Step |
Hyp |
Ref |
Expression |
1 |
|
ovnsubaddlem1.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
ovnsubaddlem1.n0 |
⊢ ( 𝜑 → 𝑋 ≠ ∅ ) |
3 |
|
ovnsubaddlem1.a |
⊢ ( 𝜑 → 𝐴 : ℕ ⟶ 𝒫 ( ℝ ↑m 𝑋 ) ) |
4 |
|
ovnsubaddlem1.e |
⊢ ( 𝜑 → 𝐸 ∈ ℝ+ ) |
5 |
|
ovnsubaddlem1.z |
⊢ 𝑍 = ( 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 ) ) ) ) ) } ) |
6 |
|
ovnsubaddlem1.c |
⊢ 𝐶 = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
7 |
|
ovnsubaddlem1.l |
⊢ 𝐿 = ( 𝑖 ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ 𝑖 ) ‘ 𝑘 ) ) ) |
8 |
|
ovnsubaddlem1.d |
⊢ 𝐷 = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) } ) ) |
9 |
|
ovnsubaddlem1.i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐼 ‘ 𝑛 ) ∈ ( ( 𝐷 ‘ ( 𝐴 ‘ 𝑛 ) ) ‘ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) |
10 |
|
ovnsubaddlem1.f |
⊢ ( 𝜑 → 𝐹 : ℕ –1-1-onto→ ( ℕ × ℕ ) ) |
11 |
|
ovnsubaddlem1.g |
⊢ 𝐺 = ( 𝑚 ∈ ℕ ↦ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
12 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐴 : ℕ ⟶ 𝒫 ( ℝ ↑m 𝑋 ) ) |
13 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ ) |
14 |
12 13
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ∈ 𝒫 ( ℝ ↑m 𝑋 ) ) |
15 |
|
elpwi |
⊢ ( ( 𝐴 ‘ 𝑛 ) ∈ 𝒫 ( ℝ ↑m 𝑋 ) → ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
16 |
14 15
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
17 |
16
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
18 |
|
iunss |
⊢ ( ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ↔ ∀ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
19 |
17 18
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ( ℝ ↑m 𝑋 ) ) |
20 |
1 19
|
ovnxrcl |
⊢ ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ∈ ℝ* ) |
21 |
|
nfv |
⊢ Ⅎ 𝑚 𝜑 |
22 |
|
nnex |
⊢ ℕ ∈ V |
23 |
22
|
a1i |
⊢ ( 𝜑 → ℕ ∈ V ) |
24 |
|
icossicc |
⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) |
25 |
|
nfv |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑚 ∈ ℕ ) |
26 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝜑 ) |
27 |
26 1
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝑋 ∈ Fin ) |
28 |
|
f1of |
⊢ ( 𝐹 : ℕ –1-1-onto→ ( ℕ × ℕ ) → 𝐹 : ℕ ⟶ ( ℕ × ℕ ) ) |
29 |
10 28
|
syl |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ( ℕ × ℕ ) ) |
30 |
29
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝐹 : ℕ ⟶ ( ℕ × ℕ ) ) |
31 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℕ ) |
32 |
30 31
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐹 ‘ 𝑚 ) ∈ ( ℕ × ℕ ) ) |
33 |
|
xp1st |
⊢ ( ( 𝐹 ‘ 𝑚 ) ∈ ( ℕ × ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) |
34 |
32 33
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) |
35 |
|
xp2nd |
⊢ ( ( 𝐹 ‘ 𝑚 ) ∈ ( ℕ × ℕ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) |
36 |
32 35
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) |
37 |
|
fvex |
⊢ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ V |
38 |
|
eleq1 |
⊢ ( 𝑗 = ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) → ( 𝑗 ∈ ℕ ↔ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) ) |
39 |
38
|
3anbi3d |
⊢ ( 𝑗 = ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) ) ) |
40 |
|
fveq2 |
⊢ ( 𝑗 = ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) = ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
41 |
40
|
feq1d |
⊢ ( 𝑗 = ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ↔ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
42 |
39 41
|
imbi12d |
⊢ ( 𝑗 = ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ↔ ( ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) ) |
43 |
|
fvex |
⊢ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ V |
44 |
|
eleq1 |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( 𝑛 ∈ ℕ ↔ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) ) |
45 |
44
|
3anbi2d |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ∧ 𝑗 ∈ ℕ ) ) ) |
46 |
|
fveq2 |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( 𝐼 ‘ 𝑛 ) = ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
47 |
46
|
fveq1d |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) = ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) ) |
48 |
47
|
feq1d |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ↔ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
49 |
45 48
|
imbi12d |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ↔ ( ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) ) |
50 |
|
sseq1 |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) ↔ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
51 |
50
|
rabbidv |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } = { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
52 |
|
ovex |
⊢ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∈ V |
53 |
52
|
rabex |
⊢ { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ∈ V |
54 |
53
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ∈ V ) |
55 |
6 51 14 54
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) = { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
56 |
|
ssrab2 |
⊢ { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ⊆ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) |
57 |
56
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ⊆ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
58 |
55 57
|
eqsstrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ⊆ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
59 |
|
fveq2 |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( 𝐶 ‘ 𝑎 ) = ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) |
60 |
59
|
eleq2d |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ↔ 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) |
