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Theorem ovnhoilem2 41610
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 2898 . . . . . . . . 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 3326 . . . . . . . . 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 267 . . . . . . . 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 208 . . . . . . 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 491 . . . . . 6 (𝑧𝑀 → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
76adantl 475 . . . . 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 1169 . . . . . . . . . 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 1169 . . . . . . . . . 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 1169 . . . . . . . . . 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 8144 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
1615ffvelrnda 6608 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖𝑛) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
17 elmapi 8144 . . . . . . . . . . . . . . . . . 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 6608 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙𝑋) → ((𝑖𝑛)‘𝑙) ∈ (ℝ × ℝ))
20 xp1st 7460 . . . . . . . . . . . . . . . 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 6634 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
23 reex 10343 . . . . . . . . . . . . . . . 16 ℝ ∈ V
2423a1i 11 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ℝ ∈ V)
25 1nn 11363 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
2625a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 1 ∈ ℕ)
2715, 26ffvelrnd 6609 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑖‘1) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
28 elmapex 8143 . . . . . . . . . . . . . . . . . 18 ((𝑖‘1) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ V))
2928simprd 491 . . . . . . . . . . . . . . . . 17 ((𝑖‘1) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → 𝑋 ∈ V)
3027, 29syl 17 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑋 ∈ V)
3130adantr 474 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ V)
32 elmapg 8135 . . . . . . . . . . . . . . 15 ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
3324, 31, 32syl2anc 581 . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . 13 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋))
3534fmpttd 6634 . . . . . . . . . . . 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 11357 . . . . . . . . . . . . . . . 16 ℕ ∈ V
3837mptex 6742 . . . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . 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 581 . . . . . . . . . . . . 13 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝐹𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
4342feq1d 6263 . . . . . . . . . . . 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 249 . . . . . . . . . . 11 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝐹𝑖):ℕ⟶(ℝ ↑𝑚 𝑋))
45443ad2ant2 1170 . . . . . . . . . 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 7461 . . . . . . . . . . . . . . . 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 6634 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
49 elmapg 8135 . . . . . . . . . . . . . . 15 ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ))
5024, 31, 49syl2anc 581 . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . 13 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ (ℝ ↑𝑚 𝑋))
5251fmpttd 6634 . . . . . . . . . . . 12 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))):ℕ⟶(ℝ ↑𝑚 𝑋))
5337mptex 6742 . . . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . 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 581 . . . . . . . . . . . . 13 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑆𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
5857feq1d 6263 . . . . . . . . . . . 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 249 . . . . . . . . . . 11 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑆𝑖):ℕ⟶(ℝ ↑𝑚 𝑋))
60593ad2ant2 1170 . . . . . . . . . 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 1174 . . . . . . . . . . . 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 6433 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑛 → (𝑖𝑗) = (𝑖𝑛))
6564fveq1d 6435 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑛 → ((𝑖𝑗)‘𝑘) = ((𝑖𝑛)‘𝑘))
6665fveq2d 6437 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → (1st ‘((𝑖𝑗)‘𝑘)) = (1st ‘((𝑖𝑛)‘𝑘)))
6765fveq2d 6437 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → (2nd ‘((𝑖𝑗)‘𝑘)) = (2nd ‘((𝑖𝑛)‘𝑘)))
6866, 67oveq12d 6923 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑛 → ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
6968ixpeq2dv 8191 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑛X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = X𝑘𝑋 ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
7069cbviunv 4779 . . . . . . . . . . . . . . . . 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))))
7215ffvelrnda 6608 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
73 elmapi 8144 . . . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝑖𝑗):𝑋⟶(ℝ × ℝ))
76 simpr 479 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
7775, 76fvovco 40189 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
7877ixpeq2dva 8190 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
7978iuneq2dv 4762 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
80 simpl 476 . . . . . . . . . . . . . . . . . . . . . . . . 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 581 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹𝑖) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
8382fveq1d 6435 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛))
84 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
85 mptexg 6740 . . . . . . . . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V)
88 eqid 2825 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
