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Theorem ovnhoi 43703
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 1921 . . . . 5 𝑘𝜑
5 ovnhoi.a . . . . . 6 (𝜑𝐴:𝑋⟶ℝ)
65ffvelrnda 6861 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
7 ovnhoi.b . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ)
87ffvelrnda 6861 . . . . . 6 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
98rexrd 10769 . . . . 5 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
104, 6, 9hoissrrn2 43678 . . . 4 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ (ℝ ↑m 𝑋))
113, 10eqsstrd 3915 . . 3 (𝜑𝐼 ⊆ (ℝ ↑m 𝑋))
121, 11ovnxrcl 43669 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) ∈ ℝ*)
13 icossxr 12906 . . 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 43682 . . 3 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))
1613, 15sseldi 3875 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ ℝ*)
17 fveq2 6674 . . . . . . . 8 (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
1817fveq1d 6676 . . . . . . 7 (𝑋 = ∅ → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
1918adantl 485 . . . . . 6 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
20 ixpeq1 8518 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)))
21 ixp0x 8536 . . . . . . . . . . . 12 X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)) = {∅}
2221a1i 11 . . . . . . . . . . 11 (𝑋 = ∅ → X𝑘 ∈ ∅ ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
2320, 22eqtrd 2773 . . . . . . . . . 10 (𝑋 = ∅ → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
2423adantl 485 . . . . . . . . 9 ((𝜑𝑋 = ∅) → X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) = {∅})
252a1i 11 . . . . . . . . 9 ((𝜑𝑋 = ∅) → 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
26 reex 10706 . . . . . . . . . . 11 ℝ ∈ V
27 mapdm0 8452 . . . . . . . . . . 11 (ℝ ∈ V → (ℝ ↑m ∅) = {∅})
2826, 27ax-mp 5 . . . . . . . . . 10 (ℝ ↑m ∅) = {∅}
2928a1i 11 . . . . . . . . 9 ((𝜑𝑋 = ∅) → (ℝ ↑m ∅) = {∅})
3024, 25, 293eqtr4d 2783 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐼 = (ℝ ↑m ∅))
31 eqimss 3933 . . . . . . . 8 (𝐼 = (ℝ ↑m ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
3230, 31syl 17 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
3332ovn0val 43650 . . . . . 6 ((𝜑𝑋 = ∅) → ((voln*‘∅)‘𝐼) = 0)
3419, 33eqtrd 2773 . . . . 5 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0)
35 0red 10722 . . . . 5 ((𝜑𝑋 = ∅) → 0 ∈ ℝ)
3634, 35eqeltrd 2833 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ)
37 eqidd 2739 . . . . 5 ((𝜑𝑋 = ∅) → 0 = 0)
38 fveq2 6674 . . . . . . . 8 (𝑋 = ∅ → (𝐿𝑋) = (𝐿‘∅))
3938oveqd 7187 . . . . . . 7 (𝑋 = ∅ → (𝐴(𝐿𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
4039adantl 485 . . . . . 6 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
415adantr 484 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
42 simpr 488 . . . . . . . . 9 ((𝜑𝑋 = ∅) → 𝑋 = ∅)
4342feq2d 6490 . . . . . . . 8 ((𝜑𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
4441, 43mpbid 235 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐴:∅⟶ℝ)
457adantr 484 . . . . . . . 8 ((𝜑𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
4642feq2d 6490 . . . . . . . 8 ((𝜑𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
4745, 46mpbid 235 . . . . . . 7 ((𝜑𝑋 = ∅) → 𝐵:∅⟶ℝ)
4814, 44, 47hoidmv0val 43683 . . . . . 6 ((𝜑𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
4940, 48eqtrd 2773 . . . . 5 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = 0)
5037, 34, 493eqtr4d 2783 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
5136, 50eqled 10821 . . 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 4437 . . . . . . . 8 (𝑛 = 𝑗 → if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩) = if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩))
