Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vonicclem2 Structured version   Visualization version   GIF version

Theorem vonicclem2 43310
 Description: The n-dimensional Lebesgue measure of closed intervals. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypotheses
Ref Expression
vonicclem2.x (𝜑𝑋 ∈ Fin)
vonicclem2.a (𝜑𝐴:𝑋⟶ℝ)
vonicclem2.b (𝜑𝐵:𝑋⟶ℝ)
vonicclem2.n (𝜑𝑋 ≠ ∅)
vonicclem2.t ((𝜑𝑘𝑋) → (𝐴𝑘) ≤ (𝐵𝑘))
vonicclem2.i 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘))
vonicclem2.c 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))))
vonicclem2.d 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
Assertion
Ref Expression
vonicclem2 (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 vonicclem2
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . 4 𝑛𝜑
2 vonicclem2.x . . . . 5 (𝜑𝑋 ∈ Fin)
32vonmea 43200 . . . 4 (𝜑 → (voln‘𝑋) ∈ Meas)
4 1zzd 12005 . . . 4 (𝜑 → 1 ∈ ℤ)
5 nnuz 12273 . . . 4 ℕ = (ℤ‘1)
62adantr 484 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
7 eqid 2801 . . . . . 6 dom (voln‘𝑋) = dom (voln‘𝑋)
8 vonicclem2.a . . . . . . 7 (𝜑𝐴:𝑋⟶ℝ)
98adantr 484 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
10 vonicclem2.b . . . . . . . . . . 11 (𝜑𝐵:𝑋⟶ℝ)
1110ffvelrnda 6832 . . . . . . . . . 10 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
1211adantlr 714 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
13 nnrecre 11671 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
1413ad2antlr 726 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / 𝑛) ∈ ℝ)
1512, 14readdcld 10663 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / 𝑛)) ∈ ℝ)
1615fmpttd 6860 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))):𝑋⟶ℝ)
17 vonicclem2.c . . . . . . . . . 10 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))))
1817a1i 11 . . . . . . . . 9 (𝜑𝐶 = (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛)))))
192mptexd 6968 . . . . . . . . . 10 (𝜑 → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) ∈ V)
2019adantr 484 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) ∈ V)
2118, 20fvmpt2d 6762 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))))
2221feq1d 6476 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝐶𝑛):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
2316, 22mpbird 260 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛):𝑋⟶ℝ)
246, 7, 9, 23hoimbl 43257 . . . . 5 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) ∈ dom (voln‘𝑋))
25 vonicclem2.d . . . . 5 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
2624, 25fmptd 6859 . . . 4 (𝜑𝐷:ℕ⟶dom (voln‘𝑋))
27 nfv 1915 . . . . . 6 𝑘(𝜑𝑛 ∈ ℕ)
28 ressxr 10678 . . . . . . . . 9 ℝ ⊆ ℝ*
298ffvelrnda 6832 . . . . . . . . 9 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3028, 29sseldi 3916 . . . . . . . 8 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ*)
3130adantlr 714 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ∈ ℝ*)
32 ovexd 7174 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / 𝑛)) ∈ V)
3321, 32fvmpt2d 6762 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) = ((𝐵𝑘) + (1 / 𝑛)))
3433, 15eqeltrd 2893 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ)
3534rexrd 10684 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ*)
369ffvelrnda 6832 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3736leidd 11199 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ≤ (𝐴𝑘))
38 1red 10635 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 1 ∈ ℝ)
39 nnre 11636 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
4039, 38readdcld 10663 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
41 peano2nn 11641 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
42 nnne0 11663 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
4341, 42syl 17 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
4438, 40, 43redivcld 11461 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
4544ad2antlr 726 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
4639ltp1d 11563 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
47 nnrp 12392 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
4841nnrpd 12421 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
4947, 48ltrecd 12441 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛)))
5046, 49mpbid 235 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) < (1 / 𝑛))
5144, 13, 50ltled 10781 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
5251ad2antlr 726 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
5345, 14, 12, 52leadd2dd 11248 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵𝑘) + (1 / 𝑛)))
54 oveq2 7147 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
5554oveq2d 7155 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((𝐵𝑘) + (1 / 𝑛)) = ((𝐵𝑘) + (1 / 𝑚)))
5655mpteq2dv 5129 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
5756cbvmptv 5136 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
