Step |
Hyp |
Ref |
Expression |
1 |
|
ovncvr2.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
ovncvr2.a |
⊢ ( 𝜑 → 𝐴 ⊆ ( ℝ ↑m 𝑋 ) ) |
3 |
|
ovncvr2.e |
⊢ ( 𝜑 → 𝐸 ∈ ℝ+ ) |
4 |
|
ovncvr2.c |
⊢ 𝐶 = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
5 |
|
ovncvr2.l |
⊢ 𝐿 = ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) |
6 |
|
ovncvr2.d |
⊢ 𝐷 = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ ( 𝑟 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑟 ) } ) ) |
7 |
|
ovncvr2.i |
⊢ ( 𝜑 → 𝐼 ∈ ( ( 𝐷 ‘ 𝐴 ) ‘ 𝐸 ) ) |
8 |
|
ovncvr2.b |
⊢ 𝐵 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
9 |
|
ovncvr2.t |
⊢ 𝑇 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
10 |
|
sseq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
11 |
10
|
rabbidv |
⊢ ( 𝑎 = 𝐴 → { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } = { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
12 |
|
ovexd |
⊢ ( 𝜑 → ( ℝ ↑m 𝑋 ) ∈ V ) |
13 |
12 2
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
14 |
|
elpwg |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↔ 𝐴 ⊆ ( ℝ ↑m 𝑋 ) ) ) |
15 |
13 14
|
syl |
⊢ ( 𝜑 → ( 𝐴 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↔ 𝐴 ⊆ ( ℝ ↑m 𝑋 ) ) ) |
16 |
2 15
|
mpbird |
⊢ ( 𝜑 → 𝐴 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ) |
17 |
|
ovex |
⊢ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∈ V |
18 |
17
|
rabex |
⊢ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ∈ V |
19 |
18
|
a1i |
⊢ ( 𝜑 → { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ∈ V ) |
20 |
4 11 16 19
|
fvmptd3 |
⊢ ( 𝜑 → ( 𝐶 ‘ 𝐴 ) = { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
21 |
|
ssrab2 |
⊢ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ⊆ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) |
22 |
21
|
a1i |
⊢ ( 𝜑 → { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ⊆ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
23 |
20 22
|
eqsstrd |
⊢ ( 𝜑 → ( 𝐶 ‘ 𝐴 ) ⊆ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
24 |
|
fveq2 |
⊢ ( 𝑎 = 𝐴 → ( 𝐶 ‘ 𝑎 ) = ( 𝐶 ‘ 𝐴 ) ) |
25 |
24
|
eleq2d |
⊢ ( 𝑎 = 𝐴 → ( 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ↔ 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ) ) |
26 |
|
fveq2 |
⊢ ( 𝑎 = 𝐴 → ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) = ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) ) |
27 |
26
|
oveq1d |
⊢ ( 𝑎 = 𝐴 → ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑟 ) = ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) ) |
28 |
27
|
breq2d |
⊢ ( 𝑎 = 𝐴 → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑟 ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) ) ) |
29 |
25 28
|
anbi12d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑟 ) ) ↔ ( 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) ) ) ) |
30 |
29
|
rabbidva2 |
⊢ ( 𝑎 = 𝐴 → { 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑟 ) } = { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) } ) |
31 |
30
|
mpteq2dv |
⊢ ( 𝑎 = 𝐴 → ( 𝑟 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑟 ) } ) = ( 𝑟 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) } ) ) |
32 |
|
rpex |
⊢ ℝ+ ∈ V |
33 |
32
|
mptex |
⊢ ( 𝑟 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) } ) ∈ V |
34 |
33
|
a1i |
⊢ ( 𝜑 → ( 𝑟 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) } ) ∈ V ) |
35 |
6 31 16 34
|
fvmptd3 |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐴 ) = ( 𝑟 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) } ) ) |
36 |
|
oveq2 |
⊢ ( 𝑟 = 𝐸 → ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) = ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) ) |
37 |
36
|
breq2d |
⊢ ( 𝑟 = 𝐸 → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) ) ) |
38 |
37
|
rabbidv |
⊢ ( 𝑟 = 𝐸 → { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) } = { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) } ) |
39 |
38
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑟 = 𝐸 ) → { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝑟 ) } = { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) } ) |
40 |
|
fvex |
⊢ ( 𝐶 ‘ 𝐴 ) ∈ V |
41 |
40
|
rabex |
⊢ { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) } ∈ V |
42 |
41
|
a1i |
⊢ ( 𝜑 → { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) } ∈ V ) |
43 |
35 39 3 42
|
fvmptd |
⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝐴 ) ‘ 𝐸 ) = { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) } ) |
44 |
7 43
|
eleqtrd |
⊢ ( 𝜑 → 𝐼 ∈ { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) } ) |
45 |
|
fveq1 |
⊢ ( 𝑖 = 𝐼 → ( 𝑖 ‘ 𝑗 ) = ( 𝐼 ‘ 𝑗 ) ) |
46 |
45
|
fveq2d |
⊢ ( 𝑖 = 𝐼 → ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) = ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) ) |
47 |
46
|
mpteq2dv |
⊢ ( 𝑖 = 𝐼 → ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) ) ) |
48 |
47
|
fveq2d |
⊢ ( 𝑖 = 𝐼 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) ) ) ) |
49 |
48
|
breq1d |
⊢ ( 𝑖 = 𝐼 → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) ) ) |
50 |
49
|
elrab |
⊢ ( 𝐼 ∈ { 𝑖 ∈ ( 𝐶 ‘ 𝐴 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) } ↔ ( 𝐼 ∈ ( 𝐶 ‘ 𝐴 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) ) ) |
51 |
44 50
|
sylib |
⊢ ( 𝜑 → ( 𝐼 ∈ ( 𝐶 ‘ 𝐴 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) ) ) |
52 |
51
|
simpld |
⊢ ( 𝜑 → 𝐼 ∈ ( 𝐶 ‘ 𝐴 ) ) |
53 |
23 52
|
sseldd |
⊢ ( 𝜑 → 𝐼 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
54 |
|
elmapi |
