Step |
Hyp |
Ref |
Expression |
0 |
|
ccpmat |
|- ConstPolyMat |
1 |
|
vn |
|- n |
2 |
|
cfn |
|- Fin |
3 |
|
vr |
|- r |
4 |
|
cvv |
|- _V |
5 |
|
vm |
|- m |
6 |
|
cbs |
|- Base |
7 |
1
|
cv |
|- n |
8 |
|
cmat |
|- Mat |
9 |
|
cpl1 |
|- Poly1 |
10 |
3
|
cv |
|- r |
11 |
10 9
|
cfv |
|- ( Poly1 ` r ) |
12 |
7 11 8
|
co |
|- ( n Mat ( Poly1 ` r ) ) |
13 |
12 6
|
cfv |
|- ( Base ` ( n Mat ( Poly1 ` r ) ) ) |
14 |
|
vi |
|- i |
15 |
|
vj |
|- j |
16 |
|
vk |
|- k |
17 |
|
cn |
|- NN |
18 |
|
cco1 |
|- coe1 |
19 |
14
|
cv |
|- i |
20 |
5
|
cv |
|- m |
21 |
15
|
cv |
|- j |
22 |
19 21 20
|
co |
|- ( i m j ) |
23 |
22 18
|
cfv |
|- ( coe1 ` ( i m j ) ) |
24 |
16
|
cv |
|- k |
25 |
24 23
|
cfv |
|- ( ( coe1 ` ( i m j ) ) ` k ) |
26 |
|
c0g |
|- 0g |
27 |
10 26
|
cfv |
|- ( 0g ` r ) |
28 |
25 27
|
wceq |
|- ( ( coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) |
29 |
28 16 17
|
wral |
|- A. k e. NN ( ( coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) |
30 |
29 15 7
|
wral |
|- A. j e. n A. k e. NN ( ( coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) |
31 |
30 14 7
|
wral |
|- A. i e. n A. j e. n A. k e. NN ( ( coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) |
32 |
31 5 13
|
crab |
|- { m e. ( Base ` ( n Mat ( Poly1 ` r ) ) ) | A. i e. n A. j e. n A. k e. NN ( ( coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) } |
33 |
1 3 2 4 32
|
cmpo |
|- ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat ( Poly1 ` r ) ) ) | A. i e. n A. j e. n A. k e. NN ( ( coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) } ) |
34 |
0 33
|
wceq |
|- ConstPolyMat = ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat ( Poly1 ` r ) ) ) | A. i e. n A. j e. n A. k e. NN ( ( coe1 ` ( i m j ) ) ` k ) = ( 0g ` r ) } ) |