Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovnhoilem2 Structured version   Visualization version   GIF version

Theorem ovnhoilem2 41137
Description: The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. Second part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
ovnhoilem2.x (𝜑𝑋 ∈ Fin)
ovnhoilem2.n (𝜑𝑋 ≠ ∅)
ovnhoilem2.a (𝜑𝐴:𝑋⟶ℝ)
ovnhoilem2.b (𝜑𝐵:𝑋⟶ℝ)
ovnhoilem2.i 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))
ovnhoilem2.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
ovnhoilem2.m 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}
ovnhoilem2.f 𝐹 = (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
ovnhoilem2.s 𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
Assertion
Ref Expression
ovnhoilem2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑖,𝑘,𝑧   𝐵,𝑎,𝑏,𝑖,𝑘,𝑧   𝑘,𝐹,𝑛   𝐼,𝑎,𝑏,𝑖,𝑛,𝑥,𝑧   𝐿,𝑎,𝑏,𝑖,𝑛,𝑥,𝑧   𝑖,𝑀,𝑧   𝑆,𝑘,𝑛   𝑋,𝑎,𝑏,𝑖,𝑗,𝑘,𝑙,𝑛   𝑥,𝑋,𝑧,𝑗,𝑘   𝜑,𝑎,𝑏,𝑖,𝑘,𝑙,𝑛   𝜑,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑗)   𝐴(𝑥,𝑗,𝑛,𝑙)   𝐵(𝑥,𝑗,𝑛,𝑙)   𝑆(𝑥,𝑧,𝑖,𝑗,𝑎,𝑏,𝑙)   𝐹(𝑥,𝑧,𝑖,𝑗,𝑎,𝑏,𝑙)   𝐼(𝑗,𝑘,𝑙)   𝐿(𝑗,𝑘,𝑙)   𝑀(𝑥,𝑗,𝑘,𝑛,𝑎,𝑏,𝑙)

Proof of Theorem ovnhoilem2
StepHypRef Expression
1 ovnhoilem2.m . . . . . . . . . 10 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}
21eleq2i 2722 . . . . . . . . 9 (𝑧𝑀𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
3 rabid 3145 . . . . . . . . 9 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ↔ (𝑧 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
42, 3bitri 264 . . . . . . . 8 (𝑧𝑀 ↔ (𝑧 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
54biimpi 206 . . . . . . 7 (𝑧𝑀 → (𝑧 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
65simprd 478 . . . . . 6 (𝑧𝑀 → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
76adantl 481 . . . . 5 ((𝜑𝑧𝑀) → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
8 ovnhoilem2.l . . . . . . . . . 10 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
9 ovnhoilem2.x . . . . . . . . . . 11 (𝜑𝑋 ∈ Fin)
1093ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝑋 ∈ Fin)
11 ovnhoilem2.a . . . . . . . . . . 11 (𝜑𝐴:𝑋⟶ℝ)
12113ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝐴:𝑋⟶ℝ)
13 ovnhoilem2.b . . . . . . . . . . 11 (𝜑𝐵:𝑋⟶ℝ)
14133ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝐵:𝑋⟶ℝ)
15 elmapi 7921 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
1615ffvelrnda 6399 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖𝑛) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
17 elmapi 7921 . . . . . . . . . . . . . . . . . 18 ((𝑖𝑛) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → (𝑖𝑛):𝑋⟶(ℝ × ℝ))
1816, 17syl 17 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖𝑛):𝑋⟶(ℝ × ℝ))
1918ffvelrnda 6399 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → ((𝑖𝑛)‘𝑙) ∈ (ℝ × ℝ))
20 xp1st 7242 . . . . . . . . . . . . . . . 16 (((𝑖𝑛)‘𝑙) ∈ (ℝ × ℝ) → (1st ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
2119, 20syl 17 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → (1st ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
22 eqid 2651 . . . . . . . . . . . . . . 15 (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))
2321, 22fmptd 6425 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
24 reex 10065 . . . . . . . . . . . . . . . 16 ℝ ∈ V
2524a1i 11 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ℝ ∈ V)
26 1nn 11069 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
2726a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 1 ∈ ℕ)
2815, 27ffvelrnd 6400 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑖‘1) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
29 elmapex 7920 . . . . . . . . . . . . . . . . . 18 ((𝑖‘1) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ V))
3029simprd 478 . . . . . . . . . . . . . . . . 17 ((𝑖‘1) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → 𝑋 ∈ V)
3128, 30syl 17 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑋 ∈ V)
3231adantr 480 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ V)
33 elmapg 7912 . . . . . . . . . . . . . . 15 ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
3425, 32, 33syl2anc 694 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
3523, 34mpbird 247 . . . . . . . . . . . . 13 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋))
36 eqid 2651 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
3735, 36fmptd 6425 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))):ℕ⟶(ℝ ↑𝑚 𝑋))
38 id 22 . . . . . . . . . . . . . 14 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
39 nnex 11064 . . . . . . . . . . . . . . . 16 ℕ ∈ V
4039mptex 6527 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) ∈ V
4140a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) ∈ V)
42 ovnhoilem2.f . . . . . . . . . . . . . . 15 𝐹 = (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
4342fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) ∈ V) → (𝐹𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
4438, 41, 43syl2anc 694 . . . . . . . . . . . . 13 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝐹𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
4544feq1d 6068 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → ((𝐹𝑖):ℕ⟶(ℝ ↑𝑚 𝑋) ↔ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))):ℕ⟶(ℝ ↑𝑚 𝑋)))
4637, 45mpbird 247 . . . . . . . . . . 11 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝐹𝑖):ℕ⟶(ℝ ↑𝑚 𝑋))
47463ad2ant2 1103 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → (𝐹𝑖):ℕ⟶(ℝ ↑𝑚 𝑋))
48 xp2nd 7243 . . . . . . . . . . . . . . . 16 (((𝑖𝑛)‘𝑙) ∈ (ℝ × ℝ) → (2nd ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
