Step |
Hyp |
Ref |
Expression |
1 |
|
ovnsubadd.1 |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
ovnsubadd.2 |
⊢ ( 𝜑 → 𝐴 : ℕ ⟶ 𝒫 ( ℝ ↑m 𝑋 ) ) |
3 |
|
fveq2 |
⊢ ( 𝑋 = ∅ → ( voln* ‘ 𝑋 ) = ( voln* ‘ ∅ ) ) |
4 |
3
|
fveq1d |
⊢ ( 𝑋 = ∅ → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) = ( ( voln* ‘ ∅ ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ) |
5 |
4
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) = ( ( voln* ‘ ∅ ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ) |
6 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐴 : ℕ ⟶ 𝒫 ( ℝ ↑m 𝑋 ) ) |
7 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ ) |
8 |
6 7
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ∈ 𝒫 ( ℝ ↑m 𝑋 ) ) |
9 |
|
elpwi |
⊢ ( ( 𝐴 ‘ 𝑛 ) ∈ 𝒫 ( ℝ ↑m 𝑋 ) → ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
10 |
8 9
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
11 |
10
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
12 |
|
iunss |
⊢ ( ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ↔ ∀ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
13 |
11 12
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
15 |
|
oveq2 |
⊢ ( 𝑋 = ∅ → ( ℝ ↑m 𝑋 ) = ( ℝ ↑m ∅ ) ) |
16 |
15
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ℝ ↑m 𝑋 ) = ( ℝ ↑m ∅ ) ) |
17 |
14 16
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m ∅ ) ) |
18 |
17
|
ovn0val |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln* ‘ ∅ ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) = 0 ) |
19 |
5 18
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) = 0 ) |
20 |
|
nnex |
⊢ ℕ ∈ V |
21 |
20
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
22 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑋 ∈ Fin ) |
23 |
22 10
|
ovncl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ∈ ( 0 [,] +∞ ) ) |
24 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) |
25 |
23 24
|
fmptd |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
26 |
21 25
|
sge0ge0 |
⊢ ( 𝜑 → 0 ≤ ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → 0 ≤ ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
28 |
19 27
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑋 = ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
29 |
1 13
|
ovnxrcl |
⊢ ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ∈ ℝ* ) |
30 |
29
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ∈ ℝ* ) |
31 |
21 25
|
sge0xrcl |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) ∈ ℝ* ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) → ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) ∈ ℝ* ) |
33 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) ∧ 𝑦 ∈ ℝ+ ) → 𝑋 ∈ Fin ) |
34 |
|
neqne |
⊢ ( ¬ 𝑋 = ∅ → 𝑋 ≠ ∅ ) |
35 |
34
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) ∧ 𝑦 ∈ ℝ+ ) → 𝑋 ≠ ∅ ) |
36 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) ∧ 𝑦 ∈ ℝ+ ) → 𝐴 : ℕ ⟶ 𝒫 ( ℝ ↑m 𝑋 ) ) |
37 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) ∧ 𝑦 ∈ ℝ+ ) → 𝑦 ∈ ℝ+ ) |
38 |
|
eqid |
⊢ ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ) = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ) |
39 |
|
sseq1 |
⊢ ( 𝑏 = 𝑎 → ( 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) ↔ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
40 |
39
|
rabbidv |
⊢ ( 𝑏 = 𝑎 → { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } = { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
41 |
40
|
cbvmptv |
⊢ ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
42 |
|
eqid |
⊢ ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) = ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) |
43 |
|
fveq2 |
⊢ ( 𝑜 = 𝑗 → ( 𝑙 ‘ 𝑜 ) = ( 𝑙 ‘ 𝑗 ) ) |
44 |
43
|
coeq2d |
⊢ ( 𝑜 = 𝑗 → ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) = ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ) |
45 |
44
|
fveq1d |
⊢ ( 𝑜 = 𝑗 → ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) = ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑑 ) ) |
46 |
45
|
ixpeq2dv |
⊢ ( 𝑜 = 𝑗 → X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) = X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑑 ) ) |
47 |
|
fveq2 |
⊢ ( 𝑑 = 𝑘 → ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑑 ) = ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
48 |
47
|
cbvixpv |
⊢ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑑 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) |
49 |
46 48
|
eqtrdi |
⊢ ( 𝑜 = 𝑗 → X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
50 |
49
|
cbviunv |
⊢ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) |
51 |
50
|
sseq2i |
⊢ ( 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) ↔ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
52 |
51
|
rabbii |
⊢ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } = { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } |
53 |
52
|
mpteq2i |
⊢ ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } ) = ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
54 |
53
|
fveq1i |
⊢ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } ) ‘ 𝑑 ) = ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑑 ) |
55 |
|
fveq2 |
⊢ ( 𝑑 = 𝑎 → ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑑 ) = ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ) |