61 |
|
fveq2 |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) = ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) |
62 |
61
|
oveq1d |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) = ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) ) |
63 |
62
|
breq2d |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) ) ) |
64 |
60 63
|
anbi12d |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( ( 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) ) ↔ ( 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) ) ) ) |
65 |
64
|
rabbidva2 |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → { 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) } = { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) } ) |
66 |
65
|
mpteq2dv |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) } ) = ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) } ) ) |
67 |
|
rpex |
⊢ ℝ+ ∈ V |
68 |
67
|
mptex |
⊢ ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) } ) ∈ V |
69 |
68
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) } ) ∈ V ) |
70 |
8 66 14 69
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐷 ‘ ( 𝐴 ‘ 𝑛 ) ) = ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) } ) ) |
71 |
|
oveq2 |
⊢ ( 𝑒 = ( 𝐸 / ( 2 ↑ 𝑛 ) ) → ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) = ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) |
72 |
71
|
breq2d |
⊢ ( 𝑒 = ( 𝐸 / ( 2 ↑ 𝑛 ) ) → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) |
73 |
72
|
rabbidv |
⊢ ( 𝑒 = ( 𝐸 / ( 2 ↑ 𝑛 ) ) → { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) } = { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) } ) |
74 |
73
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑒 = ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) → { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 𝑒 ) } = { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) } ) |
75 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐸 ∈ ℝ+ ) |
76 |
|
2nn |
⊢ 2 ∈ ℕ |
77 |
76
|
a1i |
⊢ ( 𝑛 ∈ ℕ → 2 ∈ ℕ ) |
78 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
79 |
77 78
|
nnexpcld |
⊢ ( 𝑛 ∈ ℕ → ( 2 ↑ 𝑛 ) ∈ ℕ ) |
80 |
79
|
nnrpd |
⊢ ( 𝑛 ∈ ℕ → ( 2 ↑ 𝑛 ) ∈ ℝ+ ) |
81 |
80
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2 ↑ 𝑛 ) ∈ ℝ+ ) |
82 |
75 81
|
rpdivcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐸 / ( 2 ↑ 𝑛 ) ) ∈ ℝ+ ) |
83 |
|
fvex |
⊢ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∈ V |
84 |
83
|
rabex |
⊢ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) } ∈ V |
85 |
84
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) } ∈ V ) |
86 |
70 74 82 85
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝐷 ‘ ( 𝐴 ‘ 𝑛 ) ) ‘ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) = { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) } ) |
87 |
|
ssrab2 |
⊢ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) } ⊆ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) |
88 |
87
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) } ⊆ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) |
89 |
86 88
|
eqsstrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝐷 ‘ ( 𝐴 ‘ 𝑛 ) ) ‘ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ⊆ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) |
90 |
89 9
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐼 ‘ 𝑛 ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) |
91 |
58 90
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐼 ‘ 𝑛 ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
92 |
|
elmapfn |
⊢ ( ( 𝐼 ‘ 𝑛 ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) → ( 𝐼 ‘ 𝑛 ) Fn ℕ ) |
93 |
91 92
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐼 ‘ 𝑛 ) Fn ℕ ) |
94 |
|
elmapi |
⊢ ( ( 𝐼 ‘ 𝑛 ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) → ( 𝐼 ‘ 𝑛 ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
95 |
91 94
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐼 ‘ 𝑛 ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
96 |
95
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
97 |
96
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ∀ 𝑗 ∈ ℕ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
98 |
93 97
|
jca |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝐼 ‘ 𝑛 ) Fn ℕ ∧ ∀ 𝑗 ∈ ℕ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) ) |
99 |
98
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ 𝑛 ) Fn ℕ ∧ ∀ 𝑗 ∈ ℕ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) ) |
100 |
|
ffnfv |
⊢ ( ( 𝐼 ‘ 𝑛 ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↔ ( ( 𝐼 ‘ 𝑛 ) Fn ℕ ∧ ∀ 𝑗 ∈ ℕ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) ) |
101 |
99 100
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( 𝐼 ‘ 𝑛 ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
102 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ ) |
103 |
101 102
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
104 |
|
elmapi |
⊢ ( ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) → ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
105 |
103 104
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
106 |
43 49 105
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
107 |
37 42 106
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
108 |
26 34 36 107
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
109 |
|
id |
⊢ ( 𝑚 ∈ ℕ → 𝑚 ∈ ℕ ) |
110 |
|
fvex |
⊢ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ V |
111 |
110
|
a1i |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ V ) |
112 |
11
|
