Metamath Proof Explorer


Definition df-ovoln

Description: Define the outer measure for the space of multidimensional real numbers. The cardinality of x is the dimension of the space modeled. Definition 115C of Fremlin1 p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020)

Ref Expression
Assertion df-ovoln 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 𝑘𝑥 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑥 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) ) ) )

Detailed syntax breakdown

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