5554mpteq2dv 5126 . . . . . . 7 (𝑛 = 𝑗 → (𝑘𝑋 ↦ if(𝑛 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)) = (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))
5655cbvmptv 5133 . . . . . 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 43701 . . . . 5 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
5857adantr 484 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
591adantr 484 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
60 neqne 2942 . . . . . . 7 𝑋 = ∅ → 𝑋 ≠ ∅)
6160adantl 485 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
625adantr 484 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
637adantr 484 . . . . . 6 ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
6414, 59, 61, 62, 63hoidmvn0val 43684 . . . . 5 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
6564eqcomd 2744 . . . 4 ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = (𝐴(𝐿𝑋)𝐵))
6658, 65breqtrd 5056 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
6751, 66pm2.61dan 813 . 2 (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿𝑋)𝐵))
6849, 35eqeltrd 2833 . . . 4 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ∈ ℝ)
6950eqcomd 2744 . . . 4 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) = ((voln*‘𝑋)‘𝐼))
7068, 69eqled 10821 . . 3 ((𝜑𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
71 fveq1 6673 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝑎𝑘) = (𝑐𝑘))
7271fvoveq1d 7192 . . . . . . . . . . 11 (𝑎 = 𝑐 → (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = (vol‘((𝑐𝑘)[,)(𝑏𝑘))))
7372prodeq2ad 42695 . . . . . . . . . 10 (𝑎 = 𝑐 → ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘))))
7473ifeq2d 4434 . . . . . . . . 9 (𝑎 = 𝑐 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘)))))
75 fveq1 6673 . . . . . . . . . . . . 13 (𝑏 = 𝑑 → (𝑏𝑘) = (𝑑𝑘))
7675oveq2d 7186 . . . . . . . . . . . 12 (𝑏 = 𝑑 → ((𝑐𝑘)[,)(𝑏𝑘)) = ((𝑐𝑘)[,)(𝑑𝑘)))
7776fveq2d 6678 . . . . . . . . . . 11 (𝑏 = 𝑑 → (vol‘((𝑐𝑘)[,)(𝑏𝑘))) = (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
7877prodeq2ad 42695 . . . . . . . . . 10 (𝑏 = 𝑑 → ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘))) = ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
7978ifeq2d 4434 . . . . . . . . 9 (𝑏 = 𝑑 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑏𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8074, 79cbvmpov 7263 . . . . . . . 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 7178 . . . . . . . 8 (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
83 eqeq1 2742 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
84 prodeq1 15355 . . . . . . . . 9 (𝑥 = 𝑦 → ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘))) = ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘))))
8583, 84ifbieq2d 4440 . . . . . . . 8 (𝑥 = 𝑦 → if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))) = if(𝑦 = ∅, 0, ∏𝑘𝑦 (vol‘((𝑐𝑘)[,)(𝑑𝑘)))))
8682, 82, 85mpoeq123dv 7243 . . . . . . 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 2773 . . . . . 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 5133 . . . . 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 2761 . . . 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 632 . . . . . . 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 3207 . . . . . 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 486 . . . . . . . . . . . . . . 15 (( = 𝑖𝑗 ∈ ℕ) → = 𝑖)
9493fveq1d 6676 . . . . . . . . . . . . . 14 (( = 𝑖𝑗 ∈ ℕ) → (𝑗) = (𝑖𝑗))
9594coeq2d 5705 . . . . . . . . . . . . 13 (( = 𝑖𝑗 ∈ ℕ) → ([,) ∘ (𝑗)) = ([,) ∘ (𝑖𝑗)))
9695fveq1d 6676 . . . . . . . . . . . 12 (( = 𝑖𝑗 ∈ ℕ) → (([,) ∘ (𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
9796ixpeq2dv 8523 . . . . . . . . . . 11 (( = 𝑖𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
9897iuneq2dv 4905 . . . . . . . . . 10 ( = 𝑖 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
