Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vonioolem2 Structured version   Visualization version   GIF version

Theorem vonioolem2 39572
 Description: The n-dimensional Lebesgue measure of open intervals. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypotheses
Ref Expression
vonioolem2.x (𝜑𝑋 ∈ Fin)
vonioolem2.a (𝜑𝐴:𝑋⟶ℝ)
vonioolem2.b (𝜑𝐵:𝑋⟶ℝ)
vonioolem2.n (𝜑𝑋 ≠ ∅)
vonioolem2.t ((𝜑𝑘𝑋) → (𝐴𝑘) < (𝐵𝑘))
vonioolem2.i 𝐼 = X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘))
vonioolem2.c 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))))
vonioolem2.d 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)))
Assertion
Ref Expression
vonioolem2 (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
Distinct variable groups:   𝐴,𝑘,𝑛   𝐵,𝑘,𝑛   𝐶,𝑘,𝑛   𝐷,𝑛   𝑛,𝐼   𝑘,𝑋,𝑛   𝜑,𝑘,𝑛
Allowed substitution hints:   𝐷(𝑘)   𝐼(𝑘)

Proof of Theorem vonioolem2
Dummy variables 𝑗 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vonioolem2.x . . . . 5 (𝜑𝑋 ∈ Fin)
21vonmea 39464 . . . 4 (𝜑 → (voln‘𝑋) ∈ Meas)
3 1zzd 11285 . . . 4 (𝜑 → 1 ∈ ℤ)
4 nnuz 11599 . . . 4 ℕ = (ℤ‘1)
51adantr 480 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
6 eqid 2610 . . . . . 6 dom (voln‘𝑋) = dom (voln‘𝑋)
7 vonioolem2.a . . . . . . . . . . 11 (𝜑𝐴:𝑋⟶ℝ)
87adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
98ffvelrnda 6267 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
10 nnrecre 10934 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
1110ad2antlr 759 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / 𝑛) ∈ ℝ)
129, 11readdcld 9948 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘) + (1 / 𝑛)) ∈ ℝ)
13 eqid 2610 . . . . . . . 8 (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))) = (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛)))
1412, 13fmptd 6292 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))):𝑋⟶ℝ)
15 vonioolem2.c . . . . . . . . . 10 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))))
1615a1i 11 . . . . . . . . 9 (𝜑𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛)))))
171mptexd 6391 . . . . . . . . . 10 (𝜑 → (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))) ∈ V)
1817adantr 480 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))) ∈ V)
1916, 18fvmpt2d 6202 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) = (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))))
2019feq1d 5943 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝐶𝑛):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
2114, 20mpbird 246 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛):𝑋⟶ℝ)
22 vonioolem2.b . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ)
2322adantr 480 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ)
245, 6, 21, 23hoimbl 39521 . . . . 5 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)) ∈ dom (voln‘𝑋))
25 vonioolem2.d . . . . 5 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)))
2624, 25fmptd 6292 . . . 4 (𝜑𝐷:ℕ⟶dom (voln‘𝑋))
27 nfv 1830 . . . . . 6 𝑘(𝜑𝑛 ∈ ℕ)
28 oveq2 6557 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
2928oveq2d 6565 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐴𝑘) + (1 / 𝑛)) = ((𝐴𝑘) + (1 / 𝑚)))
3029mpteq2dv 4673 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛))) = (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑚))))
3130cbvmptv 4678 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑚))))
3215, 31eqtri 2632 . . . . . . . . . . . 12 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑚))))
3332a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑚)))))
34 oveq2 6557 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
3534oveq2d 6565 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → ((𝐴𝑘) + (1 / 𝑚)) = ((𝐴𝑘) + (1 / (𝑛 + 1))))
3635mpteq2dv 4673 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑚))) = (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / (𝑛 + 1)))))
