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Theorem ovnhoi 44834
Description: The Lebesgue outer measure of a multidimensional half-open interval is its dimensional volume (the product of its length in each dimension, when the dimension is nonzero). Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
ovnhoi.x (𝜑𝑋 ∈ Fin)
ovnhoi.a (𝜑𝐴:𝑋⟶ℝ)
ovnhoi.b (𝜑𝐵:𝑋⟶ℝ)
ovnhoi.c 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))
ovnhoi.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
Assertion
Ref Expression
ovnhoi (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑘   𝐵,𝑎,𝑏,𝑘   𝑋,𝑎,𝑏,𝑘,𝑥   𝜑,𝑎,𝑏,𝑘,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐼(𝑥,𝑘,𝑎,𝑏)   𝐿(𝑥,𝑘,𝑎,𝑏)

Proof of Theorem ovnhoi
Dummy variables 𝑐 𝑑 𝑖 𝑗 𝑛 𝑧 𝑦 𝑤 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovnhoi.x . . 3 (𝜑𝑋 ∈ Fin)
2 ovnhoi.c . . . . 5 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))
32a1i 11 . . . 4 (𝜑𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
4 nfv 1917 . . . . 5 𝑘𝜑
5 ovnhoi.a . . . . . 6 (𝜑𝐴:𝑋⟶ℝ)
65ffvelcdmda 7035 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
7 ovnhoi.b . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ)
87ffvelcdmda 7035 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
98rexrd 11205 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
104, 6, 9hoissrrn2 44809 . . . 4 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑m 𝑋))
113, 10eqsstrd 3982 . . 3 (𝜑𝐼 ⊆ (ℝ ↑m 𝑋))
121, 11ovnxrcl 44800 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) ∈ ℝ*)
13 icossxr 13349 . . 3 (0[,)+∞) ⊆ ℝ*
14 ovnhoi.l . . . 4 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
1514, 1, 5, 7hoidmvcl 44813 . . 3 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))
1613, 15sselid 3942 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ ℝ*)
17 fveq2 6842 . . . . . . . 8 (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
1817fveq1d 6844 . . . . . . 7 (𝑋 = ∅ → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
1918adantl 482 . . . . . 6 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
20 ixpeq1 8846 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)))
21 ixp0x 8864 . . . . . . . . . . . 12 X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)) = {∅}
2221a1i 11 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
2320, 22eqtrd 2776 . . . . . . . . . 10 (𝑋 = ∅ → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
2423adantl 482 . . . . . . . . 9 ((𝜑𝑋 = ∅) → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
252a1i 11 . . . . . . . . 9 ((𝜑𝑋 = ∅) → 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
26 reex 11142 . . . . . . . . . . 11 ℝ ∈ V
27 mapdm0 8780 . . . . . . . . . . 11 (ℝ ∈ V → (ℝ ↑m ∅) = {∅})
2826, 27ax-mp 5 . . . . . . . . . 10 (ℝ ↑m ∅) = {∅}
2928a1i 11 . . . . . . . . 9 ((𝜑𝑋 = ∅) → (ℝ ↑m ∅) = {∅})
3024, 25, 293eqtr4d 2786 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐼 = (ℝ ↑m ∅))
31 eqimss 4000 . . . . . . . 8 (𝐼 = (ℝ ↑m ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
3230, 31syl 17 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
3332ovn0val 44781 . . . . . 6 ((𝜑𝑋 = ∅) → ((voln*‘∅)‘𝐼) = 0)
3419, 33eqtrd 2776 . . . . 5 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0)
35 0red 11158 . . . . 5 ((𝜑𝑋 = ∅) → 0 ∈ ℝ)
3634, 35eqeltrd 2838 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ)
37 eqidd 2737 . . . . 5 ((𝜑𝑋 = ∅) → 0 = 0)
38 fveq2 6842 . . . . . . . 8 (𝑋 = ∅ → (𝐿𝑋) = (𝐿‘∅))
3938oveqd 7374 . . . . . . 7 (𝑋 = ∅ → (𝐴(𝐿𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
4039adantl 482 . . . . . 6 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
415adantr 481 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
42 simpr 485 . . . . . . . . 9 ((𝜑𝑋 = ∅) → 𝑋 = ∅)
4342feq2d 6654 . . . . . . . 8 ((𝜑𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
4441, 43mpbid 231 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐴:∅⟶ℝ)
457adantr 481 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
4642feq2d 6654 . . . . . . . 8 ((𝜑𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
4745, 46mpbid 231 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐵:∅⟶ℝ)
4814, 44, 47hoidmv0val 44814 . . . . . 6 ((𝜑𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
