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Theorem vonicclem2 41470
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 2009 . . . 4 𝑛𝜑
2 vonicclem2.x . . . . 5 (𝜑𝑋 ∈ Fin)
32vonmea 41360 . . . 4 (𝜑 → (voln‘𝑋) ∈ Meas)
4 1zzd 11655 . . . 4 (𝜑 → 1 ∈ ℤ)
5 nnuz 11923 . . . 4 ℕ = (ℤ‘1)
62adantr 472 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
7 eqid 2765 . . . . . 6 dom (voln‘𝑋) = dom (voln‘𝑋)
8 vonicclem2.a . . . . . . 7 (𝜑𝐴:𝑋⟶ℝ)
98adantr 472 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
10 vonicclem2.b . . . . . . . . . . 11 (𝜑𝐵:𝑋⟶ℝ)
1110ffvelrnda 6549 . . . . . . . . . 10 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
1211adantlr 706 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
13 nnrecre 11314 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
1413ad2antlr 718 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / 𝑛) ∈ ℝ)
1512, 14readdcld 10323 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / 𝑛)) ∈ ℝ)
1615fmpttd 6575 . . . . . . 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 6680 . . . . . . . . . 10 (𝜑 → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) ∈ V)
2019adantr 472 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) ∈ V)
2118, 20fvmpt2d 6482 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))))
2221feq1d 6208 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝐶𝑛):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
2316, 22mpbird 248 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛):𝑋⟶ℝ)
246, 7, 9, 23hoimbl 41417 . . . . 5 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) ∈ dom (voln‘𝑋))
25 vonicclem2.d . . . . 5 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
2624, 25fmptd 6574 . . . 4 (𝜑𝐷:ℕ⟶dom (voln‘𝑋))
27 nfv 2009 . . . . . 6 𝑘(𝜑𝑛 ∈ ℕ)
28 ressxr 10337 . . . . . . . . 9 ℝ ⊆ ℝ*
298ffvelrnda 6549 . . . . . . . . 9 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3028, 29sseldi 3759 . . . . . . . 8 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ*)
3130adantlr 706 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ∈ ℝ*)
32 ovexd 6876 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / 𝑛)) ∈ V)
3321, 32fvmpt2d 6482 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) = ((𝐵𝑘) + (1 / 𝑛)))
3433, 15eqeltrd 2844 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ)
3534rexrd 10343 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ*)
369ffvelrnda 6549 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3736leidd 10848 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ≤ (𝐴𝑘))
38 1red 10294 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 1 ∈ ℝ)
39 nnre 11282 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
4039, 38readdcld 10323 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
41 peano2nn 11288 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
42 nnne0 11310 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
4341, 42syl 17 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
4438, 40, 43redivcld 11107 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
4544ad2antlr 718 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
4639ltp1d 11208 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
47 nnrp 12041 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
4841nnrpd 12068 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
4947, 48ltrecd 12088 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛)))
5046, 49mpbid 223 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) < (1 / 𝑛))
5144, 13, 50ltled 10439 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
5251ad2antlr 718 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
5345, 14, 12, 52leadd2dd 10896 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵𝑘) + (1 / 𝑛)))
54 oveq2 6850 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
5554oveq2d 6858 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → ((𝐵𝑘) + (1 / 𝑛)) = ((𝐵𝑘) + (1 / 𝑚)))
5655mpteq2dv 4904 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
5756cbvmptv 4909 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
5817, 57eqtri 2787 . . . . . . . . . . . 12 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
5958a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚)))))
60 oveq2 6850 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
6160oveq2d 6858 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → ((𝐵𝑘) + (1 / 𝑚)) = ((𝐵𝑘) + (1 / (𝑛 + 1))))
6261mpteq2dv 4904 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))))
6362adantl 473 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = (𝑛 + 1)) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))))
64 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
6564peano2nnd 11293 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
666mptexd 6680 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))) ∈ V)
6759, 63, 65, 66fvmptd 6477 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))))
