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Theorem vonicclem2 46699
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 1914 . . . 4 𝑛𝜑
2 vonicclem2.x . . . . 5 (𝜑𝑋 ∈ Fin)
32vonmea 46589 . . . 4 (𝜑 → (voln‘𝑋) ∈ Meas)
4 1zzd 12648 . . . 4 (𝜑 → 1 ∈ ℤ)
5 nnuz 12921 . . . 4 ℕ = (ℤ‘1)
62adantr 480 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
7 eqid 2737 . . . . . 6 dom (voln‘𝑋) = dom (voln‘𝑋)
8 vonicclem2.a . . . . . . 7 (𝜑𝐴:𝑋⟶ℝ)
98adantr 480 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
10 vonicclem2.b . . . . . . . . . . 11 (𝜑𝐵:𝑋⟶ℝ)
1110ffvelcdmda 7104 . . . . . . . . . 10 ((𝜑𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
1211adantlr 715 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑘) ∈ ℝ)
13 nnrecre 12308 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
1413ad2antlr 727 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / 𝑛) ∈ ℝ)
1512, 14readdcld 11290 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / 𝑛)) ∈ ℝ)
1615fmpttd 7135 . . . . . . 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 7244 . . . . . . . . . 10 (𝜑 → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) ∈ V)
2019adantr 480 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) ∈ V)
2118, 20fvmpt2d 7029 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))))
2221feq1d 6720 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝐶𝑛):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
2316, 22mpbird 257 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐶𝑛):𝑋⟶ℝ)
246, 7, 9, 23hoimbl 46646 . . . . 5 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) ∈ dom (voln‘𝑋))
25 vonicclem2.d . . . . 5 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
2624, 25fmptd 7134 . . . 4 (𝜑𝐷:ℕ⟶dom (voln‘𝑋))
27 nfv 1914 . . . . . 6 𝑘(𝜑𝑛 ∈ ℕ)
28 ressxr 11305 . . . . . . . . 9 ℝ ⊆ ℝ*
298ffvelcdmda 7104 . . . . . . . . 9 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3028, 29sselid 3981 . . . . . . . 8 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ*)
3130adantlr 715 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ∈ ℝ*)
32 ovexd 7466 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / 𝑛)) ∈ V)
3321, 32fvmpt2d 7029 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) = ((𝐵𝑘) + (1 / 𝑛)))
3433, 15eqeltrd 2841 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ)
3534rexrd 11311 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ*)
369ffvelcdmda 7104 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3736leidd 11829 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (𝐴𝑘) ≤ (𝐴𝑘))
38 1red 11262 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 1 ∈ ℝ)
39 nnre 12273 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
4039, 38readdcld 11290 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
41 peano2nn 12278 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
42 nnne0 12300 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
4341, 42syl 17 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
4438, 40, 43redivcld 12095 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
4544ad2antlr 727 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
4639ltp1d 12198 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
47 nnrp 13046 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
4841nnrpd 13075 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
4947, 48ltrecd 13095 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛)))
5046, 49mpbid 232 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) < (1 / 𝑛))
5144, 13, 50ltled 11409 . . . . . . . . . 10 (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
5251ad2antlr 727 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
5345, 14, 12, 52leadd2dd 11878 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵𝑘) + (1 / 𝑛)))
54 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
5554oveq2d 7447 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((𝐵𝑘) + (1 / 𝑛)) = ((𝐵𝑘) + (1 / 𝑚)))
5655mpteq2dv 5244 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛))) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