4919, 48syl 17 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → (2nd ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
50 eqid 2651 . . . . . . . . . . . . . . 15 (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))
5149, 50fmptd 6425 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
52 elmapg 7912 . . . . . . . . . . . . . . 15 ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
5325, 32, 52syl2anc 694 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
5451, 53mpbird 247 . . . . . . . . . . . . 13 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋))
55 eqid 2651 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
5654, 55fmptd 6425 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))):ℕ⟶(ℝ ↑𝑚 𝑋))
5739mptex 6527 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) ∈ V
5857a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) ∈ V)
59 ovnhoilem2.s . . . . . . . . . . . . . . 15 𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
6059fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) ∈ V) → (𝑆𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
6138, 58, 60syl2anc 694 . . . . . . . . . . . . 13 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑆𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
6261feq1d 6068 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → ((𝑆𝑖):ℕ⟶(ℝ ↑𝑚 𝑋) ↔ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))):ℕ⟶(ℝ ↑𝑚 𝑋)))
6356, 62mpbird 247 . . . . . . . . . . 11 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑆𝑖):ℕ⟶(ℝ ↑𝑚 𝑋))
64633ad2ant2 1103 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → (𝑆𝑖):ℕ⟶(ℝ ↑𝑚 𝑋))
65 simp3 1083 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
66 ovnhoilem2.i . . . . . . . . . . . . . 14 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))
6766a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
68 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑛 → (𝑖𝑗) = (𝑖𝑛))
6968fveq1d 6231 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑛 → ((𝑖𝑗)‘𝑘) = ((𝑖𝑛)‘𝑘))
7069fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → (1st ‘((𝑖𝑗)‘𝑘)) = (1st ‘((𝑖𝑛)‘𝑘)))
7169fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → (2nd ‘((𝑖𝑗)‘𝑘)) = (2nd ‘((𝑖𝑛)‘𝑘)))
7270, 71oveq12d 6708 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑛 → ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
7372ixpeq2dv 7966 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑛X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
7473cbviunv 4591 . . . . . . . . . . . . . . . . 17 𝑗 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = 𝑛 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘)))
7574a1i 11 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑗 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = 𝑛 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
7615ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
77 elmapi 7921 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → (𝑖𝑗):𝑋⟶(ℝ × ℝ))
7876, 77syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖𝑗):𝑋⟶(ℝ × ℝ))
7978adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝑖𝑗):𝑋⟶(ℝ × ℝ))
80 simpr 476 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
8179, 80fvovco 39695 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
8281ixpeq2dva 7965 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
8382iuneq2dv 4574 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
84 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
8540a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) ∈ V)
8684, 85, 43syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
8786fveq1d 6231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛))
88 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
89 mptexg 6525 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑋 ∈ V → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V)
9031, 89syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V)
9190adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V)
9236fvmpt2 6330 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℕ ∧ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9388, 91, 92syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9487, 93eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9594fveq1d 6231 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝐹𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘))
9695adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐹𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘))
97 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
98 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → 𝑙 = 𝑘)
9998fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → ((𝑖𝑛)‘𝑙) = ((𝑖𝑛)‘𝑘))
10099fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → (1st ‘((𝑖𝑛)‘𝑙)) = (1st ‘((𝑖𝑛)‘𝑘)))
101 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
102 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (1st ‘((𝑖𝑛)‘𝑘)) ∈ V)
10397, 100, 101, 102fvmptd 6327 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
104103adantlr 751 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
10596, 104eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐹𝑖)‘𝑛)‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
10661fveq1d 6231 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → ((𝑆𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛))
107106adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛))
108 mptexg 6525 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑋 ∈ V → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V)
