Step |
Hyp |
Ref |
Expression |
0 |
|
covoln |
⊢ voln* |
1 |
|
vx |
⊢ 𝑥 |
2 |
|
cfn |
⊢ Fin |
3 |
|
vy |
⊢ 𝑦 |
4 |
|
cr |
⊢ ℝ |
5 |
|
cmap |
⊢ ↑m |
6 |
1
|
cv |
⊢ 𝑥 |
7 |
4 6 5
|
co |
⊢ ( ℝ ↑m 𝑥 ) |
8 |
7
|
cpw |
⊢ 𝒫 ( ℝ ↑m 𝑥 ) |
9 |
|
c0 |
⊢ ∅ |
10 |
6 9
|
wceq |
⊢ 𝑥 = ∅ |
11 |
|
cc0 |
⊢ 0 |
12 |
|
vz |
⊢ 𝑧 |
13 |
|
cxr |
⊢ ℝ* |
14 |
|
vi |
⊢ 𝑖 |
15 |
4 4
|
cxp |
⊢ ( ℝ × ℝ ) |
16 |
15 6 5
|
co |
⊢ ( ( ℝ × ℝ ) ↑m 𝑥 ) |
17 |
|
cn |
⊢ ℕ |
18 |
16 17 5
|
co |
⊢ ( ( ( ℝ × ℝ ) ↑m 𝑥 ) ↑m ℕ ) |
19 |
3
|
cv |
⊢ 𝑦 |
20 |
|
vj |
⊢ 𝑗 |
21 |
|
vk |
⊢ 𝑘 |
22 |
|
cico |
⊢ [,) |
23 |
14
|
cv |
⊢ 𝑖 |
24 |
20
|
cv |
⊢ 𝑗 |
25 |
24 23
|
cfv |
⊢ ( 𝑖 ‘ 𝑗 ) |
26 |
22 25
|
ccom |
⊢ ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) |
27 |
21
|
cv |
⊢ 𝑘 |
28 |
27 26
|
cfv |
⊢ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) |
29 |
21 6 28
|
cixp |
⊢ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) |
30 |
20 17 29
|
ciun |
⊢ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) |
31 |
19 30
|
wss |
⊢ 𝑦 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) |
32 |
12
|
cv |
⊢ 𝑧 |
33 |
|
csumge0 |
⊢ Σ^ |
34 |
|
cvol |
⊢ vol |
35 |
28 34
|
cfv |
⊢ ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
36 |
6 35 21
|
cprod |
⊢ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
37 |
20 17 36
|
cmpt |
⊢ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
38 |
37 33
|
cfv |
⊢ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) |
39 |
32 38
|
wceq |
⊢ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) |
40 |
31 39
|
wa |
⊢ ( 𝑦 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) |
41 |
40 14 18
|
wrex |
⊢ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑥 ) ↑m ℕ ) ( 𝑦 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) |
42 |
41 12 13
|
crab |
⊢ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑥 ) ↑m ℕ ) ( 𝑦 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } |
43 |
|
clt |
⊢ < |
44 |
42 13 43
|
cinf |
⊢ inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑥 ) ↑m ℕ ) ( 𝑦 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) |
45 |
10 11 44
|
cif |
⊢ if ( 𝑥 = ∅ , 0 , inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑥 ) ↑m ℕ ) ( 𝑦 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) ) |
46 |
3 8 45
|
cmpt |
⊢ ( 𝑦 ∈ 𝒫 ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑥 ) ↑m ℕ ) ( 𝑦 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) ) ) |
47 |
1 2 46
|
cmpt |
⊢ ( 𝑥 ∈ Fin ↦ ( 𝑦 ∈ 𝒫 ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑥 ) ↑m ℕ ) ( 𝑦 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) ) ) ) |
48 |
0 47
|
wceq |
⊢ voln* = ( 𝑥 ∈ Fin ↦ ( 𝑦 ∈ 𝒫 ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑥 ) ↑m ℕ ) ( 𝑦 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑥 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) ) ) ) |