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Theorem ovnhoi 42865
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 1908 . . . . 5 𝑘𝜑
5 ovnhoi.a . . . . . 6 (𝜑𝐴:𝑋⟶ℝ)
65ffvelrnda 6844 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
7 ovnhoi.b . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ)
87ffvelrnda 6844 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
98rexrd 10683 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
104, 6, 9hoissrrn2 42840 . . . 4 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑m 𝑋))
113, 10eqsstrd 4003 . . 3 (𝜑𝐼 ⊆ (ℝ ↑m 𝑋))
121, 11ovnxrcl 42831 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) ∈ ℝ*)
13 icossxr 12813 . . 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 42844 . . 3 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))
1613, 15sseldi 3963 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ ℝ*)
17 fveq2 6663 . . . . . . . 8 (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
1817fveq1d 6665 . . . . . . 7 (𝑋 = ∅ → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
1918adantl 484 . . . . . 6 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
20 ixpeq1 8464 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)))
21 ixp0x 8482 . . . . . . . . . . . 12 X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)) = {∅}
2221a1i 11 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
2320, 22eqtrd 2854 . . . . . . . . . 10 (𝑋 = ∅ → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
2423adantl 484 . . . . . . . . 9 ((𝜑𝑋 = ∅) → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
252a1i 11 . . . . . . . . 9 ((𝜑𝑋 = ∅) → 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
26 reex 10620 . . . . . . . . . . 11 ℝ ∈ V
27 mapdm0 8413 . . . . . . . . . . 11 (ℝ ∈ V → (ℝ ↑m ∅) = {∅})
2826, 27ax-mp 5 . . . . . . . . . 10 (ℝ ↑m ∅) = {∅}
2928a1i 11 . . . . . . . . 9 ((𝜑𝑋 = ∅) → (ℝ ↑m ∅) = {∅})
3024, 25, 293eqtr4d 2864 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐼 = (ℝ ↑m ∅))
31 eqimss 4021 . . . . . . . 8 (𝐼 = (ℝ ↑m ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
3230, 31syl 17 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
3332ovn0val 42812 . . . . . 6 ((𝜑𝑋 = ∅) → ((voln*‘∅)‘𝐼) = 0)
3419, 33eqtrd 2854 . . . . 5 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0)
35 0red 10636 . . . . 5 ((𝜑𝑋 = ∅) → 0 ∈ ℝ)
3634, 35eqeltrd 2911 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ)
37 eqidd 2820 . . . . 5 ((𝜑𝑋 = ∅) → 0 = 0)
38 fveq2 6663 . . . . . . . 8 (𝑋 = ∅ → (𝐿𝑋) = (𝐿‘∅))
3938oveqd 7165 . . . . . . 7 (𝑋 = ∅ → (𝐴(𝐿𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
4039adantl 484 . . . . . 6 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
415adantr 483 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
42 simpr 487 . . . . . . . . 9 ((𝜑𝑋 = ∅) → 𝑋 = ∅)
4342feq2d 6493 . . . . . . . 8 ((𝜑𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
4441, 43mpbid 234 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐴:∅⟶ℝ)
457adantr 483 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
4642feq2d 6493 . . . . . . . 8 ((𝜑𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
4745, 46mpbid 234 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐵:∅⟶ℝ)
4814, 44, 47hoidmv0val 42845 . . . . . 6 ((𝜑𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
4940, 48eqtrd 2854 . . . . 5 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = 0)
5037, 34, 493eqtr4d 2864 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
5136, 50eqled 10735 . . 3 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
52 eqid 2819 . . . . . 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 2823 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 = 1 ↔ 𝑗 = 1))
5453ifbid 4487 . . . . . . . 8 (𝑛 = 𝑗 → if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
5554mpteq2dv 5153 . . . . . . 7 (𝑛 = 𝑗 → (𝑘𝑋 ↦ if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
5655cbvmptv 5160 . . . . . 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 42863 . . . . 5 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
5857adantr 483 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
591adantr 483 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
60 neqne 3022 . . . . . . 7 𝑋 = ∅ → 𝑋 ≠ ∅)
6160adantl 484 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
625adantr 483 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
637adantr 483 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
6414, 59, 61, 62, 63hoidmvn0val 42846 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
6564eqcomd 2825 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (𝐴(𝐿𝑋)𝐵))
6658, 65breqtrd 5083 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
6751, 66pm2.61dan 811 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
6849, 35eqeltrd 2911 . . . 4 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ∈ ℝ)
6950eqcomd 2825 . . . 4 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = ((voln*‘𝑋)‘𝐼))
7068, 69eqled 10735 . . 3 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
71 fveq1 6662 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝑎𝑘) = (𝑐𝑘))
7271fvoveq1d 7170 . . . . . . . . . . 11 (𝑎 = 𝑐 → (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = (vol‘((𝑐𝑘)[,)(𝑏𝑘))))
7372prodeq2ad 41852 . . . . . . . . . 10 (𝑎 = 𝑐 → ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘))))
