Description: The inverse transformation applied to the transformation of a matrix over a ring R results in the matrix itself. (Contributed by AV, 12-Nov-2019) (Revised by AV, 13-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | m2cpminvid.i | |
|
m2cpminvid.a | |
||
m2cpminvid.k | |
||
m2cpminvid.t | |
||
Assertion | m2cpminvid | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m2cpminvid.i | |
|
2 | m2cpminvid.a | |
|
3 | m2cpminvid.k | |
|
4 | m2cpminvid.t | |
|
5 | eqid | |
|
6 | 5 4 2 3 | m2cpm | |
7 | 1 5 | cpm2mval | |
8 | 6 7 | syld3an3 | |
9 | eqid | |
|
10 | eqid | |
|
11 | 4 2 3 9 10 | mat2pmatvalel | |
12 | 11 | 3impb | |
13 | 12 | fveq2d | |
14 | 13 | fveq1d | |
15 | simp12 | |
|
16 | eqid | |
|
17 | simp2 | |
|
18 | simp3 | |
|
19 | simp13 | |
|
20 | 2 16 3 17 18 19 | matecld | |
21 | 9 10 16 | ply1sclid | |
22 | 15 20 21 | syl2anc | |
23 | 14 22 | eqtr4d | |
24 | 23 | mpoeq3dva | |
25 | eqidd | |
|
26 | oveq12 | |
|
27 | 26 | adantl | |
28 | simprl | |
|
29 | simprr | |
|
30 | ovexd | |
|
31 | 25 27 28 29 30 | ovmpod | |
32 | 31 | ralrimivva | |
33 | simp1 | |
|
34 | simp2 | |
|
35 | 2 16 3 33 34 20 | matbas2d | |
36 | simp3 | |
|
37 | 2 3 | eqmat | |
38 | 35 36 37 | syl2anc | |
39 | 32 38 | mpbird | |
40 | 8 24 39 | 3eqtrd | |