8988fvmpt2 6538 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℕ ∧ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9084, 87, 89syl2anc 581 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9183, 90eqtrd 2861 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
9291fveq1d 6435 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝐹𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘))
9392adantr 474 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐹𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘))
94 eqidd 2826 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
95 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → 𝑙 = 𝑘)
9695fveq2d 6437 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → ((𝑖𝑛)‘𝑙) = ((𝑖𝑛)‘𝑘))
9796fveq2d 6437 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → (1st ‘((𝑖𝑛)‘𝑙)) = (1st ‘((𝑖𝑛)‘𝑘)))
98 simpr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
99 fvexd 6448 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (1st ‘((𝑖𝑛)‘𝑘)) ∈ V)
10094, 97, 98, 99fvmptd 6535 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
101100adantlr 708 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
10293, 101eqtrd 2861 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐹𝑖)‘𝑛)‘𝑘) = (1st ‘((𝑖𝑛)‘𝑘)))
10357fveq1d 6435 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → ((𝑆𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛))
104103adantr 474 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛))
105 mptexg 6740 . . . . . . . . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V)
108 eqid 2825 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
109108fvmpt2 6538 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℕ ∧ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
11084, 107, 109syl2anc 581 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
111104, 110eqtrd 2861 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
112111fveq1d 6435 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝑆𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))
113112adantr 474 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑆𝑖)‘𝑛)‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))
114 eqidd 2826 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
115 2fveq3 6438 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑘)))
116115adantl 475 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑘)))
117 fvexd 6448 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝑖𝑛)‘𝑘)) ∈ V)
118114, 116, 98, 117fvmptd 6535 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
119118adantlr 708 . . . . . . . . . . . . . . . . . . . 20 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
120113, 119eqtrd 2861 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑆𝑖)‘𝑛)‘𝑘) = (2nd ‘((𝑖𝑛)‘𝑘)))
121102, 120oveq12d 6923 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)) = ((1st ‘((𝑖𝑛)‘𝑘))[,)(2nd ‘((𝑖𝑛)‘𝑘))))
122121ixpeq2dva 8190 . . . . . . . . . . . . . . . . 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 4762 . . . . . . . . . . . . . . . 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 2871 . . . . . . . . . . . . . . 15 (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
125124adantl 475 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑘𝑋 ((((𝐹𝑖)‘𝑛)‘𝑘)[,)(((𝑆𝑖)‘𝑛)‘𝑘)))
1261253adant3 1168 . . . . . . . . . . . . 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 3859 . . . . . . . . . . . 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 224 . . . . . . . . . . 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 1237 . . . . . . . . . 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 41608 . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛 = 𝑗𝑙𝑋) → 𝑛 = 𝑗)
132131fveq2d 6437 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 = 𝑗𝑙𝑋) → (𝑖𝑛) = (𝑖𝑗))
133132fveq1d 6435 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 = 𝑗𝑙𝑋) → ((𝑖𝑛)‘𝑙) = ((𝑖𝑗)‘𝑙))
134133fveq2d 6437 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 = 𝑗𝑙𝑋) → (1st ‘((𝑖𝑛)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑙)))
135134mpteq2dva 4967 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))))
136135fveq1d 6435 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘))
137136adantr 474 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘))
138 eqidd 2826 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))))
139 2fveq3 6438 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑘)))
140139adantl 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘𝑋𝑙 = 𝑘) → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑗)‘𝑘)))
141 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋𝑘𝑋)
142 fvexd 6448 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (1st ‘((𝑖𝑗)‘𝑘)) ∈ V)
143138, 140, 141, 142fvmptd 6535 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑋 → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
144143adantl 475 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
145137, 144eqtrd 2861 . . . . . . . . . . . . . . . . . . 19 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖𝑗)‘𝑘)))
146133fveq2d 6437 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 = 𝑗𝑙𝑋) → (2nd ‘((𝑖𝑛)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑙)))
147146mpteq2dva 4967 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))))
148147fveq1d 6435 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘))
149148adantr 474 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘))
150 eqidd 2826 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))))
151 2fveq3 6438 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑘 → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑘)))
152151adantl 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘𝑋𝑙 = 𝑘) → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑗)‘𝑘)))
153 fvexd 6448 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑋 → (2nd ‘((𝑖𝑗)‘𝑘)) ∈ V)
154150, 152, 141, 153fvmptd 6535 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑋 → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
155154adantl 475 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
