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Theorem vonicclem2 42530
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 1896 . . . 4 𝑛𝜑
2 vonicclem2.x . . . . 5 (𝜑𝑋 ∈ Fin)
32vonmea 42420 . . . 4 (𝜑 → (voln‘𝑋) ∈ Meas)
4 1zzd 11867 . . . 4 (𝜑 → 1 ∈ ℤ)
5 nnuz 12134 . . . 4 ℕ = (ℤ‘1)
62adantr 481 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
7 eqid 2797 . . . . . 6 dom (voln‘𝑋) = dom (voln‘𝑋)
8 vonicclem2.a . . . . . . 7 (𝜑𝐴:𝑋⟶ℝ)
98adantr 481 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
10 vonicclem2.b . . . . . . . . . . 11 (𝜑𝐵:𝑋⟶ℝ)
1110ffvelrnda 6723 . . . . . . . . . 10 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
1211adantlr 711 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
13 nnrecre 11533 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
1413ad2antlr 723 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / 𝑛) ∈ ℝ)
1512, 14readdcld 10523 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / 𝑛)) ∈ ℝ)
1615fmpttd 6749 . . . . . . 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 6860 . . . . . . . . . 10 (𝜑 → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) ∈ V)
2019adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) ∈ V)
2118, 20fvmpt2d 6654 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))))
2221feq1d 6374 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝐶𝑛):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
2316, 22mpbird 258 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛):𝑋⟶ℝ)
246, 7, 9, 23hoimbl 42477 . . . . 5 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) ∈ dom (voln‘𝑋))
25 vonicclem2.d . . . . 5 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
2624, 25fmptd 6748 . . . 4 (𝜑𝐷:ℕ⟶dom (voln‘𝑋))
27 nfv 1896 . . . . . 6 𝑘(𝜑𝑛 ∈ ℕ)
28 ressxr 10538 . . . . . . . . 9 ℝ ⊆ ℝ*
298ffvelrnda 6723 . . . . . . . . 9 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3028, 29sseldi 3893 . . . . . . . 8 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ*)
3130adantlr 711 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ∈ ℝ*)
32 ovexd 7057 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / 𝑛)) ∈ V)
3321, 32fvmpt2d 6654 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) = ((𝐵𝑘) + (1 / 𝑛)))
3433, 15eqeltrd 2885 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ)
3534rexrd 10544 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ*)
369ffvelrnda 6723 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3736leidd 11060 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ≤ (𝐴𝑘))
38 1red 10495 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 1 ∈ ℝ)
39 nnre 11499 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
4039, 38readdcld 10523 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
41 peano2nn 11504 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
42 nnne0 11525 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
4341, 42syl 17 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
4438, 40, 43redivcld 11322 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
4544ad2antlr 723 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
4639ltp1d 11424 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
47 nnrp 12254 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
4841nnrpd 12283 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
4947, 48ltrecd 12303 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛)))
5046, 49mpbid 233 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) < (1 / 𝑛))
5144, 13, 50ltled 10641 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
5251ad2antlr 723 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
5345, 14, 12, 52leadd2dd 11109 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵𝑘) + (1 / 𝑛)))
54 oveq2 7031 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
5554oveq2d 7039 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((𝐵𝑘) + (1 / 𝑛)) = ((𝐵𝑘) + (1 / 𝑚)))
5655mpteq2dv 5063 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
5756cbvmptv 5068 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
5817, 57eqtri 2821 . . . . . . . . . . 11 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
59 oveq2 7031 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
6059oveq2d 7039 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → ((𝐵𝑘) + (1 / 𝑚)) = ((𝐵𝑘) + (1 / (𝑛 + 1))))
6160mpteq2dv 5063 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))))
62 simpr 485 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
6362peano2nnd 11509 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
