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 45828
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 2817 . . . . . . . . 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 3444 . . . . . . . . 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 275 . . . . . . . 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 215 . . . . . . 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 495 . . . . . 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 1130 . . . . . . . . . 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 1130 . . . . . . . . . 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 1130 . . . . . . . . . 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 8840 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
1615ffvelcdmda 7077 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖𝑛) ∈ ((ℝ × ℝ) ↑m 𝑋))
17 elmapi 8840 . . . . . . . . . . . . . . . . . 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))
1918ffvelcdmda 7077 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → ((𝑖𝑛)‘𝑙) ∈ (ℝ × ℝ))
20 xp1st 8001 . . . . . . . . . . . . . . . 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)
2221fmpttd 7107 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
23 reex 11198 . . . . . . . . . . . . . . . 16 ℝ ∈ V
2423a1i 11 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ℝ ∈ V)
25 1nn 12221 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
2625a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 1 ∈ ℕ)
2715, 26ffvelcdmd 7078 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑖‘1) ∈ ((ℝ × ℝ) ↑m 𝑋))
28 elmapex 8839 . . . . . . . . . . . . . . . . . 18 ((𝑖‘1) ∈ ((ℝ × ℝ) ↑m 𝑋) → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ V))
2928simprd 495 . . . . . . . . . . . . . . . . 17 ((𝑖‘1) ∈ ((ℝ × ℝ) ↑m 𝑋) → 𝑋 ∈ V)
3027, 29syl 17 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑋 ∈ V)
3130adantr 480 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ V)
32 elmapg 8830 . . . . . . . . . . . . . . 15 ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋) ↔ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
3324, 31, 32syl2anc 583 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋) ↔ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
3422, 33mpbird 257 . . . . . . . . . . . . 13 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋))
3534fmpttd 7107 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))):ℕ⟶(ℝ ↑m 𝑋))
36 id 22 . . . . . . . . . . . . . 14 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
37 nnex 12216 . . . . . . . . . . . . . . . 16 ℕ ∈ V
3837mptex 7217 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) ∈ V
3938a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) ∈ V)
40 ovnhoilem2.f . . . . . . . . . . . . . . 15 𝐹 = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
4140fvmpt2 7000 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) ∈ V) → (𝐹𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
4236, 39, 41syl2anc 583 . . . . . . . . . . . . 13 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝐹𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
4342feq1d 6693 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ((𝐹𝑖):ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))):ℕ⟶(ℝ ↑m 𝑋)))
4435, 43mpbird 257 . . . . . . . . . . 11 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝐹𝑖):ℕ⟶(ℝ ↑m 𝑋))
45443ad2ant2 1131 . . . . . . . . . 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))
46 xp2nd 8002 . . . . . . . . . . . . . . . 16 (((𝑖𝑛)‘𝑙) ∈ (ℝ × ℝ) → (2nd ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
4719, 46syl 17 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → (2nd ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
4847fmpttd 7107 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
49 elmapg 8830 . . . . . . . . . . . . . . 15 ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋) ↔ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
5024, 31, 49syl2anc 583 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋) ↔ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
5148, 50mpbird 257 . . . . . . . . . . . . 13 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋))
5251fmpttd 7107 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))):ℕ⟶(ℝ ↑m 𝑋))
5337mptex 7217 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) ∈ V
5453a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) ∈ V)
55 ovnhoilem2.s . . . . . . . . . . . . . . 15 𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
5655fvmpt2 7000 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) ∈ V) → (𝑆𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
5736, 54, 56syl2anc 583 . . . . . . . . . . . . 13 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑆𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
5857feq1d 6693 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ((𝑆𝑖):ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))):ℕ⟶(ℝ ↑m 𝑋)))
5952, 58mpbird 257 . . . . . . . . . . 11 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑆𝑖):ℕ⟶(ℝ ↑m 𝑋))
60593ad2ant2 1131 . . . . . . . . . 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))