4940, 48eqtrd 2776 . . . . 5 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = 0)
5037, 34, 493eqtr4d 2786 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
5136, 50eqled 11258 . . 3 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
52 eqid 2736 . . . . . 6 {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)))))}
53 eqeq1 2740 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 = 1 ↔ 𝑗 = 1))
5453ifbid 4509 . . . . . . . 8 (𝑛 = 𝑗 → if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
5554mpteq2dv 5207 . . . . . . 7 (𝑛 = 𝑗 → (𝑘𝑋 ↦ if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
5655cbvmptv 5218 . . . . . 6 (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))) = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
571, 5, 7, 2, 52, 56ovnhoilem1 44832 . . . . 5 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
5857adantr 481 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
591adantr 481 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
60 neqne 2951 . . . . . . 7 𝑋 = ∅ → 𝑋 ≠ ∅)
6160adantl 482 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
625adantr 481 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
637adantr 481 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
6414, 59, 61, 62, 63hoidmvn0val 44815 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
6564eqcomd 2742 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (𝐴(𝐿𝑋)𝐵))
6658, 65breqtrd 5131 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
6751, 66pm2.61dan 811 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
6849, 35eqeltrd 2838 . . . 4 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ∈ ℝ)
6950eqcomd 2742 . . . 4 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = ((voln*‘𝑋)‘𝐼))
7068, 69eqled 11258 . . 3 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
71 fveq1 6841 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝑎𝑘) = (𝑐𝑘))
7271fvoveq1d 7379 . . . . . . . . . . 11 (𝑎 = 𝑐 → (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = (vol‘((𝑐𝑘)[,)(𝑏𝑘))))
7372prodeq2ad 43823 . . . . . . . . . 10 (𝑎 = 𝑐 → ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘))))
7473ifeq2d 4506 . . . . . . . . 9 (𝑎 = 𝑐 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘)))))
75 fveq1 6841 . . . . . . . . . . . . 13 (𝑏 = 𝑑 → (𝑏𝑘) = (𝑑𝑘))
7675oveq2d 7373 . . . . . . . . . . . 12 (𝑏 = 𝑑 → ((𝑐𝑘)[,)(𝑏𝑘)) = ((𝑐𝑘)[,)(𝑑𝑘)))
7776fveq2d 6846 . . . . . . . . . . 11 (𝑏 = 𝑑 → (vol‘((𝑐𝑘)[,)(𝑏𝑘))) = (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
7877prodeq2ad 43823 . . . . . . . . . 10 (𝑏 = 𝑑 → ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
7978ifeq2d 4506 . . . . . . . . 9 (𝑏 = 𝑑 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8074, 79cbvmpov 7452 . . . . . . . 8 (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑥), 𝑑 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8180a1i 11 . . . . . . 7 (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑥), 𝑑 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))))
82 oveq2 7365 . . . . . . . 8 (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
83 eqeq1 2740 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
84 prodeq1 15792 . . . . . . . . 9 (𝑥 = 𝑦 → ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))) = ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
8583, 84ifbieq2d 4512 . . . . . . . 8 (𝑥 = 𝑦 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))) = if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8682, 82, 85mpoeq123dv 7432 . . . . . . 7 (𝑥 = 𝑦 → (𝑐 ∈ (ℝ ↑m 𝑥), 𝑑 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))))
8781, 86eqtrd 2776 . . . . . 6 (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))))
8887cbvmptv 5218 . . . . 5 (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))))) = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))))
8914, 88eqtri 2764 . . . 4 𝐿 = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))))
90 eqeq1 2740 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))))))
9190anbi2d 629 . . . . . . 7 (𝑤 = 𝑧 → ((𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ∧ 𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))))) ↔ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))))))
