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Theorem ovnhoi 44141
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)
65ffvelrnda 6961 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
7 ovnhoi.b . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ)
87ffvelrnda 6961 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
98rexrd 11025 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
104, 6, 9hoissrrn2 44116 . . . 4 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑m 𝑋))
113, 10eqsstrd 3959 . . 3 (𝜑𝐼 ⊆ (ℝ ↑m 𝑋))
121, 11ovnxrcl 44107 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) ∈ ℝ*)
13 icossxr 13164 . . 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 44120 . . 3 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))
1613, 15sselid 3919 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ ℝ*)
17 fveq2 6774 . . . . . . . 8 (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
1817fveq1d 6776 . . . . . . 7 (𝑋 = ∅ → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
1918adantl 482 . . . . . 6 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
20 ixpeq1 8696 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)))
21 ixp0x 8714 . . . . . . . . . . . 12 X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)) = {∅}
2221a1i 11 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
2320, 22eqtrd 2778 . . . . . . . . . 10 (𝑋 = ∅ → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
2423adantl 482 . . . . . . . . 9 ((𝜑𝑋 = ∅) → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
252a1i 11 . . . . . . . . 9 ((𝜑𝑋 = ∅) → 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
26 reex 10962 . . . . . . . . . . 11 ℝ ∈ V
27 mapdm0 8630 . . . . . . . . . . 11 (ℝ ∈ V → (ℝ ↑m ∅) = {∅})
2826, 27ax-mp 5 . . . . . . . . . 10 (ℝ ↑m ∅) = {∅}
2928a1i 11 . . . . . . . . 9 ((𝜑𝑋 = ∅) → (ℝ ↑m ∅) = {∅})
3024, 25, 293eqtr4d 2788 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐼 = (ℝ ↑m ∅))
31 eqimss 3977 . . . . . . . 8 (𝐼 = (ℝ ↑m ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
3230, 31syl 17 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
3332ovn0val 44088 . . . . . 6 ((𝜑𝑋 = ∅) → ((voln*‘∅)‘𝐼) = 0)
3419, 33eqtrd 2778 . . . . 5 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0)
35 0red 10978 . . . . 5 ((𝜑𝑋 = ∅) → 0 ∈ ℝ)
3634, 35eqeltrd 2839 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ)
37 eqidd 2739 . . . . 5 ((𝜑𝑋 = ∅) → 0 = 0)
38 fveq2 6774 . . . . . . . 8 (𝑋 = ∅ → (𝐿𝑋) = (𝐿‘∅))
3938oveqd 7292 . . . . . . 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 6586 . . . . . . . 8 ((𝜑𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
4441, 43mpbid 231 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐴:∅⟶ℝ)
457adantr 481 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
4642feq2d 6586 . . . . . . . 8 ((𝜑𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
4745, 46mpbid 231 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐵:∅⟶ℝ)
4814, 44, 47hoidmv0val 44121 . . . . . 6 ((𝜑𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
4940, 48eqtrd 2778 . . . . 5 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = 0)
5037, 34, 493eqtr4d 2788 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
5136, 50eqled 11078 . . 3 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
52 eqid 2738 . . . . . 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 2742 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 = 1 ↔ 𝑗 = 1))
5453ifbid 4482 . . . . . . . 8 (𝑛 = 𝑗 → if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
5554mpteq2dv 5176 . . . . . . 7 (𝑛 = 𝑗 → (𝑘𝑋 ↦ if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
5655cbvmptv 5187 . . . . . 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 44139 . . . . 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 44122 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
6564eqcomd 2744 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (𝐴(𝐿𝑋)𝐵))
6658, 65breqtrd 5100 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
6751, 66pm2.61dan 810 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
6849, 35eqeltrd 2839 . . . 4 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ∈ ℝ)
6950eqcomd 2744 . . . 4 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = ((voln*‘𝑋)‘𝐼))
7068, 69eqled 11078 . . 3 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
71 fveq1 6773 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝑎𝑘) = (𝑐𝑘))
7271fvoveq1d 7297 . . . . . . . . . . 11 (𝑎 = 𝑐 → (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = (vol‘((𝑐𝑘)[,)(𝑏𝑘))))
7372prodeq2ad 43133 . . . . . . . . . 10 (𝑎 = 𝑐 → ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘))))
7473ifeq2d 4479 . . . . . . . . 9 (𝑎 = 𝑐 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘)))))
75 fveq1 6773 . . . . . . . . . . . . 13 (𝑏 = 𝑑 → (𝑏𝑘) = (𝑑𝑘))