61 simp3 1135 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
62 ovnhoilem2.i . . . . . . . . . . . . . 14 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))
6362a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
64 fveq2 6882 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑛 → (𝑖𝑗) = (𝑖𝑛))
6564fveq1d 6884 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑛 → ((𝑖𝑗)‘𝑘) = ((𝑖𝑛)‘𝑘))
6665fveq2d 6886 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → (1st ‘((𝑖𝑗)‘𝑘)) = (1st ‘((𝑖𝑛)‘𝑘)))
6765fveq2d 6886 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → (2nd ‘((𝑖𝑗)‘𝑘)) = (2nd ‘((𝑖𝑛)‘𝑘)))
6866, 67oveq12d 7420 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑛 → ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
6968ixpeq2dv 8904 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑛X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
7069cbviunv 5034 . . . . . . . . . . . . . . . . 17 𝑗 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = 𝑛 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘)))
7170a1i 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))))
7215ffvelcdmda 7077 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
73 elmapi 8840 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝑖𝑗):𝑋⟶(ℝ × ℝ))
7472, 73syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖𝑗):𝑋⟶(ℝ × ℝ))
7574adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝑖𝑗):𝑋⟶(ℝ × ℝ))
76 simpr 484 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
7775, 76fvovco 44402 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
7877ixpeq2dva 8903 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
7978iuneq2dv 5012 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
80 simpl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
8138a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) ∈ V)
8280, 81, 41syl2anc 583 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
8382fveq1d 6884 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛))
84 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
85 mptexg 7215 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑋 ∈ V → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V)
8630, 85syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V)
8786adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V)
88 eqid 2724 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
8988fvmpt2 7000 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℕ ∧ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9084, 87, 89syl2anc 583 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9183, 90eqtrd 2764 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9291fveq1d 6884 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((𝐹𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘))
9392adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐹𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘))
94 eqidd 2725 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
95 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → 𝑙 = 𝑘)
9695fveq2d 6886 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → ((𝑖𝑛)‘𝑙) = ((𝑖𝑛)‘𝑘))
9796fveq2d 6886 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → (1st ‘((𝑖𝑛)‘𝑙)) = (1st ‘((𝑖𝑛)‘𝑘)))
98 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
99 fvexd 6897 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) → (1st ‘((𝑖𝑛)‘𝑘)) ∈ V)
10094, 97, 98, 99fvmptd 6996 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
101100adantlr 712 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
10293, 101eqtrd 2764 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐹𝑖)‘𝑛)‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
10357fveq1d 6884 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ((𝑆𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛))
104103adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛))
105 mptexg 7215 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑋 ∈ V → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V)
10630, 105syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V)
107106adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V)
108 eqid 2724 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
109108fvmpt2 7000 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℕ ∧ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
11084, 107, 109syl2anc 583 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
111104, 110eqtrd 2764 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
112111fveq1d 6884 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((𝑆𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))
113112adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑆𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))
114 eqidd 2725 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
115 2fveq3 6887 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑘)))
116115adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑘)))
117 fvexd 6897 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝑖𝑛)‘𝑘)) ∈ V)
118114, 116, 98, 117fvmptd 6996 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