646mptexd 6860 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))) ∈ V)
6558, 61, 63, 64fvmptd3 6664 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))))
66 ovexd 7057 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / (𝑛 + 1))) ∈ V)
6765, 66fvmpt2d 6654 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐵𝑘) + (1 / (𝑛 + 1))))
6867, 33breq12d 4981 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘) ↔ ((𝐵𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵𝑘) + (1 / 𝑛))))
6953, 68mpbird 258 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘))
70 icossico 12660 . . . . . . 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 835 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
7227, 71ixpssixp 40918 . . . . 5 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
73 fveq2 6545 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐶𝑛) = (𝐶𝑚))
7473fveq1d 6547 . . . . . . . . . . 11 (𝑛 = 𝑚 → ((𝐶𝑛)‘𝑘) = ((𝐶𝑚)‘𝑘))
7574oveq2d 7039 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7675ixpeq2dv 8333 . . . . . . . . 9 (𝑛 = 𝑚X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7776cbvmptv 5068 . . . . . . . 8 (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7825, 77eqtri 2821 . . . . . . 7 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
79 fveq2 6545 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (𝐶𝑚) = (𝐶‘(𝑛 + 1)))
8079fveq1d 6547 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → ((𝐶𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
8180oveq2d 7039 . . . . . . . 8 (𝑚 = (𝑛 + 1) → ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)) = ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
8281ixpeq2dv 8333 . . . . . . 7 (𝑚 = (𝑛 + 1) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
83 ovex 7055 . . . . . . . . . 10 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
8483rgenw 3119 . . . . . . . . 9 𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
85 ixpexg 8341 . . . . . . . . 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 6664 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
8925a1i 11 . . . . . . 7 (𝜑𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))))
9024elexd 3460 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) ∈ V)
9189, 90fvmpt2d 6654 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
9288, 91sseq12d 3927 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝐷‘(𝑛 + 1)) ⊆ (𝐷𝑛) ↔ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))))
9372, 92mpbird 258 . . . 4 ((𝜑𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) ⊆ (𝐷𝑛))
94 1nn 11503 . . . . . 6 1 ∈ ℕ
9594, 5eleqtri 2883 . . . . 5 1 ∈ (ℤ‘1)
9695a1i 11 . . . 4 (𝜑 → 1 ∈ (ℤ‘1))
97 fveq2 6545 . . . . . . . . . 10 (𝑛 = 1 → (𝐶𝑛) = (𝐶‘1))
9897fveq1d 6547 . . . . . . . . 9 (𝑛 = 1 → ((𝐶𝑛)‘𝑘) = ((𝐶‘1)‘𝑘))
9998oveq2d 7039 . . . . . . . 8 (𝑛 = 1 → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
10099ixpeq2dv 8333 . . . . . . 7 (𝑛 = 1 → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
10194a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℕ)
102 ovex 7055 . . . . . . . . . 10 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
103102rgenw 3119 . . . . . . . . 9 𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
104 ixpexg 8341 . . . . . . . . 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 6664 . . . . . 6 (𝜑 → (𝐷‘1) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
108107fveq2d 6549 . . . . 5 (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) = ((voln‘𝑋)‘X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘))))
109 nfv 1896 . . . . . 6 𝑘𝜑
110 simpl 483 . . . . . . 7 ((𝜑𝑘𝑋) → 𝜑)
11194a1i 11 . . . . . . 7 ((𝜑𝑘𝑋) → 1 ∈ ℕ)
112 simpr 485 . . . . . . 7 ((𝜑𝑘𝑋) → 𝑘𝑋)
11394elexi 3459 . . . . . . . 8 1 ∈ V
114 eleq1 2872 . . . . . . . . . . 11 (𝑛 = 1 → (𝑛 ∈ ℕ ↔ 1 ∈ ℕ))
115114anbi2d 628 . . . . . . . . . 10 (𝑛 = 1 → ((𝜑𝑛 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
116115anbi1d 629 . . . . . . . . 9 (𝑛 = 1 → (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) ↔ ((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋)))
11798eleq1d 2869 . . . . . . . . 9 (𝑛 = 1 → (((𝐶𝑛)‘𝑘) ∈ ℝ ↔ ((𝐶‘1)‘𝑘) ∈ ℝ))
118116, 117imbi12d 346 . . . . . . . 8 (𝑛 = 1 → ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ) ↔ (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)))
119113, 118, 34vtocl 3505 . . . . . . 7 (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
120110, 111, 112, 119syl21anc 834 . . . . . 6 ((𝜑𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
121109, 2, 29, 120vonhoire 42518 . . . . 5 (𝜑 → ((voln‘𝑋)‘X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘))) ∈ ℝ)
