Metamath Proof Explorer


Theorem ovnsubaddlem1

Description: The Lebesgue outer measure is subadditive. Proposition 115D (a)(iv) of Fremlin1 p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020)

Ref Expression
Hypotheses ovnsubaddlem1.x ( 𝜑𝑋 ∈ Fin )
ovnsubaddlem1.n0 ( 𝜑𝑋 ≠ ∅ )
ovnsubaddlem1.a ( 𝜑𝐴 : ℕ ⟶ 𝒫 ( ℝ ↑m 𝑋 ) )
ovnsubaddlem1.e ( 𝜑𝐸 ∈ ℝ+ )
ovnsubaddlem1.z 𝑍 = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝑎 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } )
ovnsubaddlem1.c 𝐶 = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∣ 𝑎 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑗 ) ) ‘ 𝑘 ) } )
ovnsubaddlem1.l 𝐿 = ( 𝑖 ∈ ( ( ℝ × ℝ ) ↑m 𝑋 ) ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ 𝑖 ) ‘ 𝑘 ) ) )
ovnsubaddlem1.d 𝐷 = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ ( 𝑒 ∈ ℝ+ ↦ { 𝑖 ∈ ( 𝐶𝑎 ) ∣ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( 𝐿 ‘ ( 𝑖𝑗 ) ) ) ) ≤ ( ( ( voln* ‘ 𝑋 ) ‘ 𝑎 ) +𝑒 𝑒 ) } ) )
ovnsubaddlem1.i ( ( 𝜑𝑛 ∈ ℕ ) → ( 𝐼𝑛 ) ∈ ( ( 𝐷 ‘ ( 𝐴𝑛 ) ) ‘ ( 𝐸 / ( 2 ↑ 𝑛 ) ) ) )
ovnsubaddlem1.f ( 𝜑𝐹 : ℕ –1-1-onto→ ( ℕ × ℕ ) )
ovnsubaddlem1.g 𝐺 = ( 𝑚 ∈ ℕ ↦ ( ( 𝐼 ‘ ( 1st ‘ ( 𝐹𝑚 ) ) ) ‘ ( 2nd ‘ ( 𝐹𝑚 ) ) ) )
Assertion ovnsubaddlem1 ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ 𝑛 ∈ ℕ ( 𝐴𝑛 ) ) ≤ ( ( Σ^ ‘ ( 𝑛 ∈ ℕ ↦ ( ( voln* ‘ 𝑋 ) ‘ ( 𝐴𝑛 ) ) ) ) +𝑒 𝐸 ) )

Proof

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