10931, 108syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V)
110109adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V)
11155fvmpt2 6330 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℕ ∧ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
11288, 110, 111syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
113107, 112eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
114113fveq1d 6231 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝑆𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))
115114adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑆𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))
116 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
117 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = 𝑘 → ((𝑖𝑛)‘𝑙) = ((𝑖𝑛)‘𝑘))
118117fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑘)))
119118adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑘)))
120 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝑖𝑛)‘𝑘)) ∈ V)
121116, 119, 101, 120fvmptd 6327 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
122121adantlr 751 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
123115, 122eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑆𝑖)‘𝑛)‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
124105, 123oveq12d 6708 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)) = ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
125124ixpeq2dva 7965 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)) = X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
126125iuneq2dv 4574 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)) = 𝑛 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
12775, 83, 1263eqtr4d 2695 . . . . . . . . . . . . . . 15 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
128127adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
1291283adant3 1101 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
13067, 129sseq12d 3667 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘))))
13165, 130mpbid 222 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
1321313adant3r 1363 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
1338, 10, 12, 14, 47, 64, 132hoidmvle 41135 . . . . . . . . 9 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (((𝐹𝑖)‘𝑛)(𝐿𝑋)((𝑆𝑖)‘𝑛)))))
134 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛 = 𝑗𝑙𝑋) → 𝑛 = 𝑗)
135134fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 = 𝑗𝑙𝑋) → (𝑖𝑛) = (𝑖𝑗))
136135fveq1d 6231 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 = 𝑗𝑙𝑋) → ((𝑖𝑛)‘𝑙) = ((𝑖𝑗)‘𝑙))
137136fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 = 𝑗𝑙𝑋) → (1st ‘((𝑖𝑛)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑙)))
138137mpteq2dva 4777 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))))
139138fveq1d 6231 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘))
140139adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘))
141 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))))
142 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = 𝑘 → ((𝑖𝑗)‘𝑙) = ((𝑖𝑗)‘𝑘))
143142fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑘)))
144143adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘𝑋𝑙 = 𝑘) → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑘)))
145 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋𝑘𝑋)
146 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (1st ‘((𝑖𝑗)‘𝑘)) ∈ V)
147141, 144, 145, 146fvmptd 6327 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑋 → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
148147adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
149140, 148eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
150136fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 = 𝑗𝑙𝑋) → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑙)))
151150mpteq2dva 4777 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))))
152151fveq1d 6231 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘))
153152adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘))
154 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))))
155142fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑘)))
156155adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘𝑋𝑙 = 𝑘) → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑘)))
157 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (2nd ‘((𝑖𝑗)‘𝑘)) ∈ V)
158154, 156, 145, 157fvmptd 6327 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑋 → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
159158adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
160153, 159eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
161149, 160oveq12d 6708 . . . . . . . . . . . . . . . . . 18 ((𝑛 = 𝑗𝑘𝑋) → (((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘)) = ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
162161fveq2d 6233 . . . . . . . . . . . . . . . . 17 ((𝑛 = 𝑗𝑘𝑋) → (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))) = (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))))
163162prodeq2dv 14697 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))) = ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))))
164163cbvmptv 4783 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘)))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))))
165164a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘)))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))))
16681eqcomd 2657 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = (([,) ∘ (𝑖𝑗))‘𝑘))