68 ovexd 6876 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / (𝑛 + 1))) ∈ V)
6967, 68fvmpt2d 6482 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐵𝑘) + (1 / (𝑛 + 1))))
7069, 33breq12d 4822 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘) ↔ ((𝐵𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵𝑘) + (1 / 𝑛))))
7153, 70mpbird 248 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘))
72 icossico 12445 . . . . . . 7 ((((𝐴𝑘) ∈ ℝ* ∧ ((𝐶𝑛)‘𝑘) ∈ ℝ*) ∧ ((𝐴𝑘) ≤ (𝐴𝑘) ∧ ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘))) → ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
7331, 35, 37, 71, 72syl22anc 867 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
7427, 73ixpssixp 39852 . . . . 5 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
75 fveq2 6375 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐶𝑛) = (𝐶𝑚))
7675fveq1d 6377 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝐶𝑛)‘𝑘) = ((𝐶𝑚)‘𝑘))
7776oveq2d 6858 . . . . . . . . . . 11 (𝑛 = 𝑚 → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7877ixpeq2dv 8129 . . . . . . . . . 10 (𝑛 = 𝑚X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7978cbvmptv 4909 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
8025, 79eqtri 2787 . . . . . . . 8 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
8180a1i 11 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘))))
82 fveq2 6375 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (𝐶𝑚) = (𝐶‘(𝑛 + 1)))
8382fveq1d 6377 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → ((𝐶𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
8483oveq2d 6858 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)) = ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
8584ixpeq2dv 8129 . . . . . . . 8 (𝑚 = (𝑛 + 1) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
8685adantl 473 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 = (𝑛 + 1)) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
87 ovex 6874 . . . . . . . . . 10 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
8887rgenw 3071 . . . . . . . . 9 𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
89 ixpexg 8137 . . . . . . . . 9 (∀𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V)
9088, 89ax-mp 5 . . . . . . . 8 X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
9190a1i 11 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V)
9281, 86, 65, 91fvmptd 6477 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
9325a1i 11 . . . . . . 7 (𝜑𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))))
9424elexd 3367 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) ∈ V)
9593, 94fvmpt2d 6482 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
9692, 95sseq12d 3794 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝐷‘(𝑛 + 1)) ⊆ (𝐷𝑛) ↔ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))))
9774, 96mpbird 248 . . . 4 ((𝜑𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) ⊆ (𝐷𝑛))
98 1nn 11287 . . . . . 6 1 ∈ ℕ
9998, 5eleqtri 2842 . . . . 5 1 ∈ (ℤ‘1)
10099a1i 11 . . . 4 (𝜑 → 1 ∈ (ℤ‘1))
101 fveq2 6375 . . . . . . . . . . 11 (𝑛 = 1 → (𝐶𝑛) = (𝐶‘1))
102101fveq1d 6377 . . . . . . . . . 10 (𝑛 = 1 → ((𝐶𝑛)‘𝑘) = ((𝐶‘1)‘𝑘))
103102oveq2d 6858 . . . . . . . . 9 (𝑛 = 1 → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
104103ixpeq2dv 8129 . . . . . . . 8 (𝑛 = 1 → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
105104adantl 473 . . . . . . 7 ((𝜑𝑛 = 1) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
10698a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℕ)
107 ovex 6874 . . . . . . . . . 10 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
108107rgenw 3071 . . . . . . . . 9 𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
109 ixpexg 8137 . . . . . . . . 9 (∀𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V)
110108, 109ax-mp 5 . . . . . . . 8 X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
111110a1i 11 . . . . . . 7 (𝜑X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V)
11293, 105, 106, 111fvmptd 6477 . . . . . 6 (𝜑 → (𝐷‘1) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
113112fveq2d 6379 . . . . 5 (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) = ((voln‘𝑋)‘X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘))))
114 nfv 2009 . . . . . 6 𝑘𝜑
115 simpl 474 . . . . . . 7 ((𝜑𝑘𝑋) → 𝜑)
11698a1i 11 . . . . . . 7 ((𝜑𝑘𝑋) → 1 ∈ ℕ)
117 simpr 477 . . . . . . 7 ((𝜑𝑘𝑋) → 𝑘𝑋)
11898elexi 3366 . . . . . . . 8 1 ∈ V
119 eleq1 2832 . . . . . . . . . . 11 (𝑛 = 1 → (𝑛 ∈ ℕ ↔ 1 ∈ ℕ))
120119anbi2d 622 . . . . . . . . . 10 (𝑛 = 1 → ((𝜑𝑛 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
121120anbi1d 623 . . . . . . . . 9 (𝑛 = 1 → (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) ↔ ((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋)))
122102eleq1d 2829 . . . . . . . . 9 (𝑛 = 1 → (((𝐶𝑛)‘𝑘) ∈ ℝ ↔ ((𝐶‘1)‘𝑘) ∈ ℝ))
