17, 6 | pmat1ovd 20502 |
. . . . . 6
|
19 | 18 | fveq2d 6195 |
. . . . 5
coe1 coe1 |
20 | 19 | fveq1d 6193 |
. . . 4
coe1 coe1 |
21 | | fvif 6204 |
. . . . . . 7
coe1 coe1 coe1 |
22 | 21 | fveq1i 6192 |
. . . . . 6
coe1 coe1 coe1 |
23 | | iffv 6205 |
. . . . . 6
coe1 coe1 coe1 coe1 |
24 | 22, 23 | eqtri 2644 |
. . . . 5
coe1 coe1 coe1 |
25 | | eqid 2622 |
. . . . . . . . . . . . 13
var1 var1 |
26 | | eqid 2622 |
. . . . . . . . . . . . 13
mulGrp mulGrp |
27 | | eqid 2622 |
. . . . . . . . . . . . 13
.gmulGrp .gmulGrp |
28 | 1, 25, 26, 27 | ply1idvr1 19663 |
. . . . . . . . . . . 12
.gmulGrpvar1 |
29 | 28 | 3ad2ant2 1083 |
. . . . . . . . . . 11
.gmulGrpvar1 |
30 | 29 | eqcomd 2628 |
. . . . . . . . . 10
.gmulGrpvar1 |
31 | 30 | fveq2d 6195 |
. . . . . . . . 9
coe1 coe1.gmulGrpvar1 |
32 | 31 | fveq1d 6193 |
. . . . . . . 8
coe1 coe1.gmulGrpvar1 |
33 | 1 | ply1lmod 19622 |
. . . . . . . . . . . . 13
|
34 | 33 | 3ad2ant2 1083 |
. . . . . . . . . . . 12
|
35 | | 0nn0 11307 |
. . . . . . . . . . . . . 14
|
36 | | eqid 2622 |
. . . . . . . . . . . . . . 15
|
37 | 1, 25, 26, 27, 36 | ply1moncl 19641 |
. . . . . . . . . . . . . 14
.gmulGrpvar1 |
38 | 35, 37 | mpan2 707 |
. . . . . . . . . . . . 13
.gmulGrpvar1 |
39 | 38 | 3ad2ant2 1083 |
. . . . . . . . . . . 12
.gmulGrpvar1 |
40 | | eqid 2622 |
. . . . . . . . . . . . 13
Scalar Scalar |
41 | | eqid 2622 |
. . . . . . . . . . . . 13
|
42 | | eqid 2622 |
. . . . . . . . . . . . 13
Scalar Scalar |
43 | 36, 40, 41, 42 | lmodvs1 18891 |
. . . . . . . . . . . 12
.gmulGrpvar1
Scalar.gmulGrpvar1 .gmulGrpvar1 |
44 | 34, 39, 43 | syl2anc 693 |
. . . . . . . . . . 11
Scalar.gmulGrpvar1 .gmulGrpvar1 |
45 | 44 | eqcomd 2628 |
. . . . . . . . . 10
.gmulGrpvar1 Scalar.gmulGrpvar1 |
46 | 45 | fveq2d 6195 |
. . . . . . . . 9
coe1.gmulGrpvar1 coe1Scalar.gmulGrpvar1 |
47 | 46 | fveq1d 6193 |
. . . . . . . 8
coe1.gmulGrpvar1 coe1Scalar.gmulGrpvar1 |
48 | | simp2 1062 |
. . . . . . . . . 10
|
49 | 1 | ply1sca 19623 |
. . . . . . . . . . . . . 14
Scalar |
50 | 49 | 3ad2ant2 1083 |
. . . . . . . . . . . . 13
Scalar |
51 | 50 | eqcomd 2628 |
. . . . . . . . . . . 12
Scalar |
52 | 51 | fveq2d 6195 |
. . . . . . . . . . 11
Scalar |
53 | | eqid 2622 |
. . . . . . . . . . . . 13
|
54 | | eqid 2622 |
. . . . . . . . . . . . 13
|
55 | 53, 54 | ringidcl 18568 |
. . . . . . . . . . . 12
|
56 | 55 | 3ad2ant2 1083 |
. . . . . . . . . . 11
|
57 | 52, 56 | eqeltrd 2701 |
. . . . . . . . . 10
Scalar |
58 | 35 | a1i 11 |
. . . . . . . . . 10
|
59 | | eqid 2622 |
. . . . . . . . . . 11
|
60 | 59, 53, 1, 25, 41, 26, 27 | coe1tm 19643 |
. . . . . . . . . 10
Scalar
coe1Scalar.gmulGrpvar1 Scalar |
61 | 48, 57, 58, 60 | syl3anc 1326 |
. . . . . . . . 9
coe1Scalar.gmulGrpvar1 Scalar |
62 | | eqeq1 2626 |
. . . . . . . . . . 11
|
63 | 62 | ifbid 4108 |