167166fveq2d 6233 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
168167prodeq2dv 14697 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
169168mpteq2dva 4777 . . . . . . . . . . . . . 14 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
170165, 169eqtrd 2685 . . . . . . . . . . . . 13 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘)))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
171170fveq2d 6233 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
1721713ad2ant2 1103 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
17394adantll 750 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
174113adantll 750 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
175173, 174oveq12d 6708 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (((𝐹𝑖)‘𝑛)(𝐿𝑋)((𝑆𝑖)‘𝑛)) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))(𝐿𝑋)(𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
1769ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
177 ovnhoilem2.n . . . . . . . . . . . . . . . . 17 (𝜑𝑋 ≠ ∅)
178177ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
17919adantlll 754 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → ((𝑖𝑛)‘𝑙) ∈ (ℝ × ℝ))
180179, 20syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → (1st ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
181180, 22fmptd 6425 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
182179, 48syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → (2nd ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
183182, 50fmptd 6425 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
1848, 176, 178, 181, 183hoidmvn0val 41119 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))(𝐿𝑋)(𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) = ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))))
185175, 184eqtrd 2685 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (((𝐹𝑖)‘𝑛)(𝐿𝑋)((𝑆𝑖)‘𝑛)) = ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))))
186185mpteq2dva 4777 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) → (𝑛 ∈ ℕ ↦ (((𝐹𝑖)‘𝑛)(𝐿𝑋)((𝑆𝑖)‘𝑛))) = (𝑛 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘)))))
187186fveq2d 6233 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) → (Σ^‘(𝑛 ∈ ℕ ↦ (((𝐹𝑖)‘𝑛)(𝐿𝑋)((𝑆𝑖)‘𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))))))
1881873adant3 1101 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (Σ^‘(𝑛 ∈ ℕ ↦ (((𝐹𝑖)‘𝑛)(𝐿𝑋)((𝑆𝑖)‘𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))))))
189 simp3 1083 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
190172, 188, 1893eqtr4d 2695 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (Σ^‘(𝑛 ∈ ℕ ↦ (((𝐹𝑖)‘𝑛)(𝐿𝑋)((𝑆𝑖)‘𝑛)))) = 𝑧)
1911903adant3l 1362 . . . . . . . . 9 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → (Σ^‘(𝑛 ∈ ℕ ↦ (((𝐹𝑖)‘𝑛)(𝐿𝑋)((𝑆𝑖)‘𝑛)))) = 𝑧)
192133, 191breqtrd 4711 . . . . . . . 8 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → (𝐴(𝐿𝑋)𝐵) ≤ 𝑧)
1931923exp 1283 . . . . . . 7 (𝜑 → (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → ((𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (𝐴(𝐿𝑋)𝐵) ≤ 𝑧)))
194193adantr 480 . . . . . 6 ((𝜑𝑧𝑀) → (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → ((𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (𝐴(𝐿𝑋)𝐵) ≤ 𝑧)))
195194rexlimdv 3059 . . . . 5 ((𝜑𝑧𝑀) → (∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (𝐴(𝐿𝑋)𝐵) ≤ 𝑧))
1967, 195mpd 15 . . . 4 ((𝜑𝑧𝑀) → (𝐴(𝐿𝑋)𝐵) ≤ 𝑧)
197196ralrimiva 2995 . . 3 (𝜑 → ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧)
198 ssrab2 3720 . . . . . 6 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ⊆ ℝ*
1991, 198eqsstri 3668 . . . . 5 𝑀 ⊆ ℝ*
200199a1i 11 . . . 4 (𝜑𝑀 ⊆ ℝ*)
201 icossxr 12296 . . . . 5 (0[,)+∞) ⊆ ℝ*
2028, 9, 11, 13hoidmvcl 41117 . . . . 5 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))
203201, 202sseldi 3634 . . . 4 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ ℝ*)
204 infxrgelb 12203 . . . 4 ((𝑀 ⊆ ℝ* ∧ (𝐴(𝐿𝑋)𝐵) ∈ ℝ*) → ((𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ) ↔ ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧))
205200, 203, 204syl2anc 694 . . 3 (𝜑 → ((𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ) ↔ ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧))
206197, 205mpbird 247 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ))
20766a1i 11 . . . . 5 (𝜑𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
208 nfv 1883 . . . . . 6 𝑘𝜑
20911ffvelrnda 6399 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
21013ffvelrnda 6399 . . . . . . 7 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
211210rexrd 10127 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
212208, 209, 211hoissrrn2 41113 . . . . 5 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
213207, 212eqsstrd 3672 . . . 4 (𝜑𝐼 ⊆ (ℝ ↑𝑚 𝑋))
2149, 177, 213, 1ovnn0val 41086 . . 3 (𝜑 → ((voln*‘𝑋)‘𝐼) = inf(𝑀, ℝ*, < ))
215214eqcomd 2657 . 2 (𝜑 → inf(𝑀, ℝ*, < ) = ((voln*‘𝑋)‘𝐼))
216206, 215breqtrd 4711 1 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  c0 3948  ifcif 4119   ciun 4552   class class class wbr 4685  cmpt 4762   × cxp 5141  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Xcixp 7950  Fincfn 7997  infcinf 8388  cr 9973  0cc0 9974  1c1 9975  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cn 11058  [,)cico 12215  cprod 14679  volcvol 23278  Σ^csumge0 40897  voln*covoln 41071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-prod 14680  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-sumge0 40898  df-ovoln 41072
This theorem is referenced by:  ovnhoi  41138
  Copyright terms: Public domain W3C validator