5817, 57eqtri 2824 . . . . . . . . . . 11 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
59 oveq2 7147 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
6059oveq2d 7155 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → ((𝐵𝑘) + (1 / 𝑚)) = ((𝐵𝑘) + (1 / (𝑛 + 1))))
6160mpteq2dv 5129 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))))
62 simpr 488 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
6362peano2nnd 11646 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
646mptexd 6968 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))) ∈ V)
6558, 61, 63, 64fvmptd3 6772 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))))
66 ovexd 7174 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / (𝑛 + 1))) ∈ V)
6765, 66fvmpt2d 6762 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐵𝑘) + (1 / (𝑛 + 1))))
6867, 33breq12d 5046 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘) ↔ ((𝐵𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵𝑘) + (1 / 𝑛))))
6953, 68mpbird 260 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘))
70 icossico 12799 . . . . . . 7 ((((𝐴𝑘) ∈ ℝ* ∧ ((𝐶𝑛)‘𝑘) ∈ ℝ*) ∧ ((𝐴𝑘) ≤ (𝐴𝑘) ∧ ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘))) → ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
7131, 35, 37, 69, 70syl22anc 837 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
7227, 71ixpssixp 41715 . . . . 5 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
73 fveq2 6649 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐶𝑛) = (𝐶𝑚))
7473fveq1d 6651 . . . . . . . . . . 11 (𝑛 = 𝑚 → ((𝐶𝑛)‘𝑘) = ((𝐶𝑚)‘𝑘))
7574oveq2d 7155 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7675ixpeq2dv 8464 . . . . . . . . 9 (𝑛 = 𝑚X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7776cbvmptv 5136 . . . . . . . 8 (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7825, 77eqtri 2824 . . . . . . 7 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
79 fveq2 6649 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (𝐶𝑚) = (𝐶‘(𝑛 + 1)))
8079fveq1d 6651 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → ((𝐶𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
8180oveq2d 7155 . . . . . . . 8 (𝑚 = (𝑛 + 1) → ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)) = ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
8281ixpeq2dv 8464 . . . . . . 7 (𝑚 = (𝑛 + 1) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
83 ovex 7172 . . . . . . . . . 10 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
8483rgenw 3121 . . . . . . . . 9 𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
85 ixpexg 8473 . . . . . . . . 9 (∀𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V)
8684, 85ax-mp 5 . . . . . . . 8 X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
8786a1i 11 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V)
8878, 82, 63, 87fvmptd3 6772 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
8925a1i 11 . . . . . . 7 (𝜑𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))))
9024elexd 3464 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) ∈ V)
9189, 90fvmpt2d 6762 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
9288, 91sseq12d 3951 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝐷‘(𝑛 + 1)) ⊆ (𝐷𝑛) ↔ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))))
9372, 92mpbird 260 . . . 4 ((𝜑𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) ⊆ (𝐷𝑛))
94 1nn 11640 . . . . . 6 1 ∈ ℕ
9594, 5eleqtri 2891 . . . . 5 1 ∈ (ℤ‘1)
9695a1i 11 . . . 4 (𝜑 → 1 ∈ (ℤ‘1))
97 fveq2 6649 . . . . . . . . . 10 (𝑛 = 1 → (𝐶𝑛) = (𝐶‘1))
9897fveq1d 6651 . . . . . . . . 9 (𝑛 = 1 → ((𝐶𝑛)‘𝑘) = ((𝐶‘1)‘𝑘))
9998oveq2d 7155 . . . . . . . 8 (𝑛 = 1 → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
10099ixpeq2dv 8464 . . . . . . 7 (𝑛 = 1 → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
10194a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℕ)
102 ovex 7172 . . . . . . . . . 10 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
103102rgenw 3121 . . . . . . . . 9 𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
104 ixpexg 8473 . . . . . . . . 9 (∀𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V)
105103, 104ax-mp 5 . . . . . . . 8 X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
106105a1i 11 . . . . . . 7 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V)
10725, 100, 101, 106fvmptd3 6772 . . . . . 6 (𝜑 → (𝐷‘1) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
108107fveq2d 6653 . . . . 5 (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) = ((voln‘𝑋)‘X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘))))
109 nfv 1915 . . . . . 6 𝑘𝜑
110 simpl 486 . . . . . . 7 ((𝜑𝑘𝑋) → 𝜑)
11194a1i 11 . . . . . . 7 ((𝜑𝑘𝑋) → 1 ∈ ℕ)
112 simpr 488 . . . . . . 7 ((𝜑𝑘𝑋) → 𝑘𝑋)
11394elexi 3463 . . . . . . . 8 1 ∈ V
114 eleq1 2880 . . . . . . . . . . 11 (𝑛 = 1 → (𝑛 ∈ ℕ ↔ 1 ∈ ℕ))
115114anbi2d 631 . . . . . . . . . 10 (𝑛 = 1 → ((𝜑𝑛 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
