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Theorem ovnhoi 40121
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 1840 . . . . 5 𝑘𝜑
5 ovnhoi.a . . . . . 6 (𝜑𝐴:𝑋⟶ℝ)
65ffvelrnda 6315 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
7 ovnhoi.b . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ)
87ffvelrnda 6315 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
98rexrd 10033 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
104, 6, 9hoissrrn2 40096 . . . 4 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
113, 10eqsstrd 3618 . . 3 (𝜑𝐼 ⊆ (ℝ ↑𝑚 𝑋))
121, 11ovnxrcl 40087 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) ∈ ℝ*)
13 icossxr 12200 . . 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 40100 . . 3 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))
1613, 15sseldi 3581 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ ℝ*)
17 fveq2 6148 . . . . . . . 8 (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
1817fveq1d 6150 . . . . . . 7 (𝑋 = ∅ → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
1918adantl 482 . . . . . 6 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
20 ixpeq1 7863 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)))
21 ixp0x 7880 . . . . . . . . . . . 12 X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)) = {∅}
2221a1i 11 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
2320, 22eqtrd 2655 . . . . . . . . . 10 (𝑋 = ∅ → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
2423adantl 482 . . . . . . . . 9 ((𝜑𝑋 = ∅) → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
252a1i 11 . . . . . . . . 9 ((𝜑𝑋 = ∅) → 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
26 reex 9971 . . . . . . . . . . 11 ℝ ∈ V
27 mapdm0 38854 . . . . . . . . . . 11 (ℝ ∈ V → (ℝ ↑𝑚 ∅) = {∅})
2826, 27ax-mp 5 . . . . . . . . . 10 (ℝ ↑𝑚 ∅) = {∅}
2928a1i 11 . . . . . . . . 9 ((𝜑𝑋 = ∅) → (ℝ ↑𝑚 ∅) = {∅})
3024, 25, 293eqtr4d 2665 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐼 = (ℝ ↑𝑚 ∅))
31 eqimss 3636 . . . . . . . 8 (𝐼 = (ℝ ↑𝑚 ∅) → 𝐼 ⊆ (ℝ ↑𝑚 ∅))
3230, 31syl 17 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑𝑚 ∅))
3332ovn0val 40068 . . . . . 6 ((𝜑𝑋 = ∅) → ((voln*‘∅)‘𝐼) = 0)
3419, 33eqtrd 2655 . . . . 5 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0)
35 0red 9985 . . . . 5 ((𝜑𝑋 = ∅) → 0 ∈ ℝ)
3634, 35eqeltrd 2698 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ)
37 eqidd 2622 . . . . 5 ((𝜑𝑋 = ∅) → 0 = 0)
38 fveq2 6148 . . . . . . . 8 (𝑋 = ∅ → (𝐿𝑋) = (𝐿‘∅))
3938oveqd 6621 . . . . . . 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 477 . . . . . . . . 9 ((𝜑𝑋 = ∅) → 𝑋 = ∅)
4342feq2d 5988 . . . . . . . 8 ((𝜑𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
4441, 43mpbid 222 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐴:∅⟶ℝ)
457adantr 481 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
4642feq2d 5988 . . . . . . . 8 ((𝜑𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
4745, 46mpbid 222 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐵:∅⟶ℝ)
4814, 44, 47hoidmv0val 40101 . . . . . 6 ((𝜑𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
4940, 48eqtrd 2655 . . . . 5 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = 0)
5037, 34, 493eqtr4d 2665 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
5136, 50eqled 10084 . . 3 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
52 eqid 2621 . . . . . 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 2625 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 = 1 ↔ 𝑗 = 1))
5453ifbid 4080 . . . . . . . 8 (𝑛 = 𝑗 → if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
5554mpteq2dv 4705 . . . . . . 7 (𝑛 = 𝑗 → (𝑘𝑋 ↦ if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
5655cbvmptv 4710 . . . . . 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 40119 . . . . 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 2798 . . . . . . 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 40102 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
6564eqcomd 2627 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (𝐴(𝐿𝑋)𝐵))
6658, 65breqtrd 4639 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
6751, 66pm2.61dan 831 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
6849, 35eqeltrd 2698 . . . 4 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ∈ ℝ)
6950eqcomd 2627 . . . 4 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = ((voln*‘𝑋)‘𝐼))
7068, 69eqled 10084 . . 3 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
71 fveq1 6147 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → (𝑎𝑘) = (𝑐𝑘))
7271oveq1d 6619 . . . . . . . . . . . 12 (𝑎 = 𝑐 → ((𝑎𝑘)[,)(𝑏𝑘)) = ((𝑐𝑘)[,)(𝑏𝑘)))
7372fveq2d 6152 . . . . . . . . . . 11 (𝑎 = 𝑐 → (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = (vol‘((𝑐𝑘)[,)(𝑏𝑘))))
7473prodeq2ad 39225 . . . . . . . . . 10 (𝑎 = 𝑐 → ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘))))
7574ifeq2d 4077 . . . . . . . . 9 (𝑎 = 𝑐 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘)))))
76 fveq1 6147 . . . . . . . . . . . . 13 (𝑏 = 𝑑 → (𝑏𝑘) = (𝑑𝑘))
7776oveq2d 6620 . . . . . . . . . . . 12 (𝑏 = 𝑑 → ((𝑐𝑘)[,)(𝑏𝑘)) = ((𝑐𝑘)[,)(𝑑𝑘)))
7877fveq2d 6152 . . . . . . . . . . 11 (𝑏 = 𝑑 → (vol‘((𝑐𝑘)[,)(𝑏𝑘))) = (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
7978prodeq2ad 39225 . . . . . . . . . 10 (𝑏 = 𝑑 → ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