fvmpt2 |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ V ) → ( 𝐺 ‘ 𝑚 ) = ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
113 |
109 111 112
|
syl2anc |
⊢ ( 𝑚 ∈ ℕ → ( 𝐺 ‘ 𝑚 ) = ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
114 |
113
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐺 ‘ 𝑚 ) = ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
115 |
114
|
feq1d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝐺 ‘ 𝑚 ) : 𝑋 ⟶ ( ℝ × ℝ ) ↔ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
116 |
108 115
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐺 ‘ 𝑚 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
117 |
25 27 7 116
|
hoiprodcl2 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐿 ‘ ( 𝐺 ‘ 𝑚 ) ) ∈ ( 0 [,) +∞ ) ) |
118 |
24 117
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐿 ‘ ( 𝐺 ‘ 𝑚 ) ) ∈ ( 0 [,] +∞ ) ) |
119 |
21 23 118
|
sge0xrclmpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑚 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐺 ‘ 𝑚 ) ) ) ) ∈ ℝ* ) |
120 |
|
nfv |
⊢ Ⅎ 𝑛 𝜑 |
121 |
|
0xr |
⊢ 0 ∈ ℝ* |
122 |
121
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 0 ∈ ℝ* ) |
123 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
124 |
123
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → +∞ ∈ ℝ* ) |
125 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑋 ∈ Fin ) |
126 |
125 16 5
|
ovnval2b |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) = if ( 𝑋 = ∅ , 0 , inf ( ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) , ℝ* , < ) ) ) |
127 |
2
|
neneqd |
⊢ ( 𝜑 → ¬ 𝑋 = ∅ ) |
128 |
127
|
iffalsed |
⊢ ( 𝜑 → if ( 𝑋 = ∅ , 0 , inf ( ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) , ℝ* , < ) ) = inf ( ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) , ℝ* , < ) ) |
129 |
128
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → if ( 𝑋 = ∅ , 0 , inf ( ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) , ℝ* , < ) ) = inf ( ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) , ℝ* , < ) ) |
130 |
126 129
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) = inf ( ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) , ℝ* , < ) ) |
131 |
|
sseq1 |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ↔ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
132 |
131
|
anbi1d |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ↔ ( ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) ) |
133 |
132
|
rexbidv |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → ( ∃ 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 ) ) ) ) ) ↔ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) ) |
134 |
133
|
rabbidv |
⊢ ( 𝑎 = ( 𝐴 ‘ 𝑛 ) → { 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 ) ) ) ) ) } = { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ) |
135 |
|
xrex |
⊢ ℝ* ∈ V |
136 |
135
|
rabex |
⊢ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ∈ V |
137 |
136
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ∈ V ) |
138 |
5 134 14 137
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) = { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ) |
139 |
|
ssrab2 |
⊢ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ⊆ ℝ* |
140 |
139
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ⊆ ℝ* ) |
141 |
138 140
|
eqsstrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) ⊆ ℝ* ) |
142 |
|
infxrcl |
⊢ ( ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) ⊆ ℝ* → inf ( ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) , ℝ* , < ) ∈ ℝ* ) |
143 |
141 142
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → inf ( ( 𝑍 ‘ ( 𝐴 ‘ 𝑛 ) ) , ℝ* , < ) ∈ ℝ* ) |
144 |
130 143
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ∈ ℝ* ) |
145 |
4
|
rpred |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
146 |
145
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐸 ∈ ℝ ) |
147 |
|
2re |
⊢ 2 ∈ ℝ |
148 |
147
|
a1i |
⊢ ( 𝑛 ∈ ℕ → 2 ∈ ℝ ) |
149 |
148 78
|
reexpcld |
⊢ ( 𝑛 ∈ ℕ → ( 2 ↑ 𝑛 ) ∈ ℝ ) |
150 |
149
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2 ↑ 𝑛 ) ∈ ℝ ) |
151 |
148
|
recnd |
⊢ ( 𝑛 ∈ ℕ → 2 ∈ ℂ ) |
152 |
|
2ne0 |
⊢ 2 ≠ 0 |
153 |
152
|
a1i |
⊢ ( 𝑛 ∈ ℕ → 2 ≠ 0 ) |
154 |
|
nnz |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℤ ) |
155 |
151 153 154
|
expne0d |
⊢ ( 𝑛 ∈ ℕ → ( 2 ↑ 𝑛 ) ≠ 0 ) |
156 |
155
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2 ↑ 𝑛 ) ≠ 0 ) |
157 |
146 150 156
|
redivcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐸 / ( 2 ↑ 𝑛 ) ) ∈ ℝ ) |
158 |
157
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐸 / ( 2 ↑ 𝑛 ) ) ∈ ℝ* ) |
159 |
144 158
|
xaddcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ∈ ℝ* ) |
160 |
125 16
|
ovncl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ∈ ( 0 [,] +∞ ) ) |
161 |
|
xrge0ge0 |
⊢ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ∈ ( 0 [,] +∞ ) → 0 ≤ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) |
162 |
160 161
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 0 ≤ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) |
163 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 0 ∈ ℝ ) |
164 |
82
|
rpgt0d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 0 < ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) |
165 |
163 157 164
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 0 ≤ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) |
166 |
157
|
ltpnfd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐸 / ( 2 ↑ 𝑛 ) ) < +∞ ) |
167 |
158 124 166
|
xrltled |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐸 / ( 2 ↑ 𝑛 ) ) ≤ +∞ ) |