56 |
54 55
|
syl5eq |
⊢ ( 𝑑 = 𝑎 → ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } ) ‘ 𝑑 ) = ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ) |
57 |
56
|
eleq2d |
⊢ ( 𝑑 = 𝑎 → ( 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } ) ‘ 𝑑 ) ↔ 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ) ) |
58 |
|
2fveq3 |
⊢ ( 𝑑 = 𝑘 → ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) = ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) |
59 |
58
|
cbvprodv |
⊢ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) |
60 |
59
|
mpteq2i |
⊢ ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) = ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) |
61 |
60
|
a1i |
⊢ ( 𝑜 = 𝑗 → ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) = ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ) |
62 |
|
fveq2 |
⊢ ( 𝑜 = 𝑗 → ( 𝑚 ‘ 𝑜 ) = ( 𝑚 ‘ 𝑗 ) ) |
63 |
61 62
|
fveq12d |
⊢ ( 𝑜 = 𝑗 → ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) = ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) |
64 |
63
|
cbvmptv |
⊢ ( 𝑜 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) |
65 |
64
|
fveq2i |
⊢ ( Σ^ ‘ ( 𝑜 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) ) |
66 |
65
|
a1i |
⊢ ( 𝑑 = 𝑎 → ( Σ^ ‘ ( 𝑜 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) ) ) |
67 |
|
fveq2 |
⊢ ( 𝑑 = 𝑎 → ( ( voln* ‘ 𝑋 ) ‘ 𝑑 ) = ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) ) |
68 |
67
|
oveq1d |
⊢ ( 𝑑 = 𝑎 → ( ( ( voln* ‘ 𝑋 ) ‘ 𝑑 ) +𝑒 𝑓 ) = ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) ) |
69 |
66 68
|
breq12d |
⊢ ( 𝑑 = 𝑎 → ( ( Σ^ ‘ ( 𝑜 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑑 ) +𝑒 𝑓 ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) ) ) |
70 |
57 69
|
anbi12d |
⊢ ( 𝑑 = 𝑎 → ( ( 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } ) ‘ 𝑑 ) ∧ ( Σ^ ‘ ( 𝑜 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑑 ) +𝑒 𝑓 ) ) ↔ ( 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) ) ) ) |
71 |
70
|
rabbidva2 |
⊢ ( 𝑑 = 𝑎 → { 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } ) ‘ 𝑑 ) ∣ ( Σ^ ‘ ( 𝑜 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑑 ) +𝑒 𝑓 ) } = { 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) } ) |
72 |
|
fveq1 |
⊢ ( 𝑚 = 𝑖 → ( 𝑚 ‘ 𝑗 ) = ( 𝑖 ‘ 𝑗 ) ) |
73 |
72
|
fveq2d |
⊢ ( 𝑚 = 𝑖 → ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) = ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) |
74 |
73
|
mpteq2dv |
⊢ ( 𝑚 = 𝑖 → ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) |
75 |
74
|
fveq2d |
⊢ ( 𝑚 = 𝑖 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ) |
76 |
75
|
breq1d |
⊢ ( 𝑚 = 𝑖 → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) ) ) |
77 |
76
|
cbvrabv |
⊢ { 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑚 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) } = { 𝑖 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) } |
78 |
71 77
|
eqtrdi |
⊢ ( 𝑑 = 𝑎 → { 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } ) ‘ 𝑑 ) ∣ ( Σ^ ‘ ( 𝑜 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑑 ) +𝑒 𝑓 ) } = { 𝑖 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) } ) |
79 |
78
|
mpteq2dv |
⊢ ( 𝑑 = 𝑎 → ( 𝑓 ∈ ℝ+ ↦ { 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } ) ‘ 𝑑 ) ∣ ( Σ^ ‘ ( 𝑜 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑑 ) +𝑒 𝑓 ) } ) = ( 𝑓 ∈ ℝ+ ↦ { 𝑖 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) } ) ) |
80 |
|
oveq2 |
⊢ ( 𝑓 = 𝑒 → ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) = ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) ) |
81 |
80
|
breq2d |
⊢ ( 𝑓 = 𝑒 → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) ) ) |
82 |
81
|
rabbidv |
⊢ ( 𝑓 = 𝑒 → { 𝑖 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) } = { 𝑖 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) } ) |
83 |
82
|
cbvmptv |
⊢ ( 𝑓 ∈ ℝ+ ↦ { 𝑖 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑓 ) } ) = ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) } ) |
84 |
79 83
|
eqtrdi |
⊢ ( 𝑑 = 𝑎 → ( 𝑓 ∈ ℝ+ ↦ { 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } ) ‘ 𝑑 ) ∣ ( Σ^ ‘ ( 𝑜 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑑 ) +𝑒 𝑓 ) } ) = ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) } ) ) |
85 |
84
|
cbvmptv |
⊢ ( 𝑑 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ ( 𝑓 ∈ ℝ+ ↦ { 𝑚 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X 𝑑 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑜 ) ) ‘ 𝑑 ) } ) ‘ 𝑑 ) ∣ ( Σ^ ‘ ( 𝑜 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑑 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑑 ) ) ) ‘ ( 𝑚 ‘ 𝑜 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑑 ) +𝑒 𝑓 ) } ) ) = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( ( 𝑏 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) } ) ) |
86 |
33 35 36 37 38 41 42 85
|
ovnsubaddlem2 |
⊢ ( ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) ∧ 𝑦 ∈ ℝ+ ) → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ≤ ( ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) +𝑒 𝑦 ) ) |
87 |
30 32 86
|
xrlexaddrp |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = ∅ ) → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |
88 |
28 87
|
pm2.61dan |
⊢ ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) ) |