8079ifeq2d 4077 . . . . . . . . 9 (𝑏 = 𝑑 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8175, 80cbvmpt2v 6688 . . . . . . . 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)))))
8281a1i 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))))))
83 oveq2 6612 . . . . . . . 8 (𝑥 = 𝑦 → (ℝ ↑𝑚 𝑥) = (ℝ ↑𝑚 𝑦))
84 eqeq1 2625 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
85 prodeq1 14564 . . . . . . . . 9 (𝑥 = 𝑦 → ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))) = ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
8684, 85ifbieq2d 4083 . . . . . . . 8 (𝑥 = 𝑦 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))) = if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8783, 83, 86mpt2eq123dv 6670 . . . . . . 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))))))
8882, 87eqtrd 2655 . . . . . 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))))))
8988cbvmptv 4710 . . . . 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))))))
9014, 89eqtri 2643 . . . 4 𝐿 = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑦), 𝑑 ∈ (ℝ ↑𝑚 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))))
91 eqeq1 2625 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))))))
9291anbi2d 739 . . . . . . 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)))))))
9392rexbidv 3045 . . . . . 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)))))))
94 simpl 473 . . . . . . . . . . . . . . 15 (( = 𝑖𝑗 ∈ ℕ) → = 𝑖)
9594fveq1d 6150 . . . . . . . . . . . . . 14 (( = 𝑖𝑗 ∈ ℕ) → (𝑗) = (𝑖𝑗))
9695coeq2d 5244 . . . . . . . . . . . . 13 (( = 𝑖𝑗 ∈ ℕ) → ([,) ∘ (𝑗)) = ([,) ∘ (𝑖𝑗)))
9796fveq1d 6150 . . . . . . . . . . . 12 (( = 𝑖𝑗 ∈ ℕ) → (([,) ∘ (𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
9897ixpeq2dv 7868 . . . . . . . . . . 11 (( = 𝑖𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
9998iuneq2dv 4508 . . . . . . . . . 10 ( = 𝑖 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
10099sseq2d 3612 . . . . . . . . 9 ( = 𝑖 → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ↔ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
101 simpl 473 . . . . . . . . . . . . . . . . 17 (( = 𝑖𝑘𝑋) → = 𝑖)
102101fveq1d 6150 . . . . . . . . . . . . . . . 16 (( = 𝑖𝑘𝑋) → (𝑗) = (𝑖𝑗))
103102coeq2d 5244 . . . . . . . . . . . . . . 15 (( = 𝑖𝑘𝑋) → ([,) ∘ (𝑗)) = ([,) ∘ (𝑖𝑗)))
104103fveq1d 6150 . . . . . . . . . . . . . 14 (( = 𝑖𝑘𝑋) → (([,) ∘ (𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
105104fveq2d 6152 . . . . . . . . . . . . 13 (( = 𝑖𝑘𝑋) → (vol‘(([,) ∘ (𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
106105prodeq2dv 14578 . . . . . . . . . . . 12 ( = 𝑖 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
107106mpteq2dv 4705 . . . . . . . . . . 11 ( = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
108107fveq2d 6152 . . . . . . . . . 10 ( = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
109108eqeq2d 2631 . . . . . . . . 9 ( = 𝑖 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
110100, 109anbi12d 746 . . . . . . . 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)))))))
111110cbvrexv 3160 . . . . . . 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))))))
112111a1i 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)))))))
11393, 112bitrd 268 . . . . 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)))))))
114113cbvrabv 3185 . . . 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)))))}
115 simpl 473 . . . . . . . . . 10 ((𝑗 = 𝑛𝑙𝑋) → 𝑗 = 𝑛)
116115fveq2d 6152 . . . . . . . . 9 ((𝑗 = 𝑛𝑙𝑋) → (𝑖𝑗) = (𝑖𝑛))
117116fveq1d 6150 . . . . . . . 8 ((𝑗 = 𝑛𝑙𝑋) → ((𝑖𝑗)‘𝑙) = ((𝑖𝑛)‘𝑙))
118117fveq2d 6152 . . . . . . 7 ((𝑗 = 𝑛𝑙𝑋) → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑛)‘𝑙)))
119118mpteq2dva 4704 . . . . . 6 (𝑗 = 𝑛 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
120119cbvmptv 4710 . . . . 5 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
121120mpteq2i 4701 . . . 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)))))
122117fveq2d 6152 . . . . . . 7 ((𝑗 = 𝑛𝑙𝑋) → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑙)))
123122mpteq2dva 4704 . . . . . 6 (𝑗 = 𝑛 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
124123cbvmptv 4710 . . . . 5 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
125124mpteq2i 4701 . . . 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)))))
12659, 61, 62, 63, 2, 90, 114, 121, 125ovnhoilem2 40120 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
12770, 126pm2.61dan 831 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
12812, 16, 67, 127xrletrid 11930 1 (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  c0 3891  ifcif 4058  {csn 4148  cop 4154   ciun 4485   class class class wbr 4613  cmpt 4673   × cxp 5072  ccom 5078  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  1st c1st 7111  2nd c2nd 7112  𝑚 cmap 7802  Xcixp 7852  Fincfn 7899  cr 9879  0cc0 9880  1c1 9881  +∞cpnf 10015  *cxr 10017  cle 10019  cn 10964  [,)cico 12119  cprod 14560  volcvol 23139  Σ^csumge0 39883  voln*covoln 40054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-rlim 14154  df-sum 14351  df-prod 14561  df-rest 16004  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-cmp 21100  df-ovol 23140  df-vol 23141  df-sumge0 39884  df-ovoln 40055
This theorem is referenced by:  vonhoi  40185
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