168 |
122 124 158 165 167
|
eliccxrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐸 / ( 2 ↑ 𝑛 ) ) ∈ ( 0 [,] +∞ ) ) |
169 |
144 168
|
xadd0ge |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) |
170 |
122 144 159 162 169
|
xrletrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 0 ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) |
171 |
|
pnfge |
⊢ ( ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ∈ ℝ* → ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ≤ +∞ ) |
172 |
159 171
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ≤ +∞ ) |
173 |
122 124 159 170 172
|
eliccxrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ∈ ( 0 [,] +∞ ) ) |
174 |
120 23 173
|
sge0xrclmpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) ∈ ℝ* ) |
175 |
|
sseq1 |
⊢ ( 𝑎 = ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) → ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) ↔ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
176 |
175
|
rabbidv |
⊢ ( 𝑎 = ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) → { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } = { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
177 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝐴 : ℕ ⟶ 𝒫 ( ℝ ↑m 𝑋 ) ) |
178 |
177 34
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ 𝒫 ( ℝ ↑m 𝑋 ) ) |
179 |
52
|
rabex |
⊢ { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ∈ V |
180 |
179
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ∈ V ) |
181 |
6 176 178 180
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) = { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
182 |
|
ssrab2 |
⊢ { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ⊆ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) |
183 |
182
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ⊆ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
184 |
181 183
|
eqsstrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ⊆ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
185 |
|
fveq2 |
⊢ ( 𝑎 = ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) → ( 𝐶 ‘ 𝑎 ) = ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) |
186 |
185
|
eleq2d |
⊢ ( 𝑎 = ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) → ( 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ↔ 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) |
187 |
|
fveq2 |
⊢ ( 𝑎 = ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) → ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) = ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) |
188 |
187
|
oveq1d |
⊢ ( 𝑎 = ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) → ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) = ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) ) |
189 |
188
|
breq2d |
⊢ ( 𝑎 = ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) ) ) |
190 |
186 189
|
anbi12d |
⊢ ( 𝑎 = ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) → ( ( 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) ) ↔ ( 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) ) ) ) |
191 |
190
|
rabbidva2 |
⊢ ( 𝑎 = ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) → { 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) } = { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) } ) |
192 |
191
|
mpteq2dv |
⊢ ( 𝑎 = ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) → ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ 𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) } ) = ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) } ) ) |
193 |
67
|
mptex |
⊢ ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) } ) ∈ V |
194 |
193
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) } ) ∈ V ) |
195 |
8 192 178 194
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐷 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) = ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) } ) ) |
196 |
|
oveq2 |
⊢ ( 𝑒 = ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) → ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) = ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) |
197 |
196
|
breq2d |
⊢ ( 𝑒 = ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) ) |
198 |
197
|
rabbidv |
⊢ ( 𝑒 = ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) → { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) } = { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) } ) |
199 |
198
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑒 = ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) → { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 𝑒 ) } = { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) } ) |
200 |
26 4
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 𝐸 ∈ ℝ+ ) |
201 |
|
2rp |
⊢ 2 ∈ ℝ+ |
202 |
201
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → 2 ∈ ℝ+ ) |
203 |
34
|
nnzd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℤ ) |
204 |
202 203
|
rpexpcld |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ ℝ+ ) |
205 |
200 204
|
rpdivcld |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∈ ℝ+ ) |
206 |
|
fvex |
⊢ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∈ V |
207 |
206
|
rabex |
⊢ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) } ∈ V |
208 |
207
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) } ∈ V ) |
209 |
195 199 205 208
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝐷 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ‘ ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) = { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) } ) |
