Step |
Hyp |
Ref |
Expression |
0 |
|
cdmat |
|- DMat |
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 |
3
|
cv |
|- r |
10 |
7 9 8
|
co |
|- ( n Mat r ) |
11 |
10 6
|
cfv |
|- ( Base ` ( n Mat r ) ) |
12 |
|
vi |
|- i |
13 |
|
vj |
|- j |
14 |
12
|
cv |
|- i |
15 |
13
|
cv |
|- j |
16 |
14 15
|
wne |
|- i =/= j |
17 |
5
|
cv |
|- m |
18 |
14 15 17
|
co |
|- ( i m j ) |
19 |
|
c0g |
|- 0g |
20 |
9 19
|
cfv |
|- ( 0g ` r ) |
21 |
18 20
|
wceq |
|- ( i m j ) = ( 0g ` r ) |
22 |
16 21
|
wi |
|- ( i =/= j -> ( i m j ) = ( 0g ` r ) ) |
23 |
22 13 7
|
wral |
|- A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) |
24 |
23 12 7
|
wral |
|- A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) |
25 |
24 5 11
|
crab |
|- { m e. ( Base ` ( n Mat r ) ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } |
26 |
1 3 2 4 25
|
cmpo |
|- ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat r ) ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } ) |
27 |
0 26
|
wceq |
|- DMat = ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat r ) ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } ) |