. . . . . . . . . 10
Scalar Scalar |
64 | 63 | adantl 482 |
. . . . . . . . 9
Scalar Scalar |
65 | | fvex 6201 |
. . . . . . . . . . 11
Scalar |
66 | | fvex 6201 |
. . . . . . . . . . 11
|
67 | 65, 66 | ifex 4156 |
. . . . . . . . . 10
Scalar |
68 | 67 | a1i 11 |
. . . . . . . . 9
Scalar |
69 | 61, 64, 9, 68 | fvmptd 6288 |
. . . . . . . 8
coe1Scalar.gmulGrpvar1 Scalar |
70 | 32, 47, 69 | 3eqtrd 2660 |
. . . . . . 7
coe1 Scalar |
71 | 1, 12, 59 | coe1z 19633 |
. . . . . . . . . 10
coe1 |
72 | 71 | 3ad2ant2 1083 |
. . . . . . . . 9
coe1 |
73 | 72 | fveq1d 6193 |
. . . . . . . 8
coe1 |
74 | 66 | a1i 11 |
. . . . . . . . 9
|
75 | | fvconst2g 6467 |
. . . . . . . . 9
|
76 | 74, 9, 75 | syl2anc 693 |
. . . . . . . 8
|
77 | 73, 76 | eqtrd 2656 |
. . . . . . 7
coe1 |
78 | 70, 77 | ifeq12d 4106 |
. . . . . 6
coe1 coe1 Scalar |
79 | 78 | 3ad2ant1 1082 |
. . . . 5
coe1 coe1 Scalar |
80 | 24, 79 | syl5eq 2668 |
. . . 4
coe1 Scalar |
81 | 20, 80 | eqtrd 2656 |
. . 3
coe1 Scalar |
82 | 81 | mpt2eq3dva 6719 |
. 2
coe1
Scalar |
83 | 50 | adantl 482 |
. . . . . . . . 9
Scalar |
84 | 83 | eqcomd 2628 |
. . . . . . . 8
Scalar |
85 | 84 | fveq2d 6195 |
. . . . . . 7
Scalar |
86 | 85 | ifeq1d 4104 |
. . . . . 6
Scalar |
87 | 86 | mpt2eq3dv 6721 |
. . . . 5
Scalar |
88 | | iftrue 4092 |
. . . . . . . 8
Scalar Scalar |
89 | 88 | ifeq1d 4104 |
. . . . . . 7
Scalar Scalar |
90 | 89 | adantr 481 |
. . . . . 6
Scalar Scalar |
91 | 90 | mpt2eq3dv 6721 |
. . . . 5
Scalar Scalar |
92 | | decpmatid.1 |
. . . . . . . 8
|
93 | | decpmatid.a |
. . . . . . . . 9
Mat |
94 | 93, 54, 59 | mat1 20253 |
. . . . . . . 8
|
95 | 92, 94 | syl5eq 2668 |
. . . . . . 7
|
96 | 95 | 3adant3 1081 |
. . . . . 6
|
97 | 96 | adantl 482 |
. . . . 5
|
98 | 87, 91, 97 | 3eqtr4d 2666 |
. . . 4
Scalar |
99 | | iftrue 4092 |
. . . . . 6
|
100 | 99 | eqcomd 2628 |
. . . . 5
|
101 | 100 | adantr 481 |
. . . 4
|
102 | 98, 101 | eqtrd 2656 |
. . 3
Scalar |
103 | | ifid 4125 |
. . . . . . 7
|
104 | 103 | a1i 11 |
. . . . . 6
|
105 | 104 | mpt2eq3dv 6721 |
. . . . 5
|
106 | | iffalse 4095 |
. . . . . . . 8
Scalar |
107 | 106 | adantr 481 |
. . . . . . 7
Scalar |
108 | 107 | ifeq1d 4104 |
. . . . . 6
Scalar |
109 | 108 | mpt2eq3dv 6721 |
. . . . 5
Scalar |
110 | | 3simpa 1058 |
. . . . . . 7
|
111 | 110 | adantl 482 |
. . . . . 6
|
112 | | decpmatid.0 |
. . . . . . 7
|
113 | 93, 59 | mat0op 20225 |
. . . . . . 7
|
114 | 112, 113 | syl5eq 2668 |
. . . . . 6
|
115 | 111, 114 | syl 17 |
. . . . 5
|
116 | 105, 109,
115 | 3eqtr4d 2666 |
. . . 4
Scalar |
117 | | iffalse 4095 |
. . . . . 6
|
118 | 117 | eqcomd 2628 |
. . . . 5
|
119 | 118 | adantr 481 |
. . . 4
|
120 | 116, 119 | eqtrd 2656 |
. . 3
Scalar |
121 | 102, 120 | pm2.61ian 831 |
. 2
Scalar |
122 | 11, 82, 121 | 3eqtrd 2660 |
1
decompPMat |