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 39291
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 2674 . . . . . . . . 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 3089 . . . . . . . . 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 262 . . . . . . . 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 204 . . . . . . 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 477 . . . . . 6 (𝑧𝑀 → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
76adantl 480 . . . . 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 1074 . . . . . . . . . 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 1074 . . . . . . . . . 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 1074 . . . . . . . . . 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 7737 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
1615ffvelrnda 6247 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖𝑛) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
17 elmapi 7737 . . . . . . . . . . . . . . . . . 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 6247 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → ((𝑖𝑛)‘𝑙) ∈ (ℝ × ℝ))
20 xp1st 7061 . . . . . . . . . . . . . . . 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 2604 . . . . . . . . . . . . . . 15 (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))
2321, 22fmptd 6272 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
24 reex 9878 . . . . . . . . . . . . . . . 16 ℝ ∈ V
2524a1i 11 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ℝ ∈ V)
26 1nn 10873 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
2726a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 1 ∈ ℕ)
2815, 27ffvelrnd 6248 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑖‘1) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
29 elmapex 7736 . . . . . . . . . . . . . . . . . 18 ((𝑖‘1) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ V))
3029simprd 477 . . . . . . . . . . . . . . . . 17 ((𝑖‘1) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → 𝑋 ∈ V)
3128, 30syl 17 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑋 ∈ V)
3231adantr 479 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ V)
33 elmapg 7729 . . . . . . . . . . . . . . 15 ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
3425, 32, 33syl2anc 690 . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . 13 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋))
36 eqid 2604 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
3735, 36fmptd 6272 . . . . . . . . . . . 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 10868 . . . . . . . . . . . . . . . 16 ℕ ∈ V
4039mptex 6363 . . . . . . . . . . . . . . 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 6180 . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . 13 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝐹𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
4544feq1d 5924 . . . . . . . . . . . 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 245 . . . . . . . . . . 11 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝐹𝑖):ℕ⟶(ℝ ↑𝑚 𝑋))
47463ad2ant2 1075 . . . . . . . . . 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 7062 . . . . . . . . . . . . . . . 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 2604 . . . . . . . . . . . . . . 15 (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))
5149, 50fmptd 6272 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
52 elmapg 7729 . . . . . . . . . . . . . . 15 ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
5325, 32, 52syl2anc 690 . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . 13 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋))
55 eqid 2604 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
5654, 55fmptd 6272 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))):ℕ⟶(ℝ ↑𝑚 𝑋))
5739mptex 6363 . . . . . . . . . . . . . . 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 6180 . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . 13 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑆𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
6261feq1d 5924 . . . . . . . . . . . 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 245 . . . . . . . . . . 11 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑆𝑖):ℕ⟶(ℝ ↑𝑚 𝑋))
64633ad2ant2 1075 . . . . . . . . . 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 1055 . . . . . . . . . . . 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 6083 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑛 → (𝑖𝑗) = (𝑖𝑛))
6968fveq1d 6085 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑛 → ((𝑖𝑗)‘𝑘) = ((𝑖𝑛)‘𝑘))
7069fveq2d 6087 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → (1st ‘((𝑖𝑗)‘𝑘)) = (1st ‘((𝑖𝑛)‘𝑘)))
7169fveq2d 6087 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → (2nd ‘((𝑖𝑗)‘𝑘)) = (2nd ‘((𝑖𝑛)‘𝑘)))
7270, 71oveq12d 6540 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑛 → ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
7372ixpeq2dv 7782 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑛X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
7473cbviunv 4484 . . . . . . . . . . . . . . . . 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 6247 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
77 elmapi 7737 . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝑖𝑗):𝑋⟶(ℝ × ℝ))
80 simpr 475 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
8179, 80fvovco 38174 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
8281ixpeq2dva 7781 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
8382iuneq2dv 4467 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
84 simpl 471 . . . . . . . . . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
8786fveq1d 6085 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛))
88 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
89 mptexg 6362 . . . . . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V)
9236fvmpt2 6180 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℕ ∧ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9388, 91, 92syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9487, 93eqtrd 2638 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9594fveq1d 6085 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝐹𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘))
9695adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐹𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘))
97 eqidd 2605 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