7473ifeq2d 4484 . . . . . . . . 9 (𝑎 = 𝑐 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘)))))
75 fveq1 6662 . . . . . . . . . . . . 13 (𝑏 = 𝑑 → (𝑏𝑘) = (𝑑𝑘))
7675oveq2d 7164 . . . . . . . . . . . 12 (𝑏 = 𝑑 → ((𝑐𝑘)[,)(𝑏𝑘)) = ((𝑐𝑘)[,)(𝑑𝑘)))
7776fveq2d 6667 . . . . . . . . . . 11 (𝑏 = 𝑑 → (vol‘((𝑐𝑘)[,)(𝑏𝑘))) = (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
7877prodeq2ad 41852 . . . . . . . . . 10 (𝑏 = 𝑑 → ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
7978ifeq2d 4484 . . . . . . . . 9 (𝑏 = 𝑑 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8074, 79cbvmpov 7241 . . . . . . . 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 7156 . . . . . . . 8 (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
83 eqeq1 2823 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
84 prodeq1 15255 . . . . . . . . 9 (𝑥 = 𝑦 → ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))) = ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
8583, 84ifbieq2d 4490 . . . . . . . 8 (𝑥 = 𝑦 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))) = if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8682, 82, 85mpoeq123dv 7221 . . . . . . 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 2854 . . . . . 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 5160 . . . . 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 2842 . . . 4 𝐿 = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))))
90 eqeq1 2823 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))))))
9190anbi2d 630 . . . . . . 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 3295 . . . . . 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 485 . . . . . . . . . . . . . . 15 (( = 𝑖𝑗 ∈ ℕ) → = 𝑖)
9493fveq1d 6665 . . . . . . . . . . . . . 14 (( = 𝑖𝑗 ∈ ℕ) → (𝑗) = (𝑖𝑗))
9594coeq2d 5726 . . . . . . . . . . . . 13 (( = 𝑖𝑗 ∈ ℕ) → ([,) ∘ (𝑗)) = ([,) ∘ (𝑖𝑗)))
9695fveq1d 6665 . . . . . . . . . . . 12 (( = 𝑖𝑗 ∈ ℕ) → (([,) ∘ (𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
9796ixpeq2dv 8469 . . . . . . . . . . 11 (( = 𝑖𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
9897iuneq2dv 4934 . . . . . . . . . 10 ( = 𝑖 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
9998sseq2d 3997 . . . . . . . . 9 ( = 𝑖 → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ↔ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
100 simpl 485 . . . . . . . . . . . . . . . . 17 (( = 𝑖𝑘𝑋) → = 𝑖)
101100fveq1d 6665 . . . . . . . . . . . . . . . 16 (( = 𝑖𝑘𝑋) → (𝑗) = (𝑖𝑗))
102101coeq2d 5726 . . . . . . . . . . . . . . 15 (( = 𝑖𝑘𝑋) → ([,) ∘ (𝑗)) = ([,) ∘ (𝑖𝑗)))
103102fveq1d 6665 . . . . . . . . . . . . . 14 (( = 𝑖𝑘𝑋) → (([,) ∘ (𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
104103fveq2d 6667 . . . . . . . . . . . . 13 (( = 𝑖𝑘𝑋) → (vol‘(([,) ∘ (𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
105104prodeq2dv 15269 . . . . . . . . . . . 12 ( = 𝑖 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
106105mpteq2dv 5153 . . . . . . . . . . 11 ( = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
107106fveq2d 6667 . . . . . . . . . 10 ( = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
108107eqeq2d 2830 . . . . . . . . 9 ( = 𝑖 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
10999, 108anbi12d 632 . . . . . . . 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 3449 . . . . . . 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 281 . . . . 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 3490 . . . 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 485 . . . . . . . . . 10 ((𝑗 = 𝑛𝑙𝑋) → 𝑗 = 𝑛)
115114fveq2d 6667 . . . . . . . . 9 ((𝑗 = 𝑛𝑙𝑋) → (𝑖𝑗) = (𝑖𝑛))
116115fveq1d 6665 . . . . . . . 8 ((𝑗 = 𝑛𝑙𝑋) → ((𝑖𝑗)‘𝑙) = ((𝑖𝑛)‘𝑙))
117116fveq2d 6667 . . . . . . 7 ((𝑗 = 𝑛𝑙𝑋) → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑛)‘𝑙)))
118117mpteq2dva 5152 . . . . . 6 (𝑗 = 𝑛 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
119118cbvmptv 5160 . . . . 5 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
120119mpteq2i 5149 . . . 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 6667 . . . . . . 7 ((𝑗 = 𝑛𝑙𝑋) → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑙)))
122121mpteq2dva 5152 . . . . . 6 (𝑗 = 𝑛 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
123122cbvmptv 5160 . . . . 5 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
124123mpteq2i 5149 . . . 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 42864 . . 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 12540 1 (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wne 3014  wrex 3137  {crab 3140  Vcvv 3493  wss 3934  c0 4289  ifcif 4465  {csn 4559  cop 4565   ciun 4910   class class class wbr 5057  cmpt 5137   × cxp 5546  ccom 5552  wf 6344  cfv 6348  (class class class)co 7148  cmpo 7150  1st c1st 7679  2nd c2nd 7680  m cmap 8398  Xcixp 8453  Fincfn 8501  cr 10528  0cc0 10529  1c1 10530  +∞cpnf 10664  *cxr 10666  cle 10668  cn 11630  [,)cico 12732  cprod 15251  volcvol 24056  Σ^csumge0 42624  voln*covoln 42798
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-seq 13362  df-exp 13422  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-sum 15035  df-prod 15252  df-rest 16688  df-topgen 16709  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-top 21494  df-topon 21511  df-bases 21546  df-cmp 21987  df-ovol 24057  df-vol 24058  df-sumge0 42625  df-ovoln 42799
This theorem is referenced by:  vonhoi  42929
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