3736adantl 481 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = (𝑛 + 1)) → (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / 𝑚))) = (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / (𝑛 + 1)))))
38 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
3938peano2nnd 10914 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
405mptexd 6391 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / (𝑛 + 1)))) ∈ V)
4133, 37, 39, 40fvmptd 6197 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘𝑋 ↦ ((𝐴𝑘) + (1 / (𝑛 + 1)))))
42 ovex 6577 . . . . . . . . . . 11 ((𝐴𝑘) + (1 / (𝑛 + 1))) ∈ V
4342a1i 11 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘) + (1 / (𝑛 + 1))) ∈ V)
4441, 43fvmpt2d 6202 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐴𝑘) + (1 / (𝑛 + 1))))
45 1red 9934 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 1 ∈ ℝ)
46 nnre 10904 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
4746, 45readdcld 9948 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
48 peano2nn 10909 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
49 nnne0 10930 . . . . . . . . . . . . 13 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
5048, 49syl 17 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
5145, 47, 50redivcld 10732 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
5251ad2antlr 759 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
539, 52readdcld 9948 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘) + (1 / (𝑛 + 1))) ∈ ℝ)
5444, 53eqeltrd 2688 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ)
5554rexrd 9968 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ*)
56 ressxr 9962 . . . . . . . . 9 ℝ ⊆ ℝ*
5722ffvelrnda 6267 . . . . . . . . 9 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
5856, 57sseldi 3566 . . . . . . . 8 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
5958adantlr 747 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑘) ∈ ℝ*)
6046ltp1d 10833 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
61 nnrp 11718 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
6248nnrpd 11746 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
6361, 62ltrecd 11766 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛)))
6460, 63mpbid 221 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) < (1 / 𝑛))
6551, 10, 64ltled 10064 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
6665ad2antlr 759 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
6752, 11, 9, 66leadd2dd 10521 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐴𝑘) + (1 / 𝑛)))
6812elexd 3187 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘) + (1 / 𝑛)) ∈ V)
6919, 68fvmpt2d 6202 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) = ((𝐴𝑘) + (1 / 𝑛)))
7044, 69breq12d 4596 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘) ↔ ((𝐴𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐴𝑘) + (1 / 𝑛))))
7167, 70mpbird 246 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘))
7257adantlr 747 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
73 eqidd 2611 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑘) = (𝐵𝑘))
7472, 73eqled 10019 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑘) ≤ (𝐵𝑘))
75 icossico 12114 . . . . . . 7 (((((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ* ∧ (𝐵𝑘) ∈ ℝ*) ∧ (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘) ∧ (𝐵𝑘) ≤ (𝐵𝑘))) → (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)) ⊆ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)))
7655, 59, 71, 74, 75syl22anc 1319 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)) ⊆ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)))
7727, 76ixpssixp 38297 . . . . 5 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)) ⊆ X𝑘𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)))
