Step |
Hyp |
Ref |
Expression |
0 |
|
cmvmul |
|- maVecMul |
1 |
|
vr |
|- r |
2 |
|
cvv |
|- _V |
3 |
|
vo |
|- o |
4 |
|
c1st |
|- 1st |
5 |
3
|
cv |
|- o |
6 |
5 4
|
cfv |
|- ( 1st ` o ) |
7 |
|
vm |
|- m |
8 |
|
c2nd |
|- 2nd |
9 |
5 8
|
cfv |
|- ( 2nd ` o ) |
10 |
|
vn |
|- n |
11 |
|
vx |
|- x |
12 |
|
cbs |
|- Base |
13 |
1
|
cv |
|- r |
14 |
13 12
|
cfv |
|- ( Base ` r ) |
15 |
|
cmap |
|- ^m |
16 |
7
|
cv |
|- m |
17 |
10
|
cv |
|- n |
18 |
16 17
|
cxp |
|- ( m X. n ) |
19 |
14 18 15
|
co |
|- ( ( Base ` r ) ^m ( m X. n ) ) |
20 |
|
vy |
|- y |
21 |
14 17 15
|
co |
|- ( ( Base ` r ) ^m n ) |
22 |
|
vi |
|- i |
23 |
|
cgsu |
|- gsum |
24 |
|
vj |
|- j |
25 |
22
|
cv |
|- i |
26 |
11
|
cv |
|- x |
27 |
24
|
cv |
|- j |
28 |
25 27 26
|
co |
|- ( i x j ) |
29 |
|
cmulr |
|- .r |
30 |
13 29
|
cfv |
|- ( .r ` r ) |
31 |
20
|
cv |
|- y |
32 |
27 31
|
cfv |
|- ( y ` j ) |
33 |
28 32 30
|
co |
|- ( ( i x j ) ( .r ` r ) ( y ` j ) ) |
34 |
24 17 33
|
cmpt |
|- ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) |
35 |
13 34 23
|
co |
|- ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) |
36 |
22 16 35
|
cmpt |
|- ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) |
37 |
11 20 19 21 36
|
cmpo |
|- ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) |
38 |
10 9 37
|
csb |
|- [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) |
39 |
7 6 38
|
csb |
|- [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) |
40 |
1 3 2 2 39
|
cmpo |
|- ( r e. _V , o e. _V |-> [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) ) |
41 |
0 40
|
wceq |
|- maVecMul = ( r e. _V , o e. _V |-> [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) ) |