123121, 122imbi12d 335 . . . . . . . 8 (𝑛 = 1 → ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ) ↔ (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)))
124118, 123, 34vtocl 3411 . . . . . . 7 (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
125115, 116, 117, 124syl21anc 866 . . . . . 6 ((𝜑𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
126114, 2, 29, 125vonhoire 41458 . . . . 5 (𝜑 → ((voln‘𝑋)‘X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘))) ∈ ℝ)
127113, 126eqeltrd 2844 . . . 4 (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) ∈ ℝ)
128 eqid 2765 . . . 4 (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))
1291, 3, 4, 5, 26, 97, 100, 127, 128meaiininc 41273 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)))
130114, 29, 11iinhoiicc 41460 . . . . . . 7 (𝜑 𝑛 ∈ ℕ X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))) = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘)))
13133oveq2d 6858 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
132131ixpeq2dva 8128 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
13395, 132eqtrd 2799 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
134133iineq2dv 4699 . . . . . . 7 (𝜑 𝑛 ∈ ℕ (𝐷𝑛) = 𝑛 ∈ ℕ X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
135 vonicclem2.i . . . . . . . 8 𝐼 = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘))
136135a1i 11 . . . . . . 7 (𝜑𝐼 = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘)))
137130, 134, 1363eqtr4d 2809 . . . . . 6 (𝜑 𝑛 ∈ ℕ (𝐷𝑛) = 𝐼)
138137eqcomd 2771 . . . . 5 (𝜑𝐼 = 𝑛 ∈ ℕ (𝐷𝑛))
139138fveq2d 6379 . . . 4 (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)))
140139eqcomd 2771 . . 3 (𝜑 → ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)) = ((voln‘𝑋)‘𝐼))
141129, 140breqtrd 4835 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
142 2fveq3 6380 . . . . 5 (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷𝑛)) = ((voln‘𝑋)‘(𝐷𝑚)))
143142cbvmptv 4909 . . . 4 (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚)))
144143a1i 11 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))))
145 vonicclem2.n . . . 4 (𝜑𝑋 ≠ ∅)
146 vonicclem2.t . . . 4 ((𝜑𝑘𝑋) → (𝐴𝑘) ≤ (𝐵𝑘))
147143eqcomi 2774 . . . 4 (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))
1482, 8, 10, 145, 146, 17, 25, 147vonicclem1 41469 . . 3 (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
149144, 148eqbrtrd 4831 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
150 climuni 14570 . 2 (((𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘𝐼) ∧ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘))) → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
151141, 149, 150syl2anc 579 1 (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  Vcvv 3350  wss 3732  c0 4079   ciin 4677   class class class wbr 4809  cmpt 4888  dom cdm 5277  wf 6064  cfv 6068  (class class class)co 6842  Xcixp 8113  Fincfn 8160  cr 10188  0cc0 10189  1c1 10190   + caddc 10192  *cxr 10327   < clt 10328  cle 10329  cmin 10520   / cdiv 10938  cn 11274  cuz 11886  [,)cico 12379  [,]cicc 12380  cli 14502  cprod 14920  volncvoln 41324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cc 9510  ax-ac2 9538  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-disj 4778  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-tpos 7555  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-omul 7769  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-fi 8524  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-acn 9019  df-ac 9190  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-ioo 12381  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-clim 14506  df-rlim 14507  df-sum 14704  df-prod 14921  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-mulr 16230  df-starv 16231  df-sca 16232  df-vsca 16233  df-ip 16234  df-tset 16235  df-ple 16236  df-ds 16238  df-unif 16239  df-hom 16240  df-cco 16241  df-rest 16351  df-topn 16352  df-0g 16370  df-gsum 16371  df-topgen 16372  df-pt 16373  df-prds 16376  df-xrs 16430  df-qtop 16435  df-imas 16436  df-xps 16438  df-mre 16514  df-mrc 16515  df-acs 16517  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-submnd 17604  df-grp 17694  df-minusg 17695  df-mulg 17810  df-subg 17857  df-cntz 18015  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-cring 18817  df-oppr 18890  df-dvdsr 18908  df-unit 18909  df-invr 18939  df-dvr 18950  df-drng 19018  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-cnfld 20020  df-top 20978  df-topon 20995  df-topsp 21017  df-bases 21030  df-cn 21311  df-cnp 21312  df-cmp 21470  df-tx 21645  df-hmeo 21838  df-xms 22404  df-ms 22405  df-tms 22406  df-cncf 22960  df-ovol 23522  df-vol 23523  df-salg 41098  df-sumge0 41149  df-mea 41236  df-ome 41276  df-caragen 41278  df-ovoln 41323  df-voln 41325
This theorem is referenced by:  vonicc  41471
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