7825a1i 11 . . . . . . 7 (𝜑𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘))))
7924elexd 3187 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)) ∈ V)
8078, 79fvmpt2d 6202 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) = X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)))
81 fveq2 6103 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐶𝑛) = (𝐶𝑚))
8281fveq1d 6105 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝐶𝑛)‘𝑘) = ((𝐶𝑚)‘𝑘))
8382oveq1d 6564 . . . . . . . . . . 11 (𝑛 = 𝑚 → (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)) = (((𝐶𝑚)‘𝑘)[,)(𝐵𝑘)))
8483ixpeq2dv 7810 . . . . . . . . . 10 (𝑛 = 𝑚X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)) = X𝑘𝑋 (((𝐶𝑚)‘𝑘)[,)(𝐵𝑘)))
8584cbvmptv 4678 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑚)‘𝑘)[,)(𝐵𝑘)))
8625, 85eqtri 2632 . . . . . . . 8 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑚)‘𝑘)[,)(𝐵𝑘)))
8786a1i 11 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘𝑋 (((𝐶𝑚)‘𝑘)[,)(𝐵𝑘))))
88 fveq2 6103 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (𝐶𝑚) = (𝐶‘(𝑛 + 1)))
8988fveq1d 6105 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → ((𝐶𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
9089oveq1d 6564 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → (((𝐶𝑚)‘𝑘)[,)(𝐵𝑘)) = (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)))
9190ixpeq2dv 7810 . . . . . . . 8 (𝑚 = (𝑛 + 1) → X𝑘𝑋 (((𝐶𝑚)‘𝑘)[,)(𝐵𝑘)) = X𝑘𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)))
9291adantl 481 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = (𝑛 + 1)) → X𝑘𝑋 (((𝐶𝑚)‘𝑘)[,)(𝐵𝑘)) = X𝑘𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)))
93 ovex 6577 . . . . . . . . . 10 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)) ∈ V
9493rgenw 2908 . . . . . . . . 9 𝑘𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)) ∈ V
95 ixpexg 7818 . . . . . . . . 9 (∀𝑘𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)) ∈ V → X𝑘𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)) ∈ V)
9694, 95ax-mp 5 . . . . . . . 8 X𝑘𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)) ∈ V
9796a1i 11 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)) ∈ V)
9887, 92, 39, 97fvmptd 6197 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘)))
9980, 98sseq12d 3597 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝐷𝑛) ⊆ (𝐷‘(𝑛 + 1)) ↔ X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)) ⊆ X𝑘𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵𝑘))))
10077, 99mpbird 246 . . . 4 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) ⊆ (𝐷‘(𝑛 + 1)))
1011, 6, 7, 22hoimbl 39521 . . . . 5 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ∈ dom (voln‘𝑋))
102 nfv 1830 . . . . . 6 𝑘𝜑
1037ffvelrnda 6267 . . . . . 6 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
104102, 1, 103, 57vonhoire 39563 . . . . 5 (𝜑 → ((voln‘𝑋)‘X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))) ∈ ℝ)
105 vonioolem2.i . . . . . . 7 𝐼 = X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘))
106105a1i 11 . . . . . 6 (𝜑𝐼 = X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘)))
107 nftru 1721 . . . . . . . . 9 𝑘
108 ioossico 12133 . . . . . . . . . 10 ((𝐴𝑘)(,)(𝐵𝑘)) ⊆ ((𝐴𝑘)[,)(𝐵𝑘))
109108a1i 11 . . . . . . . . 9 ((⊤ ∧ 𝑘𝑋) → ((𝐴𝑘)(,)(𝐵𝑘)) ⊆ ((𝐴𝑘)[,)(𝐵𝑘)))
110107, 109ixpssixp 38297 . . . . . . . 8 (⊤ → X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
111110trud 1484 . . . . . . 7 X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))
112111a1i 11 . . . . . 6 (𝜑X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
113106, 112eqsstrd 3602 . . . . 5 (𝜑𝐼X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)))
11456a1i 11 . . . . . . . 8 (𝜑 → ℝ ⊆ ℝ*)
1157, 114fssd 5970 . . . . . . 7 (𝜑𝐴:𝑋⟶ℝ*)
11622, 114fssd 5970 . . . . . . 7 (𝜑𝐵:𝑋⟶ℝ*)