119118adantlr 712 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
120113, 119eqtrd 2764 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑆𝑖)‘𝑛)‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
121102, 120oveq12d 7420 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)) = ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
122121ixpeq2dva 8903 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)) = X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
123122iuneq2dv 5012 . . . . . . . . . . . . . . . 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))))
12471, 79, 1233eqtr4d 2774 . . . . . . . . . . . . . . 15 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
125124adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
1261253adant3 1129 . . . . . . . . . . . . 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)))
12763, 126sseq12d 4008 . . . . . . . . . . . 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))))
12861, 127mpbid 231 . . . . . . . . . . 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)))
1291283adant3r 1178 . . . . . . . . . 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)))
1308, 10, 12, 14, 45, 60, 129hoidmvle 45826 . . . . . . . . 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)))))
131 simpl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛 = 𝑗𝑙𝑋) → 𝑛 = 𝑗)
132131fveq2d 6886 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 = 𝑗𝑙𝑋) → (𝑖𝑛) = (𝑖𝑗))
133132fveq1d 6884 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 = 𝑗𝑙𝑋) → ((𝑖𝑛)‘𝑙) = ((𝑖𝑗)‘𝑙))
134133fveq2d 6886 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 = 𝑗𝑙𝑋) → (1st ‘((𝑖𝑛)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑙)))
135134mpteq2dva 5239 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))))
136135fveq1d 6884 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘))
137136adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘))
138 eqidd 2725 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))))
139 2fveq3 6887 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑘)))
140139adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘𝑋𝑙 = 𝑘) → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑘)))
141 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋𝑘𝑋)
142 fvexd 6897 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (1st ‘((𝑖𝑗)‘𝑘)) ∈ V)
143138, 140, 141, 142fvmptd 6996 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑋 → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
144143adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
145137, 144eqtrd 2764 . . . . . . . . . . . . . . . . . . 19 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
146133fveq2d 6886 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 = 𝑗𝑙𝑋) → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑙)))
147146mpteq2dva 5239 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))))
148147fveq1d 6884 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘))
149148adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘))
150 eqidd 2725 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))))
151 2fveq3 6887 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑘)))
152151adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘𝑋𝑙 = 𝑘) → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑘)))
153 fvexd 6897 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (2nd ‘((𝑖𝑗)‘𝑘)) ∈ V)
154150, 152, 141, 153fvmptd 6996 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑋 → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
155154adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
156149, 155eqtrd 2764 . . . . . . . . . . . . . . . . . . 19 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
157145, 156oveq12d 7420 . . . . . . . . . . . . . . . . . 18 ((𝑛 = 𝑗𝑘𝑋) → (((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘)) = ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
158157fveq2d 6886 . . . . . . . . . . . . . . . . 17 ((𝑛 = 𝑗𝑘𝑋) → (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))) = (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))))
159158prodeq2dv 15865 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))) = ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))))
160159cbvmptv 5252 . . . . . . . . . . . . . . 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)))))
161160a1i 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))))))
16277eqcomd 2730 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = (([,) ∘ (𝑖𝑗))‘𝑘))
163162fveq2d 6886 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
164163prodeq2dv 15865 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
165164mpteq2dva 5239 . . . . . . . . . . . . . 14 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
166161, 165eqtrd 2764 . . . . . . . . . . . . 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))))
167166fveq2d 6886 . . . . . . . . . . . 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)))))
1681673ad2ant2 1131 . . . . . . . . . . 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)))))
16991adantll 711 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
