Step |
Hyp |
Ref |
Expression |
0 |
|
clsm |
⊢ LSSum |
1 |
|
vw |
⊢ 𝑤 |
2 |
|
cvv |
⊢ V |
3 |
|
vt |
⊢ 𝑡 |
4 |
|
cbs |
⊢ Base |
5 |
1
|
cv |
⊢ 𝑤 |
6 |
5 4
|
cfv |
⊢ ( Base ‘ 𝑤 ) |
7 |
6
|
cpw |
⊢ 𝒫 ( Base ‘ 𝑤 ) |
8 |
|
vu |
⊢ 𝑢 |
9 |
|
vx |
⊢ 𝑥 |
10 |
3
|
cv |
⊢ 𝑡 |
11 |
|
vy |
⊢ 𝑦 |
12 |
8
|
cv |
⊢ 𝑢 |
13 |
9
|
cv |
⊢ 𝑥 |
14 |
|
cplusg |
⊢ +g |
15 |
5 14
|
cfv |
⊢ ( +g ‘ 𝑤 ) |
16 |
11
|
cv |
⊢ 𝑦 |
17 |
13 16 15
|
co |
⊢ ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) |
18 |
9 11 10 12 17
|
cmpo |
⊢ ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) ) |
19 |
18
|
crn |
⊢ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) ) |
20 |
3 8 7 7 19
|
cmpo |
⊢ ( 𝑡 ∈ 𝒫 ( Base ‘ 𝑤 ) , 𝑢 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) ) ) |
21 |
1 2 20
|
cmpt |
⊢ ( 𝑤 ∈ V ↦ ( 𝑡 ∈ 𝒫 ( Base ‘ 𝑤 ) , 𝑢 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) ) ) ) |
22 |
0 21
|
wceq |
⊢ LSSum = ( 𝑤 ∈ V ↦ ( 𝑡 ∈ 𝒫 ( Base ‘ 𝑤 ) , 𝑢 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) ) ) ) |