1171, 6, 115, 116ioovonmbl 39568 . . . . . 6 (𝜑X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘)) ∈ dom (voln‘𝑋))
118105, 117syl5eqel 2692 . . . . 5 (𝜑𝐼 ∈ dom (voln‘𝑋))
1192, 101, 104, 113, 118meassre 39370 . . . 4 (𝜑 → ((voln‘𝑋)‘𝐼) ∈ ℝ)
1202adantr 480 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (voln‘𝑋) ∈ Meas)
12180, 24eqeltrd 2688 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) ∈ dom (voln‘𝑋))
122118adantr 480 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝐼 ∈ dom (voln‘𝑋))
12356, 103sseldi 3566 . . . . . . . . 9 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ*)
124123adantlr 747 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ∈ ℝ*)
12561rpreccld 11758 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ+)
126125ad2antlr 759 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / 𝑛) ∈ ℝ+)
1279, 126ltaddrpd 11781 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) < ((𝐴𝑘) + (1 / 𝑛)))
128 icossioo 12135 . . . . . . . 8 ((((𝐴𝑘) ∈ ℝ* ∧ (𝐵𝑘) ∈ ℝ*) ∧ ((𝐴𝑘) < ((𝐴𝑘) + (1 / 𝑛)) ∧ (𝐵𝑘) ≤ (𝐵𝑘))) → (((𝐴𝑘) + (1 / 𝑛))[,)(𝐵𝑘)) ⊆ ((𝐴𝑘)(,)(𝐵𝑘)))
129124, 59, 127, 74, 128syl22anc 1319 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐴𝑘) + (1 / 𝑛))[,)(𝐵𝑘)) ⊆ ((𝐴𝑘)(,)(𝐵𝑘)))
13027, 129ixpssixp 38297 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 (((𝐴𝑘) + (1 / 𝑛))[,)(𝐵𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘)))
13169oveq1d 6564 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)) = (((𝐴𝑘) + (1 / 𝑛))[,)(𝐵𝑘)))
132131ixpeq2dva 7809 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 (((𝐶𝑛)‘𝑘)[,)(𝐵𝑘)) = X𝑘𝑋 (((𝐴𝑘) + (1 / 𝑛))[,)(𝐵𝑘)))
13380, 132eqtrd 2644 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) = X𝑘𝑋 (((𝐴𝑘) + (1 / 𝑛))[,)(𝐵𝑘)))
134105a1i 11 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝐼 = X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘)))
135133, 134sseq12d 3597 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐷𝑛) ⊆ 𝐼X𝑘𝑋 (((𝐴𝑘) + (1 / 𝑛))[,)(𝐵𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘))))
136130, 135mpbird 246 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) ⊆ 𝐼)
137120, 6, 121, 122, 136meassle 39356 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐷𝑛)) ≤ ((voln‘𝑋)‘𝐼))
138 eqid 2610 . . . 4 (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))
1392, 3, 4, 26, 100, 119, 137, 138meaiuninc2 39375 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)))
140102, 1, 103, 58iunhoiioo 39567 . . . . . . 7 (𝜑 𝑛 ∈ ℕ X𝑘𝑋 (((𝐴𝑘) + (1 / 𝑛))[,)(𝐵𝑘)) = X𝑘𝑋 ((𝐴𝑘)(,)(𝐵𝑘)))
141133iuneq2dv 4478 . . . . . . 7 (𝜑 𝑛 ∈ ℕ (𝐷𝑛) = 𝑛 ∈ ℕ X𝑘𝑋 (((𝐴𝑘) + (1 / 𝑛))[,)(𝐵𝑘)))
142140, 141, 1063eqtr4d 2654 . . . . . 6 (𝜑 𝑛 ∈ ℕ (𝐷𝑛) = 𝐼)
143142eqcomd 2616 . . . . 5 (𝜑𝐼 = 𝑛 ∈ ℕ (𝐷𝑛))
144143fveq2d 6107 . . . 4 (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)))
145144eqcomd 2616 . . 3 (𝜑 → ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)) = ((voln‘𝑋)‘𝐼))
146139, 145breqtrd 4609 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
147 fveq2 6103 . . . . . 6 (𝑛 = 𝑚 → (𝐷𝑛) = (𝐷𝑚))
148147fveq2d 6107 . . . . 5 (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷𝑛)) = ((voln‘𝑋)‘(𝐷𝑚)))
149148cbvmptv 4678 . . . 4 (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚)))
150149a1i 11 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))))
151 vonioolem2.n . . . 4 (𝜑𝑋 ≠ ∅)
152 vonioolem2.t . . . 4 ((𝜑𝑘𝑋) → (𝐴𝑘) < (𝐵𝑘))
153149eqcomi 2619 . . . 4 (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))
154 eqcom 2617 . . . . . . . . . 10 (𝑛 = 𝑚𝑚 = 𝑛)
155154imbi1i 338 . . . . . . . . 9 ((𝑛 = 𝑚 → ((𝐶𝑛)‘𝑘) = ((𝐶𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶𝑛)‘𝑘) = ((𝐶𝑚)‘𝑘)))
156 eqcom 2617 . . . . . . . . . 10 (((𝐶𝑛)‘𝑘) = ((𝐶𝑚)‘𝑘) ↔ ((𝐶𝑚)‘𝑘) = ((𝐶𝑛)‘𝑘))
157156imbi2i 325 . . . . . . . . 9 ((𝑚 = 𝑛 → ((𝐶𝑛)‘𝑘) = ((𝐶𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶𝑚)‘𝑘) = ((𝐶𝑛)‘𝑘)))