170111adantll 711 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
171169, 170oveq12d 7420 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (((𝐹𝑖)‘𝑛)(𝐿𝑋)((𝑆𝑖)‘𝑛)) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))(𝐿𝑋)(𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
1729ad2antrr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
173 ovnhoilem2.n . . . . . . . . . . . . . . . . 17 (𝜑𝑋 ≠ ∅)
174173ad2antrr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
17519adantlll 715 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → ((𝑖𝑛)‘𝑙) ∈ (ℝ × ℝ))
176175, 20syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → (1st ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
177176fmpttd 7107 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
178175, 46syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → (2nd ‘((𝑖𝑛)‘𝑙)) ∈ ℝ)
179178fmpttd 7107 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
1808, 172, 174, 177, 179hoidmvn0val 45810 . . . . . . . . . . . . . . 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))))
181171, 180eqtrd 2764 . . . . . . . . . . . . . 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))))
182181mpteq2dva 5239 . . . . . . . . . . . . 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)))))
183182fveq2d 6886 . . . . . . . . . . . 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))))))
1841833adant3 1129 . . . . . . . . . . 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))))))
185 simp3 1135 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
186168, 184, 1853eqtr4d 2774 . . . . . . . . . 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)
1871863adant3l 1177 . . . . . . . . 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)
188130, 187breqtrd 5165 . . . . . . . 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)
1891883exp 1116 . . . . . . 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)))
190189adantr 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)))
191190rexlimdv 3145 . . . . 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))
1927, 191mpd 15 . . . 4 ((𝜑𝑧𝑀) → (𝐴(𝐿𝑋)𝐵) ≤ 𝑧)
193192ralrimiva 3138 . . 3 (𝜑 → ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧)
194 ssrab2 4070 . . . . . 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*
1951, 194eqsstri 4009 . . . . 5 𝑀 ⊆ ℝ*
196195a1i 11 . . . 4 (𝜑𝑀 ⊆ ℝ*)
197 icossxr 13407 . . . . 5 (0[,)+∞) ⊆ ℝ*
1988, 9, 11, 13hoidmvcl 45808 . . . . 5 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))
199197, 198sselid 3973 . . . 4 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ ℝ*)
200 infxrgelb 13312 . . . 4 ((𝑀 ⊆ ℝ* ∧ (𝐴(𝐿𝑋)𝐵) ∈ ℝ*) → ((𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ) ↔ ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧))
201196, 199, 200syl2anc 583 . . 3 (𝜑 → ((𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ) ↔ ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧))
202193, 201mpbird 257 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ))
20362a1i 11 . . . . 5 (𝜑𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
204 nfv 1909 . . . . . 6 𝑘𝜑
20511ffvelcdmda 7077 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
20613ffvelcdmda 7077 . . . . . . 7 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
207206rexrd 11262 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
208204, 205, 207hoissrrn2 45804 . . . . 5 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑m 𝑋))
209203, 208eqsstrd 4013 . . . 4 (𝜑𝐼 ⊆ (ℝ ↑m 𝑋))
2109, 173, 209, 1ovnn0val 45777 . . 3 (𝜑 → ((voln*‘𝑋)‘𝐼) = inf(𝑀, ℝ*, < ))
211210eqcomd 2730 . 2 (𝜑 → inf(𝑀, ℝ*, < ) = ((voln*‘𝑋)‘𝐼))
212202, 211breqtrd 5165 1 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2932  wral 3053  wrex 3062  {crab 3424  Vcvv 3466  wss 3941  c0 4315  ifcif 4521   ciun 4988   class class class wbr 5139  cmpt 5222   × cxp 5665  ccom 5671  wf 6530  cfv 6534  (class class class)co 7402  cmpo 7404  1st c1st 7967  2nd c2nd 7968  m cmap 8817  Xcixp 8888  Fincfn 8936  infcinf 9433  cr 11106  0cc0 11107  1c1 11108  +∞cpnf 11243  *cxr 11245   < clt 11246  cle 11247  cn 12210  [,)cico 13324  cprod 15847  volcvol 25316  Σ^csumge0 45588  voln*covoln 45762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8700  df-map 8819  df-pm 8820  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9502  df-dju 9893  df-card 9931  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-div 11870  df-nn 12211  df-2 12273  df-3 12274  df-n0 12471  df-z 12557  df-uz 12821  df-q 12931  df-rp 12973  df-xneg 13090  df-xadd 13091  df-xmul 13092  df-ioo 13326  df-ico 13328  df-icc 13329  df-fz 13483  df-fzo 13626  df-fl 13755  df-seq 13965  df-exp 14026  df-hash 14289  df-cj 15044  df-re 15045  df-im 15046  df-sqrt 15180  df-abs 15181  df-clim 15430  df-rlim 15431  df-sum 15631  df-prod 15848  df-rest 17369  df-topgen 17390  df-psmet 21222  df-xmet 21223  df-met 21224  df-bl 21225  df-mopn 21226  df-top 22720  df-topon 22737  df-bases 22773  df-cmp 23215  df-ovol 25317  df-vol 25318  df-sumge0 45589  df-ovoln 45763
This theorem is referenced by:  ovnhoi  45829
  Copyright terms: Public domain W3C validator