7675oveq2d 7291 . . . . . . . . . . . 12 (𝑏 = 𝑑 → ((𝑐𝑘)[,)(𝑏𝑘)) = ((𝑐𝑘)[,)(𝑑𝑘)))
7776fveq2d 6778 . . . . . . . . . . 11 (𝑏 = 𝑑 → (vol‘((𝑐𝑘)[,)(𝑏𝑘))) = (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
7877prodeq2ad 43133 . . . . . . . . . 10 (𝑏 = 𝑑 → ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
7978ifeq2d 4479 . . . . . . . . 9 (𝑏 = 𝑑 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8074, 79cbvmpov 7370 . . . . . . . 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 7283 . . . . . . . 8 (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
83 eqeq1 2742 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
84 prodeq1 15619 . . . . . . . . 9 (𝑥 = 𝑦 → ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))) = ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
8583, 84ifbieq2d 4485 . . . . . . . 8 (𝑥 = 𝑦 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))) = if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8682, 82, 85mpoeq123dv 7350 . . . . . . 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 2778 . . . . . 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 5187 . . . . 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 2766 . . . 4 𝐿 = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))))
90 eqeq1 2742 . . . . . . . 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 3226 . . . . . 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 6776 . . . . . . . . . . . . . 14 (( = 𝑖𝑗 ∈ ℕ) → (𝑗) = (𝑖𝑗))
9594coeq2d 5771 . . . . . . . . . . . . 13 (( = 𝑖𝑗 ∈ ℕ) → ([,) ∘ (𝑗)) = ([,) ∘ (𝑖𝑗)))
9695fveq1d 6776 . . . . . . . . . . . 12 (( = 𝑖𝑗 ∈ ℕ) → (([,) ∘ (𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
9796ixpeq2dv 8701 . . . . . . . . . . 11 (( = 𝑖𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
9897iuneq2dv 4948 . . . . . . . . . 10 ( = 𝑖 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
9998sseq2d 3953 . . . . . . . . 9 ( = 𝑖 → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ↔ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
100 simpl 483 . . . . . . . . . . . . . . . . 17 (( = 𝑖𝑘𝑋) → = 𝑖)
101100fveq1d 6776 . . . . . . . . . . . . . . . 16 (( = 𝑖𝑘𝑋) → (𝑗) = (𝑖𝑗))
102101coeq2d 5771 . . . . . . . . . . . . . . 15 (( = 𝑖𝑘𝑋) → ([,) ∘ (𝑗)) = ([,) ∘ (𝑖𝑗)))
103102fveq1d 6776 . . . . . . . . . . . . . 14 (( = 𝑖𝑘𝑋) → (([,) ∘ (𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
104103fveq2d 6778 . . . . . . . . . . . . 13 (( = 𝑖𝑘𝑋) → (vol‘(([,) ∘ (𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
105104prodeq2dv 15633 . . . . . . . . . . . 12 ( = 𝑖 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
106105mpteq2dv 5176 . . . . . . . . . . 11 ( = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
107106fveq2d 6778 . . . . . . . . . 10 ( = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
108107eqeq2d 2749 . . . . . . . . 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 3384 . . . . . . 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 3426 . . . 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 6778 . . . . . . . . 9 ((𝑗 = 𝑛𝑙𝑋) → (𝑖𝑗) = (𝑖𝑛))
116115fveq1d 6776 . . . . . . . 8 ((𝑗 = 𝑛𝑙𝑋) → ((𝑖𝑗)‘𝑙) = ((𝑖𝑛)‘𝑙))
117116fveq2d 6778 . . . . . . 7 ((𝑗 = 𝑛𝑙𝑋) → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑛)‘𝑙)))
118117mpteq2dva 5174 . . . . . 6 (𝑗 = 𝑛 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
119118cbvmptv 5187 . . . . 5 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
120119mpteq2i 5179 . . . 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 6778 . . . . . . 7 ((𝑗 = 𝑛𝑙𝑋) → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑙)))
122121mpteq2dva 5174 . . . . . 6 (𝑗 = 𝑛 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
123122cbvmptv 5187 . . . . 5 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
124123mpteq2i 5179 . . . 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 44140 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
12670, 125pm2.61dan 810 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
12712, 16, 67, 126xrletrid 12889 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 1539  wcel 2106  wne 2943  wrex 3065  {crab 3068  Vcvv 3432  wss 3887  c0 4256  ifcif 4459  {csn 4561  cop 4567   ciun 4924   class class class wbr 5074  cmpt 5157   × cxp 5587  ccom 5593  wf 6429  cfv 6433  (class class class)co 7275  cmpo 7277  1st c1st 7829  2nd c2nd 7830  m cmap 8615  Xcixp 8685  Fincfn 8733  cr 10870  0cc0 10871  1c1 10872  +∞cpnf 11006  *cxr 11008  cle 11010  cn 11973  [,)cico 13081  cprod 15615  volcvol 24627  Σ^csumge0 43900  voln*covoln 44074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-prod 15616  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-bases 22096  df-cmp 22538  df-ovol 24628  df-vol 24629  df-sumge0 43901  df-ovoln 44075
This theorem is referenced by:  vonhoi  44205
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