210 |
|
ssrab2 |
⊢ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) } ⊆ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
211 |
210
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) +𝑒 ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) } ⊆ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) |
212 |
209 211
|
eqsstrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝐷 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ‘ ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ⊆ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) |
213 |
44
|
anbi2d |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ↔ ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) ) ) |
214 |
|
2fveq3 |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( 𝐷 ‘ ( 𝐴 ‘ 𝑛 ) ) = ( 𝐷 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) |
215 |
|
oveq2 |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( 2 ↑ 𝑛 ) = ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
216 |
215
|
oveq2d |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( 𝐸 / ( 2 ↑ 𝑛 ) ) = ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) |
217 |
214 216
|
fveq12d |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( 𝐷 ‘ ( 𝐴 ‘ 𝑛 ) ) ‘ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) = ( ( 𝐷 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ‘ ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) |
218 |
46 217
|
eleq12d |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( 𝐼 ‘ 𝑛 ) ∈ ( ( 𝐷 ‘ ( 𝐴 ‘ 𝑛 ) ) ‘ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ↔ ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ ( ( 𝐷 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ‘ ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) ) |
219 |
213 218
|
imbi12d |
⊢ ( 𝑛 = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) → ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐼 ‘ 𝑛 ) ∈ ( ( 𝐷 ‘ ( 𝐴 ‘ 𝑛 ) ) ‘ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ↔ ( ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ ( ( 𝐷 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ‘ ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) ) ) |
220 |
43 219 9
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ∈ ℕ ) → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ ( ( 𝐷 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ‘ ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) |
221 |
26 34 220
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ ( ( 𝐷 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ‘ ( 𝐸 / ( 2 ↑ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) |
222 |
212 221
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) |
223 |
184 222
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) |
224 |
|
elmapfn |
⊢ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) Fn ℕ ) |
225 |
223 224
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) Fn ℕ ) |
226 |
|
elmapi |
⊢ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
227 |
223 226
|
syl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
228 |
227
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
229 |
228
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ∀ 𝑗 ∈ ℕ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
230 |
225 229
|
jca |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) Fn ℕ ∧ ∀ 𝑗 ∈ ℕ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) ) |
231 |
|
ffnfv |
⊢ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↔ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) Fn ℕ ∧ ∀ 𝑗 ∈ ℕ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ 𝑗 ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) ) |
232 |
230 231
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
233 |
232 36
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
234 |
233 11
|
fmptd |
⊢ ( 𝜑 → 𝐺 : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) ) |
235 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝜑 ) |
236 |
9 86
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐼 ‘ 𝑛 ) ∈ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) } ) |
237 |
87 236
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐼 ‘ 𝑛 ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) |
238 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ ( 𝐼 ‘ 𝑛 ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) → ( 𝐼 ‘ 𝑛 ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) |
239 |
55
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ ( 𝐼 ‘ 𝑛 ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) → ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) = { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
240 |
238 239
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ ( 𝐼 ‘ 𝑛 ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) → ( 𝐼 ‘ 𝑛 ) ∈ { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ) |
241 |
|
fveq1 |
⊢ ( ℎ = ( 𝐼 ‘ 𝑛 ) → ( ℎ ‘ 𝑗 ) = ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) |
242 |
241
|
coeq2d |
⊢ ( ℎ = ( 𝐼 ‘ 𝑛 ) → ( [,) ∘ ( ℎ ‘ 𝑗 ) ) = ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) |
243 |
242
|
fveq1d |
⊢ ( ℎ = ( 𝐼 ‘ 𝑛 ) → ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
244 |
243
|
ixpeq2dv |
⊢ ( ℎ = ( 𝐼 ‘ 𝑛 ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
245 |
244
|
iuneq2d |
⊢ ( ℎ = ( 𝐼 ‘ 𝑛 ) → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) = ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
246 |
245
|
sseq2d |
⊢ ( ℎ = ( 𝐼 ‘ 𝑛 ) → ( ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) ↔ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