156149, 155eqtrd 2861 . . . . . . . . . . . . . . . . . . 19 ((𝑛 = 𝑗𝑘𝑋) → ((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖𝑗)‘𝑘)))
157145, 156oveq12d 6923 . . . . . . . . . . . . . . . . . 18 ((𝑛 = 𝑗𝑘𝑋) → (((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘)) = ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))))
158157fveq2d 6437 . . . . . . . . . . . . . . . . 17 ((𝑛 = 𝑗𝑘𝑋) → (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))) = (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))))
159158prodeq2dv 15026 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → ∏𝑘𝑋 (vol‘(((𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))‘𝑘)[,)((𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))‘𝑘))) = ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))))
160159cbvmptv 4973 . . . . . . . . . . . . . . 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 2831 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘))) = (([,) ∘ (𝑖𝑗))‘𝑘))
163162fveq2d 6437 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
164163prodeq2dv 15026 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘((1st ‘((𝑖𝑗)‘𝑘))[,)(2nd ‘((𝑖𝑗)‘𝑘)))) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
165164mpteq2dva 4967 . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . 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 6437 . . . . . . . . . . . 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 1170 . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝐹𝑖)‘𝑛) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
170111adantll 707 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑆𝑖)‘𝑛) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
171169, 170oveq12d 6923 . . . . . . . . . . . . . . 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 719 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
173 ovnhoilem2.n . . . . . . . . . . . . . . . . 17 (𝜑𝑋 ≠ ∅)
174173ad2antrr 719 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
17519adantlll 711 . . . . . . . . . . . . . . . . . 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 6634 . . . . . . . . . . . . . . . 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 6634 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))):𝑋⟶ℝ)
1808, 172, 174, 177, 179hoidmvn0val 41592 . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . 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 4967 . . . . . . . . . . . . 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 6437 . . . . . . . . . . . 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 1168 . . . . . . . . . . 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 1174 . . . . . . . . . . 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 2871 . . . . . . . . . 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 1235 . . . . . . . . 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 4899 . . . . . . . 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 1154 . . . . . . 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 474 . . . . . 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 3239 . . . . 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 3175 . . 3 (𝜑 → ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧)
194 ssrab2 3912 . . . . . 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 3860 . . . . 5 𝑀 ⊆ ℝ*
196195a1i 11 . . . 4 (𝜑𝑀 ⊆ ℝ*)
197 icossxr 12546 . . . . 5 (0[,)+∞) ⊆ ℝ*
1988, 9, 11, 13hoidmvcl 41590 . . . . 5 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))
199197, 198sseldi 3825 . . . 4 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ ℝ*)
200 infxrgelb 12453 . . . 4 ((𝑀 ⊆ ℝ* ∧ (𝐴(𝐿𝑋)𝐵) ∈ ℝ*) → ((𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ) ↔ ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧))
201196, 199, 200syl2anc 581 . . 3 (𝜑 → ((𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ) ↔ ∀𝑧𝑀 (𝐴(𝐿𝑋)𝐵) ≤ 𝑧))
202193, 201mpbird 249 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ))
20362a1i 11 . . . . 5 (𝜑𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
204 nfv 2015 . . . . . 6 𝑘𝜑
20511ffvelrnda 6608 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
20613ffvelrnda 6608 . . . . . . 7 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
207206rexrd 10406 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
208204, 205, 207hoissrrn2 41586 . . . . 5 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
209203, 208eqsstrd 3864 . . . 4 (𝜑𝐼 ⊆ (ℝ ↑𝑚 𝑋))
2109, 173, 209, 1ovnn0val 41559 . . 3 (𝜑 → ((voln*‘𝑋)‘𝐼) = inf(𝑀, ℝ*, < ))
211210eqcomd 2831 . 2 (𝜑 → inf(𝑀, ℝ*, < ) = ((voln*‘𝑋)‘𝐼))
212202, 211breqtrd 4899 1 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wne 2999  wral 3117  wrex 3118  {crab 3121  Vcvv 3414  wss 3798  c0 4144  ifcif 4306   ciun 4740   class class class wbr 4873  cmpt 4952   × cxp 5340  ccom 5346  wf 6119  cfv 6123  (class class class)co 6905  cmpt2 6907  1st c1st 7426  2nd c2nd 7427  𝑚 cmap 8122  Xcixp 8175  Fincfn 8222  infcinf 8616  cr 10251  0cc0 10252  1c1 10253  +∞cpnf 10388  *cxr 10390   < clt 10391  cle 10392  cn 11350  [,)cico 12465  cprod 15008  volcvol 23629  Σ^csumge0 41370  voln*covoln 41544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-pre-sup 10330
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fi 8586  df-sup 8617  df-inf 8618  df-oi 8684  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-2 11414  df-3 11415  df-n0 11619  df-z 11705  df-uz 11969  df-q 12072  df-rp 12113  df-xneg 12232  df-xadd 12233  df-xmul 12234  df-ioo 12467  df-ico 12469  df-icc 12470  df-fz 12620  df-fzo 12761  df-fl 12888  df-seq 13096  df-exp 13155  df-hash 13411  df-cj 14216  df-re 14217  df-im 14218  df-sqrt 14352  df-abs 14353  df-clim 14596  df-rlim 14597  df-sum 14794  df-prod 15009  df-rest 16436  df-topgen 16457  df-psmet 20098  df-xmet 20099  df-met 20100  df-bl 20101  df-mopn 20102  df-top 21069  df-topon 21086  df-bases 21121  df-cmp 21561  df-ovol 23630  df-vol 23631  df-sumge0 41371  df-ovoln 41545
This theorem is referenced by:  ovnhoi  41611
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