98 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → 𝑙 = 𝑘)
9998fveq2d 6087 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → ((𝑖𝑛)‘𝑙) = ((𝑖𝑛)‘𝑘))
10099fveq2d 6087 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → (1st ‘((𝑖𝑛)‘𝑙)) = (1st ‘((𝑖𝑛)‘𝑘)))
101 simpr 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
102 fvex 6093 . . . . . . . . . . . . . . . . . . . . . . 23 (1st ‘((𝑖𝑛)‘𝑘)) ∈ V
103102a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (1st ‘((𝑖𝑛)‘𝑘)) ∈ V)
10497, 100, 101, 103fvmptd 6177 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
105104adantlr 746 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
10696, 105eqtrd 2638 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐹𝑖)‘𝑛)‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
10761fveq1d 6085 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → ((𝑆𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛))
108107adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛))
109 mptexg 6362 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑋 ∈ V → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V)
11031, 109syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V)
111110adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V)
11255fvmpt2 6180 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℕ ∧ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
11388, 111, 112syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
114108, 113eqtrd 2638 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
115114fveq1d 6085 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝑆𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))
116115adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑆𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))
117 eqidd 2605 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
118 fveq2 6083 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = 𝑘 → ((𝑖𝑛)‘𝑙) = ((𝑖𝑛)‘𝑘))
119118fveq2d 6087 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑘)))
120119adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑘)))
121 fvex 6093 . . . . . . . . . . . . . . . . . . . . . . 23 (2nd ‘((𝑖𝑛)‘𝑘)) ∈ V
122121a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝑖𝑛)‘𝑘)) ∈ V)
123117, 120, 101, 122fvmptd 6177 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
124123adantlr 746 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
125116, 124eqtrd 2638 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑆𝑖)‘𝑛)‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
126106, 125oveq12d 6540 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)) = ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
127126ixpeq2dva 7781 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)) = X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
128127iuneq2dv 4467 . . . . . . . . . . . . . . . 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))))
12975, 83, 1283eqtr4d 2648 . . . . . . . . . . . . . . 15 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
130129adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
1311303adant3 1073 . . . . . . . . . . . . 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)))
13267, 131sseq12d 3591 . . . . . . . . . . . 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))))
13365, 132mpbid 220 . . . . . . . . . . 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)))
1341333adant3r 1314 . . . . . . . . . 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)))
1358, 10, 12, 14, 47, 64, 134hoidmvle 39289 . . . . . . . . 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)))))
136 simpl 471 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛 = 𝑗𝑙𝑋) → 𝑛 = 𝑗)
137136fveq2d 6087 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 = 𝑗𝑙𝑋) → (𝑖𝑛) = (𝑖𝑗))
138137fveq1d 6085 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 = 𝑗𝑙𝑋) → ((𝑖𝑛)‘𝑙) = ((𝑖𝑗)‘𝑙))
139138fveq2d 6087 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 = 𝑗𝑙𝑋) → (1st ‘((𝑖𝑛)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑙)))
140139mpteq2dva 4661 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))))
141140fveq1d 6085 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘))
142141adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘))
143 eqidd 2605 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))))
144 fveq2 6083 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = 𝑘 → ((𝑖𝑗)‘𝑙) = ((𝑖𝑗)‘𝑘))
145144fveq2d 6087 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑘)))
146145adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘𝑋𝑙 = 𝑘) → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑘)))
147 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋𝑘𝑋)
148 fvex 6093 . . . . . . . . . . . . . . . . . . . . . . 23 (1st ‘((𝑖𝑗)‘𝑘)) ∈ V
149148a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (1st ‘((𝑖𝑗)‘𝑘)) ∈ V)
150143, 146, 147, 149fvmptd 6177 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑋 → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
151150adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
152142, 151eqtrd 2638 . . . . . . . . . . . . . . . . . . 19 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
153138fveq2d 6087 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 = 𝑗𝑙𝑋) → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑙)))
154153mpteq2dva 4661 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))))
155154fveq1d 6085 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘))
156155adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘))
157 eqidd 2605 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))))
158144fveq2d 6087 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑘)))
159158adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘𝑋𝑙 = 𝑘) → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑘)))
160 fvex 6093 . . . . . . . . . . . . . . . . . . . . . . 23 (2nd ‘((𝑖𝑗)‘𝑘)) ∈ V
161160a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (2nd ‘((𝑖𝑗)‘𝑘)) ∈ V)
162157, 159, 147, 161fvmptd 6177 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑋 → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
163162adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
164156, 163eqtrd 2638 . . . . . . . . . . . . . . . . . . 19 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
165152, 164oveq12d 6540 . . . . . . . . . . . . . . . . . 18 ((𝑛 = 𝑗𝑘𝑋) → (((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘)) = ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