5756cbvmptv 5255 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
5817, 57eqtri 2765 . . . . . . . . . . 11 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))))
59 oveq2 7439 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
6059oveq2d 7447 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → ((𝐵𝑘) + (1 / 𝑚)) = ((𝐵𝑘) + (1 / (𝑛 + 1))))
6160mpteq2dv 5244 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / 𝑚))) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))))
62 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
6362peano2nnd 12283 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
646mptexd 7244 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))) ∈ V)
6558, 61, 63, 64fvmptd3 7039 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘𝑋 ↦ ((𝐵𝑘) + (1 / (𝑛 + 1)))))
66 ovexd 7466 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑘) + (1 / (𝑛 + 1))) ∈ V)
6765, 66fvmpt2d 7029 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐵𝑘) + (1 / (𝑛 + 1))))
6867, 33breq12d 5156 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘) ↔ ((𝐵𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵𝑘) + (1 / 𝑛))))
6953, 68mpbird 257 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶𝑛)‘𝑘))
70 icossico 13457 . . . . . . 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 839 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
7227, 71ixpssixp 45097 . . . . 5 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
73 fveq2 6906 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐶𝑛) = (𝐶𝑚))
7473fveq1d 6908 . . . . . . . . . . 11 (𝑛 = 𝑚 → ((𝐶𝑛)‘𝑘) = ((𝐶𝑚)‘𝑘))
7574oveq2d 7447 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7675ixpeq2dv 8953 . . . . . . . . 9 (𝑛 = 𝑚X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7776cbvmptv 5255 . . . . . . . 8 (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
7825, 77eqtri 2765 . . . . . . 7 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)))
79 fveq2 6906 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → (𝐶𝑚) = (𝐶‘(𝑛 + 1)))
8079fveq1d 6908 . . . . . . . . 9 (𝑚 = (𝑛 + 1) → ((𝐶𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
8180oveq2d 7447 . . . . . . . 8 (𝑚 = (𝑛 + 1) → ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)) = ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
8281ixpeq2dv 8953 . . . . . . 7 (𝑚 = (𝑛 + 1) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑚)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
83 ovex 7464 . . . . . . . . . 10 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
8483rgenw 3065 . . . . . . . . 9 𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
85 ixpexg 8962 . . . . . . . . 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 7039 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
8925a1i 11 . . . . . . 7 (𝜑𝐷 = (𝑛 ∈ ℕ ↦ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))))
9024elexd 3504 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) ∈ V)
9189, 90fvmpt2d 7029 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)))
9288, 91sseq12d 4017 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝐷‘(𝑛 + 1)) ⊆ (𝐷𝑛) ↔ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘))))
9372, 92mpbird 257 . . . 4 ((𝜑𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) ⊆ (𝐷𝑛))
94 1nn 12277 . . . . . 6 1 ∈ ℕ
9594, 5eleqtri 2839 . . . . 5 1 ∈ (ℤ‘1)
9695a1i 11 . . . 4 (𝜑 → 1 ∈ (ℤ‘1))
97 fveq2 6906 . . . . . . . . . 10 (𝑛 = 1 → (𝐶𝑛) = (𝐶‘1))
9897fveq1d 6908 . . . . . . . . 9 (𝑛 = 1 → ((𝐶𝑛)‘𝑘) = ((𝐶‘1)‘𝑘))
9998oveq2d 7447 . . . . . . . 8 (𝑛 = 1 → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
10099ixpeq2dv 8953 . . . . . . 7 (𝑛 = 1 → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
10194a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℕ)
102 ovex 7464 . . . . . . . . . 10 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
103102rgenw 3065 . . . . . . . . 9 𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