247 |
246
|
elrab |
⊢ ( ( 𝐼 ‘ 𝑛 ) ∈ { ℎ ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ℎ ‘ 𝑗 ) ) ‘ 𝑘 ) } ↔ ( ( 𝐼 ‘ 𝑛 ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
248 |
240 247
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ ( 𝐼 ‘ 𝑛 ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) → ( ( 𝐼 ‘ 𝑛 ) ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) |
249 |
248
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ ( 𝐼 ‘ 𝑛 ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ) → ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
250 |
235 13 237 249
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ) |
251 |
|
f1ofo |
⊢ ( 𝐹 : ℕ –1-1-onto→ ( ℕ × ℕ ) → 𝐹 : ℕ –onto→ ( ℕ × ℕ ) ) |
252 |
10 251
|
syl |
⊢ ( 𝜑 → 𝐹 : ℕ –onto→ ( ℕ × ℕ ) ) |
253 |
252
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → 𝐹 : ℕ –onto→ ( ℕ × ℕ ) ) |
254 |
|
opelxpi |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) → 〈 𝑛 , 𝑗 〉 ∈ ( ℕ × ℕ ) ) |
255 |
13 254
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → 〈 𝑛 , 𝑗 〉 ∈ ( ℕ × ℕ ) ) |
256 |
|
foelrni |
⊢ ( ( 𝐹 : ℕ –onto→ ( ℕ × ℕ ) ∧ 〈 𝑛 , 𝑗 〉 ∈ ( ℕ × ℕ ) ) → ∃ 𝑚 ∈ ℕ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) |
257 |
253 255 256
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → ∃ 𝑚 ∈ ℕ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) |
258 |
|
nfv |
⊢ Ⅎ 𝑚 ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) |
259 |
|
nfre1 |
⊢ Ⅎ 𝑚 ∃ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) |
260 |
|
simpl |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → 𝑚 ∈ ℕ ) |
261 |
|
fveq2 |
⊢ ( ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 → ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) = ( 1st ‘ 〈 𝑛 , 𝑗 〉 ) ) |
262 |
|
op1stg |
⊢ ( ( 𝑛 ∈ V ∧ 𝑗 ∈ V ) → ( 1st ‘ 〈 𝑛 , 𝑗 〉 ) = 𝑛 ) |
263 |
262
|
el2v |
⊢ ( 1st ‘ 〈 𝑛 , 𝑗 〉 ) = 𝑛 |
264 |
263
|
a1i |
⊢ ( ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 → ( 1st ‘ 〈 𝑛 , 𝑗 〉 ) = 𝑛 ) |
265 |
261 264
|
eqtrd |
⊢ ( ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 → ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) = 𝑛 ) |
266 |
265
|
adantl |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) = 𝑛 ) |
267 |
260 266
|
jca |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → ( 𝑚 ∈ ℕ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) = 𝑛 ) ) |
268 |
|
2fveq3 |
⊢ ( 𝑖 = 𝑚 → ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) |
269 |
268
|
eqeq1d |
⊢ ( 𝑖 = 𝑚 → ( ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 ↔ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) = 𝑛 ) ) |
270 |
269
|
elrab |
⊢ ( 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } ↔ ( 𝑚 ∈ ℕ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) = 𝑛 ) ) |
271 |
267 270
|
sylibr |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } ) |
272 |
271
|
3adant1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) ∧ 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } ) |
273 |
260 113
|
syl |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → ( 𝐺 ‘ 𝑚 ) = ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
274 |
265
|
fveq2d |
⊢ ( ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 → ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) = ( 𝐼 ‘ 𝑛 ) ) |
275 |
|
vex |
⊢ 𝑛 ∈ V |
276 |
|
vex |
⊢ 𝑗 ∈ V |
277 |
275 276
|
op2ndd |
⊢ ( ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 → ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) = 𝑗 ) |
278 |
274 277
|
fveq12d |
⊢ ( ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) = ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) |
279 |
278
|
adantl |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) = ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) |
280 |
273 279
|
eqtr2d |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) = ( 𝐺 ‘ 𝑚 ) ) |
281 |
280
|
coeq2d |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) = ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ) |
282 |
281
|
fveq1d |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
283 |
282
|
ixpeq2dv |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
284 |
|
eqimss |
⊢ ( X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
285 |
283 284
|
syl |
⊢ ( ( 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
286 |
285
|
3adant1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) ∧ 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
287 |
|
rspe |
⊢ ( ( 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } ∧ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) → ∃ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
288 |
272 286 287
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) ∧ 𝑚 ∈ ℕ ∧ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 ) → ∃ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
289 |
288
|
3exp |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → ( 𝑚 ∈ ℕ → ( ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 → ∃ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) ) ) |
290 |
258 259 289
|
rexlimd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → ( ∃ 𝑚 ∈ ℕ ( 𝐹 ‘ 𝑚 ) = 〈 𝑛 , 𝑗 〉 → ∃ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) ) |
291 |
257 290
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → ∃ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
292 |
291
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ∀ 𝑗 ∈ ℕ ∃ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
293 |
|
iunss2 |
⊢ ( ∀ 𝑗 ∈ ℕ ∃ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ ∪ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
