Description: The solution of a system of linear equations represented by a matrix with a unit as determinant is the multiplication of the inverse of the matrix with the right-hand side vector. (Contributed by AV, 11-Feb-2019) (Revised by AV, 28-Feb-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | slesolex.a | |
|
slesolex.b | |
||
slesolex.v | |
||
slesolex.x | |
||
slesolex.d | |
||
slesolinv.i | |
||
Assertion | slesolinvbi | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slesolex.a | |
|
2 | slesolex.b | |
|
3 | slesolex.v | |
|
4 | slesolex.x | |
|
5 | slesolex.d | |
|
6 | slesolinv.i | |
|
7 | simpl1 | |
|
8 | simpl2 | |
|
9 | simp3 | |
|
10 | 9 | anim1i | |
11 | 1 2 3 4 5 6 | slesolinv | |
12 | 7 8 10 11 | syl3anc | |
13 | oveq2 | |
|
14 | simpr | |
|
15 | 1 2 | matrcl | |
16 | 15 | simpld | |
17 | 16 | adantr | |
18 | 14 17 | anim12ci | |
19 | 18 | 3adant3 | |
20 | eqid | |
|
21 | 1 20 | matmulr | |
22 | 19 21 | syl | |
23 | 22 | oveqd | |
24 | crngring | |
|
25 | 24 | adantl | |
26 | 25 17 | anim12ci | |
27 | 26 | 3adant3 | |
28 | 1 | matring | |
29 | 27 28 | syl | |
30 | eqid | |
|
31 | eqid | |
|
32 | 1 5 2 30 31 | matunit | |
33 | 32 | ad2ant2lr | |
34 | 33 | biimp3ar | |
35 | eqid | |
|
36 | eqid | |
|
37 | 30 6 35 36 | unitrinv | |
38 | 29 34 37 | syl2anc | |
39 | 23 38 | eqtrd | |
40 | 39 | oveq1d | |
41 | eqid | |
|
42 | 25 | 3ad2ant1 | |
43 | 17 | 3ad2ant2 | |
44 | 3 | eleq2i | |
45 | 44 | biimpi | |
46 | 45 | adantl | |
47 | 46 | 3ad2ant2 | |
48 | 2 | eleq2i | |
49 | 48 | biimpi | |
50 | 49 | adantr | |
51 | 50 | 3ad2ant2 | |
52 | eqid | |
|
53 | 30 6 52 | ringinvcl | |
54 | 29 34 53 | syl2anc | |
55 | 1 41 4 42 43 47 20 51 54 | mavmulass | |
56 | 1 41 4 42 43 47 | 1mavmul | |
57 | 40 55 56 | 3eqtr3d | |
58 | 13 57 | sylan9eqr | |
59 | 12 58 | impbida | |