116115anbi1d 632 . . . . . . . . 9 (𝑛 = 1 → (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) ↔ ((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋)))
11798eleq1d 2877 . . . . . . . . 9 (𝑛 = 1 → (((𝐶𝑛)‘𝑘) ∈ ℝ ↔ ((𝐶‘1)‘𝑘) ∈ ℝ))
118116, 117imbi12d 348 . . . . . . . 8 (𝑛 = 1 → ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ) ↔ (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)))
119113, 118, 34vtocl 3510 . . . . . . 7 (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
120110, 111, 112, 119syl21anc 836 . . . . . 6 ((𝜑𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
121109, 2, 29, 120vonhoire 43298 . . . . 5 (𝜑 → ((voln‘𝑋)‘X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘))) ∈ ℝ)
122108, 121eqeltrd 2893 . . . 4 (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) ∈ ℝ)
123 eqid 2801 . . . 4 (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))
1241, 3, 4, 5, 26, 93, 96, 122, 123meaiininc 43113 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)))
125109, 29, 11iinhoiicc 43300 . . . . . . 7 (𝜑 𝑛 ∈ ℕ X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))) = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘)))
12633oveq2d 7155 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
127126ixpeq2dva 8463 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
12891, 127eqtrd 2836 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
129128iineq2dv 4909 . . . . . . 7 (𝜑 𝑛 ∈ ℕ (𝐷𝑛) = 𝑛 ∈ ℕ X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
130 vonicclem2.i . . . . . . . 8 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘))
131130a1i 11 . . . . . . 7 (𝜑𝐼 = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘)))
132125, 129, 1313eqtr4d 2846 . . . . . 6 (𝜑 𝑛 ∈ ℕ (𝐷𝑛) = 𝐼)
133132eqcomd 2807 . . . . 5 (𝜑𝐼 = 𝑛 ∈ ℕ (𝐷𝑛))
134133fveq2d 6653 . . . 4 (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)))
135134eqcomd 2807 . . 3 (𝜑 → ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)) = ((voln‘𝑋)‘𝐼))
136124, 135breqtrd 5059 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
137 2fveq3 6654 . . . . 5 (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷𝑛)) = ((voln‘𝑋)‘(𝐷𝑚)))
138137cbvmptv 5136 . . . 4 (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚)))
139138a1i 11 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))))
140 vonicclem2.n . . . 4 (𝜑𝑋 ≠ ∅)
141 vonicclem2.t . . . 4 ((𝜑𝑘𝑋) → (𝐴𝑘) ≤ (𝐵𝑘))
142138eqcomi 2810 . . . 4 (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))
1432, 8, 10, 140, 141, 17, 25, 142vonicclem1 43309 . . 3 (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
144139, 143eqbrtrd 5055 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
145 climuni 14904 . 2 (((𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘𝐼) ∧ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘))) → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
146136, 144, 145syl2anc 587 1 (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  Vcvv 3444   ⊆ wss 3884  ∅c0 4246  ∩ ciin 4885   class class class wbr 5033   ↦ cmpt 5113  dom cdm 5523  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  Xcixp 8448  Fincfn 8496  ℝcr 10529  0cc0 10530  1c1 10531   + caddc 10533  ℝ*cxr 10667   < clt 10668   ≤ cle 10669   − cmin 10863   / cdiv 11290  ℕcn 11629  ℤ≥cuz 12235  [,)cico 12732  [,]cicc 12733   ⇝ cli 14836  ∏cprod 15254  volncvoln 43164 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cc 9850  ax-ac2 9878  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609  ax-mulf 10610 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-disj 4999  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-ac 9531  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  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 12890  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-rlim 14841  df-sum 15038  df-prod 15255  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-starv 16575  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-unif 16583  df-hom 16584  df-cco 16585  df-rest 16691  df-topn 16692  df-0g 16710  df-gsum 16711  df-topgen 16712  df-pt 16713  df-prds 16716  df-xrs 16770  df-qtop 16775  df-imas 16776  df-xps 16778  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-grp 18101  df-minusg 18102  df-mulg 18220  df-subg 18271  df-cntz 18442  df-cmn 18903  df-abl 18904  df-mgp 19236  df-ur 19248  df-ring 19295  df-cring 19296  df-oppr 19372  df-dvdsr 19390  df-unit 19391  df-invr 19421  df-dvr 19432  df-drng 19500  df-psmet 20086  df-xmet 20087  df-met 20088  df-bl 20089  df-mopn 20090  df-cnfld 20095  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cn 21835  df-cnp 21836  df-cmp 21995  df-tx 22170  df-hmeo 22363  df-xms 22930  df-ms 22931  df-tms 22932  df-cncf 23486  df-ovol 24071  df-vol 24072  df-salg 42938  df-sumge0 42989  df-mea 43076  df-ome 43116  df-caragen 43118  df-ovoln 43163  df-voln 43165 This theorem is referenced by:  vonicc  43311
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