294 |
292 293
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ‘ 𝑘 ) ⊆ ∪ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
295 |
250 294
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
296 |
|
ssrab2 |
⊢ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } ⊆ ℕ |
297 |
|
iunss1 |
⊢ ( { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } ⊆ ℕ → ∪ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ⊆ ∪ 𝑚 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
298 |
296 297
|
ax-mp |
⊢ ∪ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ⊆ ∪ 𝑚 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) |
299 |
298
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ∪ 𝑚 ∈ { 𝑖 ∈ ℕ ∣ ( 1st ‘ ( 𝐹 ‘ 𝑖 ) ) = 𝑛 } X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ⊆ ∪ 𝑚 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
300 |
295 299
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑚 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
301 |
300
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑚 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
302 |
|
iunss |
⊢ ( ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑚 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ↔ ∀ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑚 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
303 |
301 302
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ⊆ ∪ 𝑚 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝐺 ‘ 𝑚 ) ) ‘ 𝑘 ) ) |
304 |
1 2 7 234 303
|
ovnlecvr |
⊢ ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ≤ ( Σ^ ‘ ( 𝑚 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐺 ‘ 𝑚 ) ) ) ) ) |
305 |
114
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐿 ‘ ( 𝐺 ‘ 𝑚 ) ) = ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) |
306 |
305
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑚 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐺 ‘ 𝑚 ) ) ) = ( 𝑚 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) |
307 |
306
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑚 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐺 ‘ 𝑚 ) ) ) ) = ( Σ^ ‘ ( 𝑚 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) ) |
308 |
|
nfv |
⊢ Ⅎ 𝑝 𝜑 |
309 |
|
2fveq3 |
⊢ ( 𝑝 = ( 𝐹 ‘ 𝑚 ) → ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) = ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
310 |
|
fveq2 |
⊢ ( 𝑝 = ( 𝐹 ‘ 𝑚 ) → ( 2nd ‘ 𝑝 ) = ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) |
311 |
309 310
|
fveq12d |
⊢ ( 𝑝 = ( 𝐹 ‘ 𝑚 ) → ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) = ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) |
312 |
311
|
fveq2d |
⊢ ( 𝑝 = ( 𝐹 ‘ 𝑚 ) → ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) ) = ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) |
313 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℕ ) → ( 𝐹 ‘ 𝑚 ) = ( 𝐹 ‘ 𝑚 ) ) |
314 |
|
nfv |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑝 ∈ ( ℕ × ℕ ) ) |
315 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ℕ × ℕ ) ) → 𝑋 ∈ Fin ) |
316 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ℕ × ℕ ) ) → 𝜑 ) |
317 |
|
xp1st |
⊢ ( 𝑝 ∈ ( ℕ × ℕ ) → ( 1st ‘ 𝑝 ) ∈ ℕ ) |
318 |
317
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ℕ × ℕ ) ) → ( 1st ‘ 𝑝 ) ∈ ℕ ) |
319 |
|
xp2nd |
⊢ ( 𝑝 ∈ ( ℕ × ℕ ) → ( 2nd ‘ 𝑝 ) ∈ ℕ ) |
320 |
319
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ℕ × ℕ ) ) → ( 2nd ‘ 𝑝 ) ∈ ℕ ) |
321 |
|
fvex |
⊢ ( 2nd ‘ 𝑝 ) ∈ V |
322 |
|
eleq1 |
⊢ ( 𝑗 = ( 2nd ‘ 𝑝 ) → ( 𝑗 ∈ ℕ ↔ ( 2nd ‘ 𝑝 ) ∈ ℕ ) ) |
323 |
322
|
3anbi3d |
⊢ ( 𝑗 = ( 2nd ‘ 𝑝 ) → ( ( 𝜑 ∧ ( 1st ‘ 𝑝 ) ∈ ℕ ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ ( 1st ‘ 𝑝 ) ∈ ℕ ∧ ( 2nd ‘ 𝑝 ) ∈ ℕ ) ) ) |
324 |
|
fveq2 |
⊢ ( 𝑗 = ( 2nd ‘ 𝑝 ) → ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ 𝑗 ) = ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) ) |
325 |
324
|
feq1d |
⊢ ( 𝑗 = ( 2nd ‘ 𝑝 ) → ( ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ↔ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
326 |
323 325
|
imbi12d |
⊢ ( 𝑗 = ( 2nd ‘ 𝑝 ) → ( ( ( 𝜑 ∧ ( 1st ‘ 𝑝 ) ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ↔ ( ( 𝜑 ∧ ( 1st ‘ 𝑝 ) ∈ ℕ ∧ ( 2nd ‘ 𝑝 ) ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) ) |
327 |
|
fvex |
⊢ ( 1st ‘ 𝑝 ) ∈ V |
328 |
|
eleq1 |
⊢ ( 𝑛 = ( 1st ‘ 𝑝 ) → ( 𝑛 ∈ ℕ ↔ ( 1st ‘ 𝑝 ) ∈ ℕ ) ) |
329 |
328
|
3anbi2d |
⊢ ( 𝑛 = ( 1st ‘ 𝑝 ) → ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ ( 1st ‘ 𝑝 ) ∈ ℕ ∧ 𝑗 ∈ ℕ ) ) ) |
330 |
|
fveq2 |
⊢ ( 𝑛 = ( 1st ‘ 𝑝 ) → ( 𝐼 ‘ 𝑛 ) = ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ) |
331 |
330
|
fveq1d |
⊢ ( 𝑛 = ( 1st ‘ 𝑝 ) → ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) = ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ 𝑗 ) ) |
332 |
331
|
feq1d |
⊢ ( 𝑛 = ( 1st ‘ 𝑝 ) → ( ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ↔ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) |
333 |
329 332
|
imbi12d |
⊢ ( 𝑛 = ( 1st ‘ 𝑝 ) → ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ↔ ( ( 𝜑 ∧ ( 1st ‘ 𝑝 ) ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) ) ) |
334 |
327 333 105
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 1st ‘ 𝑝 ) ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
335 |
321 326 334
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 1st ‘ 𝑝 ) ∈ ℕ ∧ ( 2nd ‘ 𝑝 ) ∈ ℕ ) → ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
336 |
316 318 320 335
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ℕ × ℕ ) ) → ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
337 |
314 315 7 336
|