122108, 121eqeltrd 2885 . . . 4 (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) ∈ ℝ)
123 eqid 2797 . . . 4 (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))
1241, 3, 4, 5, 26, 93, 96, 122, 123meaiininc 42333 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)))
125109, 29, 11iinhoiicc 42520 . . . . . . 7 (𝜑 𝑛 ∈ ℕ X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))) = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘)))
12633oveq2d 7039 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
127126ixpeq2dva 8332 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
12891, 127eqtrd 2833 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
129128iineq2dv 4855 . . . . . . 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 2843 . . . . . 6 (𝜑 𝑛 ∈ ℕ (𝐷𝑛) = 𝐼)
133132eqcomd 2803 . . . . 5 (𝜑𝐼 = 𝑛 ∈ ℕ (𝐷𝑛))
134133fveq2d 6549 . . . 4 (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)))
135134eqcomd 2803 . . 3 (𝜑 → ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)) = ((voln‘𝑋)‘𝐼))
136124, 135breqtrd 4994 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
137 2fveq3 6550 . . . . 5 (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷𝑛)) = ((voln‘𝑋)‘(𝐷𝑚)))
138137cbvmptv 5068 . . . 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 2806 . . . 4 (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))
1432, 8, 10, 140, 141, 17, 25, 142vonicclem1 42529 . . 3 (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
144139, 143eqbrtrd 4990 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
145 climuni 14747 . 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 584 1 (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wcel 2083  wne 2986  wral 3107  Vcvv 3440  wss 3865  c0 4217   ciin 4832   class class class wbr 4968  cmpt 5047  dom cdm 5450  wf 6228  cfv 6232  (class class class)co 7023  Xcixp 8317  Fincfn 8364  cr 10389  0cc0 10390  1c1 10391   + caddc 10393  *cxr 10527   < clt 10528  cle 10529  cmin 10723   / cdiv 11151  cn 11492  cuz 12097  [,)cico 12594  [,]cicc 12595  cli 14679  cprod 15096  volncvoln 42384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cc 9710  ax-ac2 9738  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468  ax-addf 10469  ax-mulf 10470
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-disj 4937  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-of 7274  df-om 7444  df-1st 7552  df-2nd 7553  df-supp 7689  df-tpos 7750  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-2o 7961  df-oadd 7964  df-omul 7965  df-er 8146  df-map 8265  df-pm 8266  df-ixp 8318  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-fsupp 8687  df-fi 8728  df-sup 8759  df-inf 8760  df-oi 8827  df-dju 9183  df-card 9221  df-acn 9224  df-ac 9395  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-4 11556  df-5 11557  df-6 11558  df-7 11559  df-8 11560  df-9 11561  df-n0 11752  df-z 11836  df-dec 11953  df-uz 12098  df-q 12202  df-rp 12244  df-xneg 12361  df-xadd 12362  df-xmul 12363  df-ioo 12596  df-ico 12598  df-icc 12599  df-fz 12747  df-fzo 12888  df-fl 13016  df-seq 13224  df-exp 13284  df-hash 13545  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-clim 14683  df-rlim 14684  df-sum 14881  df-prod 15097  df-struct 16318  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-ress 16324  df-plusg 16411  df-mulr 16412  df-starv 16413  df-sca 16414  df-vsca 16415  df-ip 16416  df-tset 16417  df-ple 16418  df-ds 16420  df-unif 16421  df-hom 16422  df-cco 16423  df-rest 16529  df-topn 16530  df-0g 16548  df-gsum 16549  df-topgen 16550  df-pt 16551  df-prds 16554  df-xrs 16608  df-qtop 16613  df-imas 16614  df-xps 16616  df-mre 16690  df-mrc 16691  df-acs 16693  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-submnd 17779  df-grp 17868  df-minusg 17869  df-mulg 17986  df-subg 18034  df-cntz 18192  df-cmn 18639  df-abl 18640  df-mgp 18934  df-ur 18946  df-ring 18993  df-cring 18994  df-oppr 19067  df-dvdsr 19085  df-unit 19086  df-invr 19116  df-dvr 19127  df-drng 19198  df-psmet 20223  df-xmet 20224  df-met 20225  df-bl 20226  df-mopn 20227  df-cnfld 20232  df-top 21190  df-topon 21207  df-topsp 21229  df-bases 21242  df-cn 21523  df-cnp 21524  df-cmp 21683  df-tx 21858  df-hmeo 22051  df-xms 22617  df-ms 22618  df-tms 22619  df-cncf 23173  df-ovol 23752  df-vol 23753  df-salg 42158  df-sumge0 42209  df-mea 42296  df-ome 42336  df-caragen 42338  df-ovoln 42383  df-voln 42385
This theorem is referenced by:  vonicc  42531
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