9998sseq2d 3909 . . . . . . . . 9 ( = 𝑖 → (𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘) ↔ 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
100 simpl 486 . . . . . . . . . . . . . . . . 17 (( = 𝑖𝑘𝑋) → = 𝑖)
101100fveq1d 6676 . . . . . . . . . . . . . . . 16 (( = 𝑖𝑘𝑋) → (𝑗) = (𝑖𝑗))
102101coeq2d 5705 . . . . . . . . . . . . . . 15 (( = 𝑖𝑘𝑋) → ([,) ∘ (𝑗)) = ([,) ∘ (𝑖𝑗)))
103102fveq1d 6676 . . . . . . . . . . . . . 14 (( = 𝑖𝑘𝑋) → (([,) ∘ (𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
104103fveq2d 6678 . . . . . . . . . . . . 13 (( = 𝑖𝑘𝑋) → (vol‘(([,) ∘ (𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
105104prodeq2dv 15369 . . . . . . . . . . . 12 ( = 𝑖 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
106105mpteq2dv 5126 . . . . . . . . . . 11 ( = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
107106fveq2d 6678 . . . . . . . . . 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 634 . . . . . . . 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 3350 . . . . . . 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 282 . . . . 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 3393 . . . 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 486 . . . . . . . . . 10 ((𝑗 = 𝑛𝑙𝑋) → 𝑗 = 𝑛)
115114fveq2d 6678 . . . . . . . . 9 ((𝑗 = 𝑛𝑙𝑋) → (𝑖𝑗) = (𝑖𝑛))
116115fveq1d 6676 . . . . . . . 8 ((𝑗 = 𝑛𝑙𝑋) → ((𝑖𝑗)‘𝑙) = ((𝑖𝑛)‘𝑙))
117116fveq2d 6678 . . . . . . 7 ((𝑗 = 𝑛𝑙𝑋) → (1st ‘((𝑖𝑗)‘𝑙)) = (1st ‘((𝑖𝑛)‘𝑙)))
118117mpteq2dva 5125 . . . . . 6 (𝑗 = 𝑛 → (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
119118cbvmptv 5133 . . . . 5 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙))))
120119mpteq2i 5122 . . . 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 6678 . . . . . . 7 ((𝑗 = 𝑛𝑙𝑋) → (2nd ‘((𝑖𝑗)‘𝑙)) = (2nd ‘((𝑖𝑛)‘𝑙)))
122121mpteq2dva 5125 . . . . . 6 (𝑗 = 𝑛 → (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙))) = (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
123122cbvmptv 5133 . . . . 5 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙))))
124123mpteq2i 5122 . . . 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 43702 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
12670, 125pm2.61dan 813 . 2 (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
12712, 16, 67, 126xrletrid 12631 1 (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wne 2934  wrex 3054  {crab 3057  Vcvv 3398  wss 3843  c0 4211  ifcif 4414  {csn 4516  cop 4522   ciun 4881   class class class wbr 5030  cmpt 5110   × cxp 5523  ccom 5529  wf 6335  cfv 6339  (class class class)co 7170  cmpo 7172  1st c1st 7712  2nd c2nd 7713  m cmap 8437  Xcixp 8507  Fincfn 8555  cr 10614  0cc0 10615  1c1 10616  +∞cpnf 10750  *cxr 10752  cle 10754  cn 11716  [,)cico 12823  cprod 15351  volcvol 24215  Σ^csumge0 43462  voln*covoln 43636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-2o 8132  df-er 8320  df-map 8439  df-pm 8440  df-ixp 8508  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-fi 8948  df-sup 8979  df-inf 8980  df-oi 9047  df-dju 9403  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-n0 11977  df-z 12063  df-uz 12325  df-q 12431  df-rp 12473  df-xneg 12590  df-xadd 12591  df-xmul 12592  df-ioo 12825  df-ico 12827  df-icc 12828  df-fz 12982  df-fzo 13125  df-fl 13253  df-seq 13461  df-exp 13522  df-hash 13783  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-abs 14685  df-clim 14935  df-rlim 14936  df-sum 15136  df-prod 15352  df-rest 16799  df-topgen 16820  df-psmet 20209  df-xmet 20210  df-met 20211  df-bl 20212  df-mopn 20213  df-top 21645  df-topon 21662  df-bases 21697  df-cmp 22138  df-ovol 24216  df-vol 24217  df-sumge0 43463  df-ovoln 43637
This theorem is referenced by:  vonhoi  43767
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