hoiprodcl2 |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ℕ × ℕ ) ) → ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) ) ∈ ( 0 [,) +∞ ) ) |
338 |
24 337
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑝 ∈ ( ℕ × ℕ ) ) → ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) ) ∈ ( 0 [,] +∞ ) ) |
339 |
308 21 312 23 10 313 338
|
sge0f1o |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑝 ∈ ( ℕ × ℕ ) ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) ) ) ) = ( Σ^ ‘ ( 𝑚 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹 ‘ 𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹 ‘ 𝑚 ) ) ) ) ) ) ) |
340 |
307 339
|
eqtr4d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑚 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐺 ‘ 𝑚 ) ) ) ) = ( Σ^ ‘ ( 𝑝 ∈ ( ℕ × ℕ ) ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) ) ) ) ) |
341 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
342 |
275 276
|
op1std |
⊢ ( 𝑝 = 〈 𝑛 , 𝑗 〉 → ( 1st ‘ 𝑝 ) = 𝑛 ) |
343 |
342
|
fveq2d |
⊢ ( 𝑝 = 〈 𝑛 , 𝑗 〉 → ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) = ( 𝐼 ‘ 𝑛 ) ) |
344 |
275 276
|
op2ndd |
⊢ ( 𝑝 = 〈 𝑛 , 𝑗 〉 → ( 2nd ‘ 𝑝 ) = 𝑗 ) |
345 |
343 344
|
fveq12d |
⊢ ( 𝑝 = 〈 𝑛 , 𝑗 〉 → ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) |
346 |
345
|
fveq2d |
⊢ ( 𝑝 = 〈 𝑛 , 𝑗 〉 → ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) ) = ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) |
347 |
|
nfv |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) |
348 |
125
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → 𝑋 ∈ Fin ) |
349 |
96 104
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) : 𝑋 ⟶ ( ℝ × ℝ ) ) |
350 |
347 348 7 349
|
hoiprodcl2 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ∈ ( 0 [,) +∞ ) ) |
351 |
24 350
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑗 ∈ ℕ ) → ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ∈ ( 0 [,] +∞ ) ) |
352 |
351
|
3impa |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ ) → ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ∈ ( 0 [,] +∞ ) ) |
353 |
341 346 23 23 352
|
sge0xp |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) ) ) ) = ( Σ^ ‘ ( 𝑝 ∈ ( ℕ × ℕ ) ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) ) ) ) ) |
354 |
353
|
eqcomd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑝 ∈ ( ℕ × ℕ ) ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) ) ) ) = ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) ) ) ) ) |
355 |
22
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ℕ ∈ V ) |
356 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) |
357 |
351 356
|
fmptd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) : ℕ ⟶ ( 0 [,] +∞ ) ) |
358 |
355 357
|
sge0cl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) ) ∈ ( 0 [,] +∞ ) ) |
359 |
|
fveq1 |
⊢ ( 𝑖 = ( 𝐼 ‘ 𝑛 ) → ( 𝑖 ‘ 𝑗 ) = ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) |
360 |
359
|
fveq2d |
⊢ ( 𝑖 = ( 𝐼 ‘ 𝑛 ) → ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) = ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) |
361 |
360
|
mpteq2dv |
⊢ ( 𝑖 = ( 𝐼 ‘ 𝑛 ) → ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) ) |
362 |
361
|
fveq2d |
⊢ ( 𝑖 = ( 𝐼 ‘ 𝑛 ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) ) ) |
363 |
362
|
breq1d |
⊢ ( 𝑖 = ( 𝐼 ‘ 𝑛 ) → ( ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ↔ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) |
364 |
363
|
elrab |
⊢ ( ( 𝐼 ‘ 𝑛 ) ∈ { 𝑖 ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖 ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) } ↔ ( ( 𝐼 ‘ 𝑛 ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) |
365 |
236 364
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝐼 ‘ 𝑛 ) ∈ ( 𝐶 ‘ ( 𝐴 ‘ 𝑛 ) ) ∧ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) |
366 |
365
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) |
367 |
120 23 358 173 366
|
sge0lempt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ 𝑛 ) ‘ 𝑗 ) ) ) ) ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) ) |
368 |
354 367
|
eqbrtrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑝 ∈ ( ℕ × ℕ ) ↦ ( 𝐿 ‘ ( ( 𝐼 ‘ ( 1st ‘ 𝑝 ) ) ‘ ( 2nd ‘ 𝑝 ) ) ) ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) ) |
369 |
340 368
|
eqbrtrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑚 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝐺 ‘ 𝑚 ) ) ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) ) |
370 |
20 119 174 304 369
|
xrletrd |
⊢ ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) ) |
371 |
120 23 160 168
|
sge0xadd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) +𝑒 ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) ) |
372 |
121
|
a1i |
⊢ ( 𝜑 → 0 ∈ ℝ* ) |
373 |
123
|
a1i |
⊢ ( 𝜑 → +∞ ∈ ℝ* ) |
374 |
145
|
rexrd |
⊢ ( 𝜑 → 𝐸 ∈ ℝ* ) |
375 |
4
|
rpge0d |
⊢ ( 𝜑 → 0 ≤ 𝐸 ) |
376 |
145
|
ltpnfd |
⊢ ( 𝜑 → 𝐸 < +∞ ) |
377 |
372 373 374 375 376
|
elicod |
⊢ ( 𝜑 → 𝐸 ∈ ( 0 [,) +∞ ) ) |
378 |
377
|
sge0ad2en |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) = 𝐸 ) |
379 |
378
|
oveq2d |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) +𝑒 ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) +𝑒 𝐸 ) ) |
380 |
371 379
|
eqtrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) +𝑒 ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) ) ) = ( ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) +𝑒 𝐸 ) ) |
381 |
370 380
|
breqtrd |
⊢ ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ ∪ 𝑛 ∈ ℕ ( 𝐴 ‘ 𝑛 ) ) ≤ ( ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴 ‘ 𝑛 ) ) ) ) +𝑒 𝐸 ) ) |