104 ixpexg 8962 . . . . . . . . 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 7039 . . . . . 6 (𝜑 → (𝐷‘1) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘)))
108107fveq2d 6910 . . . . 5 (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) = ((voln‘𝑋)‘X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘))))
109 nfv 1914 . . . . . 6 𝑘𝜑
110 simpl 482 . . . . . . 7 ((𝜑𝑘𝑋) → 𝜑)
11194a1i 11 . . . . . . 7 ((𝜑𝑘𝑋) → 1 ∈ ℕ)
112 simpr 484 . . . . . . 7 ((𝜑𝑘𝑋) → 𝑘𝑋)
11394elexi 3503 . . . . . . . 8 1 ∈ V
114 eleq1 2829 . . . . . . . . . . 11 (𝑛 = 1 → (𝑛 ∈ ℕ ↔ 1 ∈ ℕ))
115114anbi2d 630 . . . . . . . . . 10 (𝑛 = 1 → ((𝜑𝑛 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
116115anbi1d 631 . . . . . . . . 9 (𝑛 = 1 → (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) ↔ ((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋)))
11798eleq1d 2826 . . . . . . . . 9 (𝑛 = 1 → (((𝐶𝑛)‘𝑘) ∈ ℝ ↔ ((𝐶‘1)‘𝑘) ∈ ℝ))
118116, 117imbi12d 344 . . . . . . . 8 (𝑛 = 1 → ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶𝑛)‘𝑘) ∈ ℝ) ↔ (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)))
119113, 118, 34vtocl 3558 . . . . . . 7 (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
120110, 111, 112, 119syl21anc 838 . . . . . 6 ((𝜑𝑘𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
121109, 2, 29, 120vonhoire 46687 . . . . 5 (𝜑 → ((voln‘𝑋)‘X𝑘𝑋 ((𝐴𝑘)[,)((𝐶‘1)‘𝑘))) ∈ ℝ)
122108, 121eqeltrd 2841 . . . 4 (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) ∈ ℝ)
123 eqid 2737 . . . 4 (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))
1241, 3, 4, 5, 26, 93, 96, 122, 123meaiininc 46502 . . 3 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)))
125109, 29, 11iinhoiicc 46689 . . . . . . 7 (𝜑 𝑛 ∈ ℕ X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))) = X𝑘𝑋 ((𝐴𝑘)[,](𝐵𝑘)))
12633oveq2d 7447 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
127126ixpeq2dva 8952 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → X𝑘𝑋 ((𝐴𝑘)[,)((𝐶𝑛)‘𝑘)) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
12891, 127eqtrd 2777 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) = X𝑘𝑋 ((𝐴𝑘)[,)((𝐵𝑘) + (1 / 𝑛))))
129128iineq2dv 5017 . . . . . . 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 2787 . . . . . 6 (𝜑 𝑛 ∈ ℕ (𝐷𝑛) = 𝐼)
133132eqcomd 2743 . . . . 5 (𝜑𝐼 = 𝑛 ∈ ℕ (𝐷𝑛))
134133fveq2d 6910 . . . 4 (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)))
135134eqcomd 2743 . . 3 (𝜑 → ((voln‘𝑋)‘ 𝑛 ∈ ℕ (𝐷𝑛)) = ((voln‘𝑋)‘𝐼))
136124, 135breqtrd 5169 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
137 2fveq3 6911 . . . . 5 (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷𝑛)) = ((voln‘𝑋)‘(𝐷𝑚)))
138137cbvmptv 5255 . . . 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 2746 . . . 4 (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛)))
1432, 8, 10, 140, 141, 17, 25, 142vonicclem1 46698 . . 3 (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑚))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
144139, 143eqbrtrd 5165 . 2 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷𝑛))) ⇝ ∏𝑘𝑋 ((𝐵𝑘) − (𝐴𝑘)))
145 climuni 15588 . 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 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  wss 3951  c0 4333   ciin 4992   class class class wbr 5143  cmpt 5225  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  Xcixp 8937  Fincfn 8985  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  *cxr 11294   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  cn 12266  cuz 12878  [,)cico 13389  [,]cicc 13390  cli 15520  cprod 15939  volncvoln 46553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-prod 15940  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cn 23235  df-cnp 23236  df-cmp 23395  df-tx 23570  df-hmeo 23763  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-ovol 25499  df-vol 25500  df-salg 46324  df-sumge0 46378  df-mea 46465  df-ome 46505  df-caragen 46507  df-ovoln 46552  df-voln 46554
This theorem is referenced by:  vonicc  46700
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