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 × 𝑛 ) |
19 |
14 18 15
|
co |
⊢ ( ( Base ‘ 𝑟 ) ↑m ( 𝑚 × 𝑛 ) ) |
20 |
|
vy |
⊢ 𝑦 |
21 |
14 17 15
|
co |
⊢ ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) |
22 |
|
vi |
⊢ 𝑖 |
23 |
|
cgsu |
⊢ Σg |
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 ∈ 𝑛 ↦ ( ( 𝑖 𝑥 𝑗 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑗 ) ) ) |
35 |
13 34 23
|
co |
⊢ ( 𝑟 Σg ( 𝑗 ∈ 𝑛 ↦ ( ( 𝑖 𝑥 𝑗 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑗 ) ) ) ) |
36 |
22 16 35
|
cmpt |
⊢ ( 𝑖 ∈ 𝑚 ↦ ( 𝑟 Σg ( 𝑗 ∈ 𝑛 ↦ ( ( 𝑖 𝑥 𝑗 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑗 ) ) ) ) ) |
37 |
11 20 19 21 36
|
cmpo |
⊢ ( 𝑥 ∈ ( ( Base ‘ 𝑟 ) ↑m ( 𝑚 × 𝑛 ) ) , 𝑦 ∈ ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) ↦ ( 𝑖 ∈ 𝑚 ↦ ( 𝑟 Σg ( 𝑗 ∈ 𝑛 ↦ ( ( 𝑖 𝑥 𝑗 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑗 ) ) ) ) ) ) |
38 |
10 9 37
|
csb |
⊢ ⦋ ( 2nd ‘ 𝑜 ) / 𝑛 ⦌ ( 𝑥 ∈ ( ( Base ‘ 𝑟 ) ↑m ( 𝑚 × 𝑛 ) ) , 𝑦 ∈ ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) ↦ ( 𝑖 ∈ 𝑚 ↦ ( 𝑟 Σg ( 𝑗 ∈ 𝑛 ↦ ( ( 𝑖 𝑥 𝑗 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑗 ) ) ) ) ) ) |
39 |
7 6 38
|
csb |
⊢ ⦋ ( 1st ‘ 𝑜 ) / 𝑚 ⦌ ⦋ ( 2nd ‘ 𝑜 ) / 𝑛 ⦌ ( 𝑥 ∈ ( ( Base ‘ 𝑟 ) ↑m ( 𝑚 × 𝑛 ) ) , 𝑦 ∈ ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) ↦ ( 𝑖 ∈ 𝑚 ↦ ( 𝑟 Σg ( 𝑗 ∈ 𝑛 ↦ ( ( 𝑖 𝑥 𝑗 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑗 ) ) ) ) ) ) |
40 |
1 3 2 2 39
|
cmpo |
⊢ ( 𝑟 ∈ V , 𝑜 ∈ V ↦ ⦋ ( 1st ‘ 𝑜 ) / 𝑚 ⦌ ⦋ ( 2nd ‘ 𝑜 ) / 𝑛 ⦌ ( 𝑥 ∈ ( ( Base ‘ 𝑟 ) ↑m ( 𝑚 × 𝑛 ) ) , 𝑦 ∈ ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) ↦ ( 𝑖 ∈ 𝑚 ↦ ( 𝑟 Σg ( 𝑗 ∈ 𝑛 ↦ ( ( 𝑖 𝑥 𝑗 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑗 ) ) ) ) ) ) ) |
41 |
0 40
|
wceq |
⊢ maVecMul = ( 𝑟 ∈ V , 𝑜 ∈ V ↦ ⦋ ( 1st ‘ 𝑜 ) / 𝑚 ⦌ ⦋ ( 2nd ‘ 𝑜 ) / 𝑛 ⦌ ( 𝑥 ∈ ( ( Base ‘ 𝑟 ) ↑m ( 𝑚 × 𝑛 ) ) , 𝑦 ∈ ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) ↦ ( 𝑖 ∈ 𝑚 ↦ ( 𝑟 Σg ( 𝑗 ∈ 𝑛 ↦ ( ( 𝑖 𝑥 𝑗 ) ( .r ‘ 𝑟 ) ( 𝑦 ‘ 𝑗 ) ) ) ) ) ) ) |