⊢ ( 𝐼 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) → 𝐼 : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
55 |
53 54
|
syl |
⊢ ( 𝜑 → 𝐼 : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
56 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝐼 : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
57 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ ) |
58 |
56 57
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐼 ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
59 |
|
elmapi |
⊢ ( ( 𝐼 ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) → ( 𝐼 ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
60 |
58 59
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐼 ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
61 |
60
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ( ℝ × ℝ ) ) |
62 |
|
xp1st |
⊢ ( ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ( ℝ × ℝ ) → ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ∈ ℝ ) |
63 |
61 62
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ∈ ℝ ) |
64 |
63
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) : 𝑋 ⟶ ℝ ) |
65 |
|
reex |
⊢ ℝ ∈ V |
66 |
65
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
67 |
|
elmapg |
⊢ ( ( ℝ ∈ V ∧ 𝑋 ∈ Fin ) → ( ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ( ℝ ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) : 𝑋 ⟶ ℝ ) ) |
68 |
66 1 67
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ( ℝ ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) : 𝑋 ⟶ ℝ ) ) |
69 |
68
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ( ℝ ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) : 𝑋 ⟶ ℝ ) ) |
70 |
64 69
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ( ℝ ↑m 𝑋 ) ) |
71 |
70
|
fmpttd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) |
72 |
8
|
a1i |
⊢ ( 𝜑 → 𝐵 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) |
73 |
72
|
feq1d |
⊢ ( 𝜑 → ( 𝐵 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ↔ ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) ) |
74 |
71 73
|
mpbird |
⊢ ( 𝜑 → 𝐵 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) |
75 |
|
xp2nd |
⊢ ( ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ( ℝ × ℝ ) → ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ∈ ℝ ) |
76 |
61 75
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ∈ ℝ ) |
77 |
76
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) : 𝑋 ⟶ ℝ ) |
78 |
|
elmapg |
⊢ ( ( ℝ ∈ V ∧ 𝑋 ∈ Fin ) → ( ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ( ℝ ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) : 𝑋 ⟶ ℝ ) ) |
79 |
66 1 78
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ( ℝ ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) : 𝑋 ⟶ ℝ ) ) |
80 |
79
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ( ℝ ↑m 𝑋 ) ↔ ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) : 𝑋 ⟶ ℝ ) ) |
81 |
77 80
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ( ℝ ↑m 𝑋 ) ) |
82 |
81
|
fmpttd |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) |
83 |
9
|
a1i |
⊢ ( 𝜑 → 𝑇 = ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) |
84 |
83
|
feq1d |
⊢ ( 𝜑 → ( 𝑇 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ↔ ( 𝑗 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) ) |
85 |
82 84
|
mpbird |
⊢ ( 𝜑 → 𝑇 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) |
86 |
74 85
|
jca |
⊢ ( 𝜑 → ( 𝐵 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ∧ 𝑇 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) ) |
87 |
52 20
|
eleqtrd |
⊢ ( 𝜑 → 𝐼 ∈ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
88 |
|
fveq1 |
⊢ ( 𝑙 = 𝐼 → ( 𝑙 ‘ 𝑗 ) = ( 𝐼 ‘ 𝑗 ) ) |
89 |
88
|
coeq2d |
⊢ ( 𝑙 = 𝐼 → ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) = ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ) |
90 |
89
|
fveq1d |
⊢ ( 𝑙 = 𝐼 → ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
91 |
90
|
ixpeq2dv |
⊢ ( 𝑙 = 𝐼 → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
92 |
91
|
adantr |
⊢ ( ( 𝑙 = 𝐼 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
93 |
92
|
iuneq2dv |
⊢ ( 𝑙 = 𝐼 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
94 |
93
|
sseq2d |
⊢ ( 𝑙 = 𝐼 → ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
95 |
94
|
elrab |
⊢ ( 𝐼 ∈ { 𝑙 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑙 ‘ 𝑗 ) ) ‘ 𝑘 ) } ↔ ( 𝐼 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
96 |
87 95
|
sylib |
⊢ ( 𝜑 → ( 𝐼 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
97 |
96
|
simprd |
⊢ ( 𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
98 |
60
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝐼 ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
99 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → 𝑘 ∈ 𝑋 ) |
100 |
98 99
|
fvovco |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
101 |
|
mptexg |
⊢ ( 𝑋 ∈ Fin → ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ V ) |
102 |
1 101
|
syl |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ V ) |
103 |
102
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ V ) |
104 |
72 103
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐵 ‘ 𝑗 ) = ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
105 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ∈ V ) |
106 |
104 105
|
fvmpt2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) = ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
107 |
106
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) ) |
108 |
|
mptexg |
⊢ ( 𝑋 ∈ Fin → ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ V ) |