166165fveq2d 6087 . . . . . . . . . . . . . . . . 17 ((𝑛 = 𝑗𝑘𝑋) → (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))) = (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))))
167166prodeq2dv 14433 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))) = ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))))
168167cbvmptv 4667 . . . . . . . . . . . . . . 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)))))
169168a1i 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))))))
17081eqcomd 2610 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = (([,) ∘ (𝑖𝑗))‘𝑘))
171170fveq2d 6087 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
172171prodeq2dv 14433 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
173172mpteq2dva 4661 . . . . . . . . . . . . . 14 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
174169, 173eqtrd 2638 . . . . . . . . . . . . 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))))
175174fveq2d 6087 . . . . . . . . . . . 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)))))
1761753ad2ant2 1075 . . . . . . . . . . 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)))))
17794adantll 745 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
178114adantll 745 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
179177, 178oveq12d 6540 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (((𝐹𝑖)‘𝑛)(𝐿𝑋)((𝑆𝑖)‘𝑛)) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))(𝐿𝑋)(𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
1809ad2antrr 757 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
181 ovnhoilem2.n . . . . . . . . . . . . . . . . 17 (𝜑𝑋 ≠ ∅)
182181ad2antrr 757 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
18319adantlll 749 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → ((𝑖𝑛)‘𝑙) ∈ (ℝ × ℝ))
184183, 20syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → (1st ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
185184, 22fmptd 6272 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
186183, 48syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → (2nd ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
187186, 50fmptd 6272 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
1888, 180, 182, 185, 187hoidmvn0val 39273 . . . . . . . . . . . . . . 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))))
189179, 188eqtrd 2638 . . . . . . . . . . . . . 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))))
190189mpteq2dva 4661 . . . . . . . . . . . . 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)))))
191190fveq2d 6087 . . . . . . . . . . . 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))))))
1921913adant3 1073 . . . . . . . . . . 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))))))
193 simp3 1055 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
194176, 192, 1933eqtr4d 2648 . . . . . . . . . 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)
1951943adant3l 1313 . . . . . . . . 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)
196135, 195breqtrd 4598 . . . . . . . 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)
1971963exp 1255 . . . . . . 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)))
198197adantr 479 . . . . . 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)))
199198rexlimdv 3006 . . . . 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))
2007, 199mpd 15 . . . 4 ((𝜑𝑧𝑀) → (𝐴(𝐿𝑋)𝐵) ≤ 𝑧)
201200ralrimiva 2943 . . 3 (𝜑 → ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧)
202 ssrab2 3644 . . . . . 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*
2031, 202eqsstri 3592 . . . . 5 𝑀 ⊆ ℝ*
204203a1i 11 . . . 4 (𝜑𝑀 ⊆ ℝ*)
205 icossxr 12080 . . . . 5 (0[,)+∞) ⊆ ℝ*
2068, 9, 11, 13hoidmvcl 39271 . . . . 5 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))
207205, 206sseldi 3560 . . . 4 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ ℝ*)
208 infxrgelb 11988 . . . 4 ((𝑀 ⊆ ℝ* ∧ (𝐴(𝐿𝑋)𝐵) ∈ ℝ*) → ((𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ) ↔ ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧))
209204, 207, 208syl2anc 690 . . 3 (𝜑 → ((𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ) ↔ ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧))
210201, 209mpbird 245 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ))
21166a1i 11 . . . . 5 (𝜑𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
212 nfv 1828 . . . . . 6 𝑘𝜑
21311ffvelrnda 6247 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
21413ffvelrnda 6247 . . . . . . 7 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
215214rexrd 9940 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
216212, 213, 215hoissrrn2 39267 . . . . 5 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
217211, 216eqsstrd 3596 . . . 4 (𝜑𝐼 ⊆ (ℝ ↑𝑚 𝑋))
2189, 181, 217, 1ovnn0val 39240 . . 3 (𝜑 → ((voln*‘𝑋)‘𝐼) = inf(𝑀, ℝ*, < ))
219218eqcomd 2610 . 2 (𝜑 → inf(𝑀, ℝ*, < ) = ((voln*‘𝑋)‘𝐼))
220210, 219breqtrd 4598 1 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774  wral 2890  wrex 2891  {crab 2894  Vcvv 3167  wss 3534  c0 3868  ifcif 4030   ciun 4444   class class class wbr 4572  cmpt 4632   × cxp 5021  ccom 5027  wf 5781  cfv 5785  (class class class)co 6522  cmpt2 6524  1st c1st 7029  2nd c2nd 7030  𝑚 cmap 7716  Xcixp 7766  Fincfn 7813  infcinf 8202  cr 9786  0cc0 9787  1c1 9788  +∞cpnf 9922  *cxr 9924   < clt 9925  cle 9926  cn 10862  [,)cico 11999  cprod 14415  volcvol 22951  Σ^csumge0 39054  voln*covoln 39225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864  ax-pre-sup 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-of 6767  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-2o 7420  df-oadd 7423  df-er 7601  df-map 7718  df-pm 7719  df-ixp 7767  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-fi 8172  df-sup 8203  df-inf 8204  df-oi 8270  df-card 8620  df-cda 8845  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-div 10529  df-nn 10863  df-2 10921  df-3 10922  df-n0 11135  df-z 11206  df-uz 11515  df-q 11616  df-rp 11660  df-xneg 11773  df-xadd 11774  df-xmul 11775  df-ioo 12001  df-ico 12003  df-icc 12004  df-fz 12148  df-fzo 12285  df-fl 12405  df-seq 12614  df-exp 12673  df-hash 12930  df-cj 13628  df-re 13629  df-im 13630  df-sqrt 13764  df-abs 13765  df-clim 14008  df-rlim 14009  df-sum 14206  df-prod 14416  df-rest 15847  df-topgen 15868  df-psmet 19500  df-xmet 19501  df-met 19502  df-bl 19503  df-mopn 19504  df-top 20458  df-bases 20459  df-topon 20460  df-cmp 20937  df-ovol 22952  df-vol 22953  df-sumge0 39055  df-ovoln 39226
This theorem is referenced by:  ovnhoi  39292
  Copyright terms: Public domain W3C validator