9291rexbidv 3175 . . . . . 6 (𝑤 = 𝑧 → (∃ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ∧ 𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))))) ↔ ∃ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))))))
93 simpl 483 . . . . . . . . . . . . . . 15 (( = 𝑖𝑗 ∈ ℕ) → = 𝑖)
9493fveq1d 6844 . . . . . . . . . . . . . 14 (( = 𝑖𝑗 ∈ ℕ) → (𝑗) = (𝑖𝑗))
9594coeq2d 5818 . . . . . . . . . . . . 13 (( = 𝑖𝑗 ∈ ℕ) → ([,) ∘ (𝑗)) = ([,) ∘ (𝑖𝑗)))
9695fveq1d 6844 . . . . . . . . . . . 12 (( = 𝑖𝑗 ∈ ℕ) → (([,) ∘ (𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
9796ixpeq2dv 8851 . . . . . . . . . . 11 (( = 𝑖𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
9897iuneq2dv 4978 . . . . . . . . . 10 ( = 𝑖 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
9998sseq2d 3976 . . . . . . . . 9 ( = 𝑖 → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ↔ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
100 simpl 483 . . . . . . . . . . . . . . . . 17 (( = 𝑖𝑘𝑋) → = 𝑖)
101100fveq1d 6844 . . . . . . . . . . . . . . . 16 (( = 𝑖𝑘𝑋) → (𝑗) = (𝑖𝑗))
102101coeq2d 5818 . . . . . . . . . . . . . . 15 (( = 𝑖𝑘𝑋) → ([,) ∘ (𝑗)) = ([,) ∘ (𝑖𝑗)))
103102fveq1d 6844 . . . . . . . . . . . . . 14 (( = 𝑖𝑘𝑋) → (([,) ∘ (𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
104103fveq2d 6846 . . . . . . . . . . . . 13 (( = 𝑖𝑘𝑋) → (vol‘(([,) ∘ (𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
105104prodeq2dv 15806 . . . . . . . . . . . 12 ( = 𝑖 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
106105mpteq2dv 5207 . . . . . . . . . . 11 ( = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
107106fveq2d 6846 . . . . . . . . . 10 ( = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
108107eqeq2d 2747 . . . . . . . . 9 ( = 𝑖 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
10999, 108anbi12d 631 . . . . . . . 8 ( = 𝑖 → ((𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))))) ↔ (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
110109cbvrexvw 3226 . . . . . . 7 (∃ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
111110a1i 11 . . . . . 6 (𝑤 = 𝑧 → (∃ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
11292, 111bitrd 278 . . . . 5 (𝑤 = 𝑧 → (∃ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ∧ 𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
113112cbvrabv 3417 . . . 4 {𝑤 ∈ ℝ* ∣ ∃ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ∧ 𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}
114 simpl 483 . . . . . . . . . 10 ((𝑗 = 𝑛𝑙𝑋) → 𝑗 = 𝑛)
115114fveq2d 6846 . . . . . . . . 9 ((𝑗 = 𝑛𝑙𝑋) → (𝑖𝑗) = (𝑖𝑛))
116115fveq1d 6844 . . . . . . . 8 ((𝑗 = 𝑛𝑙𝑋) → ((𝑖𝑗)‘𝑙) = ((𝑖𝑛)‘𝑙))
117116fveq2d 6846 . . . . . . 7 ((𝑗 = 𝑛𝑙𝑋) → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑛)‘𝑙)))
118117mpteq2dva 5205 . . . . . 6 (𝑗 = 𝑛 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
119118cbvmptv 5218 . . . . 5 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
120119mpteq2i 5210 . . . 4 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))))) = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))
121116fveq2d 6846 . . . . . . 7 ((𝑗 = 𝑛𝑙𝑋) → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑙)))
122121mpteq2dva 5205 . . . . . 6 (𝑗 = 𝑛 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
123122cbvmptv 5218 . . . . 5 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
124123mpteq2i 5210 . . . 4 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))))) = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))
12559, 61, 62, 63, 2, 89, 113, 120, 124ovnhoilem2 44833 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
12670, 125pm2.61dan 811 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
12712, 16, 67, 126xrletrid 13074 1 (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  c0 4282  ifcif 4486  {csn 4586  cop 4592   ciun 4954   class class class wbr 5105  cmpt 5188   × cxp 5631  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  m cmap 8765  Xcixp 8835  Fincfn 8883  cr 11050  0cc0 11051  1c1 11052  +∞cpnf 11186  *cxr 11188  cle 11190  cn 12153  [,)cico 13266  cprod 15788  volcvol 24827  Σ^csumge0 44593  voln*covoln 44767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-prod 15789  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-sumge0 44594  df-ovoln 44768
This theorem is referenced by:  vonhoi  44898
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