109 |
1 108
|
syl |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ V ) |
110 |
109
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ V ) |
111 |
83 110
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑇 ‘ 𝑗 ) = ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
112 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ∈ V ) |
113 |
111 112
|
fvmpt2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) = ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
114 |
113
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) |
115 |
107 114
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
116 |
100 115
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
117 |
116
|
ixpeq2dva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
118 |
117
|
iuneq2dv |
⊢ ( 𝜑 → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
119 |
97 118
|
sseqtrd |
⊢ ( 𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
120 |
5
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝐿 = ( ℎ ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) ) ) |
121 |
|
coeq2 |
⊢ ( ℎ = ( 𝐼 ‘ 𝑗 ) → ( [,) ∘ ℎ ) = ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ) |
122 |
121
|
fveq1d |
⊢ ( ℎ = ( 𝐼 ‘ 𝑗 ) → ( ( [,) ∘ ℎ ) ‘ 𝑘 ) = ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
123 |
122
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ℎ = ( 𝐼 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ℎ ) ‘ 𝑘 ) = ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
124 |
123
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ℎ = ( 𝐼 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ℎ ) ‘ 𝑘 ) = ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
125 |
100
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ℎ = ( 𝐼 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ( 𝐼 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
126 |
115
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ℎ = ( 𝐼 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) [,) ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
127 |
124 125 126
|
3eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ℎ = ( 𝐼 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ ℎ ) ‘ 𝑘 ) = ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) |
128 |
127
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ℎ = ( 𝐼 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) = ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
129 |
128
|
prodeq2dv |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ℎ = ( 𝐼 ‘ 𝑗 ) ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ℎ ) ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
130 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑋 ∈ Fin ) |
131 |
8
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ℕ ∧ ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ V ) → ( 𝐵 ‘ 𝑗 ) = ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
132 |
57 103 131
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐵 ‘ 𝑗 ) = ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
133 |
132
|
feq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐵 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ↔ ( 𝑘 ∈ 𝑋 ↦ ( 1st ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) : 𝑋 ⟶ ℝ ) ) |
134 |
64 133
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐵 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
135 |
134
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
136 |
135 99
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ ) |
137 |
9
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ℕ ∧ ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ V ) → ( 𝑇 ‘ 𝑗 ) = ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
138 |
57 110 137
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑇 ‘ 𝑗 ) = ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
139 |
138
|
feq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑇 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ↔ ( 𝑘 ∈ 𝑋 ↦ ( 2nd ‘ ( ( 𝐼 ‘ 𝑗 ) ‘ 𝑘 ) ) ) : 𝑋 ⟶ ℝ ) ) |
140 |
77 139
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑇 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
141 |
140
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝑇 ‘ 𝑗 ) : 𝑋 ⟶ ℝ ) |
142 |
141 99
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ ) |
143 |
|
volicore |
⊢ ( ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ ∧ ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ∈ ℝ ) → ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ℝ ) |
144 |
136 142 143
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ℝ ) |
145 |
130 144
|
fprodrecl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∈ ℝ ) |
146 |
120 129 58 145
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) |
147 |
146
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) ) |
148 |
147
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) ) ) |
149 |
148
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) ) ) ) |
150 |
51
|
simprd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐼 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) ) |
151 |
149 150
|
eqbrtrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) ) |
152 |
86 119 151
|
jca31 |
⊢ ( 𝜑 → ( ( ( 𝐵 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ∧ 𝑇 : ℕ ⟶ ( ℝ ↑m 𝑋 ) ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐵 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝑇 ‘ 𝑗 ) ‘ 𝑘 ) ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) +𝑒 𝐸 ) ) ) |