158155, 157bitri 263 . . . . . . . 8 ((𝑛 = 𝑚 → ((𝐶𝑛)‘𝑘) = ((𝐶𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶𝑚)‘𝑘) = ((𝐶𝑛)‘𝑘)))
15982, 158mpbi 219 . . . . . . 7 (𝑚 = 𝑛 → ((𝐶𝑚)‘𝑘) = ((𝐶𝑛)‘𝑘))
160159oveq2d 6565 . . . . . 6 (𝑚 = 𝑛 → ((𝐵𝑘) − ((𝐶𝑚)‘𝑘)) = ((𝐵𝑘) − ((𝐶𝑛)‘𝑘)))
161160prodeq2ad 38659 . . . . 5 (𝑚 = 𝑛 → ∏𝑘𝑋 ((𝐵𝑘) − ((𝐶𝑚)‘𝑘)) = ∏𝑘𝑋 ((𝐵𝑘) − ((𝐶𝑛)‘𝑘)))
162161cbvmptv 4678 . . . 4 (𝑚 ∈ ℕ ↦ ∏𝑘𝑋 ((𝐵𝑘) − ((𝐶𝑚)‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑘𝑋 ((𝐵𝑘) − ((𝐶𝑛)‘𝑘)))
163 eqid 2610 . . . 4 inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < ) = inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < )
164 eqid 2610 . . . 4 ((⌊‘(1 / inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < ))) + 1) = ((⌊‘(1 / inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < ))) + 1)
165 fveq2 6103 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐵𝑗) = (𝐵𝑘))
166 fveq2 6103 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐴𝑗) = (𝐴𝑘))
167165, 166oveq12d 6567 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐵𝑗) − (𝐴𝑗)) = ((𝐵𝑘) − (𝐴𝑘)))
168167cbvmptv 4678 . . . . . . . . . 10 (𝑗𝑋 ↦ ((𝐵𝑗) − (𝐴𝑗))) = (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘)))
169168rneqi 5273 . . . . . . . . 9 ran (𝑗𝑋 ↦ ((𝐵𝑗) − (𝐴𝑗))) = ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘)))
170169infeq1i 8267 . . . . . . . 8 inf(ran (𝑗𝑋 ↦ ((𝐵𝑗) − (𝐴𝑗))), ℝ, < ) = inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < )
171170oveq2i 6560 . . . . . . 7 (1 / inf(ran (𝑗𝑋 ↦ ((𝐵𝑗) − (𝐴𝑗))), ℝ, < )) = (1 / inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < ))
172171fveq2i 6106 . . . . . 6 (⌊‘(1 / inf(ran (𝑗𝑋 ↦ ((𝐵𝑗) − (𝐴𝑗))), ℝ, < ))) = (⌊‘(1 / inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < )))
173172oveq1i 6559 . . . . 5 ((⌊‘(1 / inf(ran (𝑗𝑋 ↦ ((𝐵𝑗) − (𝐴𝑗))), ℝ, < ))) + 1) = ((⌊‘(1 / inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < ))) + 1)
174173fveq2i 6106 . . . 4 (ℤ‘((⌊‘(1 / inf(ran (𝑗𝑋 ↦ ((𝐵𝑗) − (𝐴𝑗))), ℝ, < ))) + 1)) = (ℤ‘((⌊‘(1 / inf(ran (𝑘𝑋 ↦ ((𝐵𝑘) − (𝐴𝑘))), ℝ, < ))) + 1))
1751, 7, 22, 151, 152, 15, 25, 153, 162, 163, 164, 174vonioolem1 39571 . . 3 (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
176150, 175eqbrtrd 4605 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
177 climuni 14131 . 2 (((𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘𝐼) ∧ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘))) → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
178146, 176, 177syl2anc 691 1 (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Xcixp 7794  Fincfn 7841  infcinf 8230  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  ℤ≥cuz 11563  ℝ+crp 11708  (,)cioo 12046  [,)cico 12048  ⌊cfl 12453   ⇝ cli 14063  ∏cprod 14474  Meascmea 39342  volncvoln 39428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-prod 14475  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-pws 15933  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-rnghom 18538  df-drng 18572  df-field 18573  df-subrg 18601  df-abv 18640  df-staf 18668  df-srng 18669  df-lmod 18688  df-lss 18754  df-lmhm 18843  df-lvec 18924  df-sra 18993  df-rgmod 18994  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-refld 19770  df-phl 19790  df-dsmm 19895  df-frlm 19910  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-cnp 20842  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-xms 21935  df-ms 21936  df-tms 21937  df-nm 22197  df-ngp 22198  df-tng 22199  df-nrg 22200  df-nlm 22201  df-cncf 22489  df-clm 22671  df-cph 22776  df-tch 22777  df-rrx 22981  df-ovol 23040  df-vol 23041  df-salg 39205  df-sumge0 39256  df-mea 39343  df-ome 39380  df-caragen 39382  df-ovoln 39427  df-voln 39429 This theorem is referenced by:  vonioo  39573
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