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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  cr 10529  0cc0 10530  1c1 10531   + caddc 10533  *cxr 10667   < clt 10668  cle 10669  cmin 10863   / cdiv 11290  cn 11629  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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