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Theorem ovoliunlem2 24867
Description: Lemma for ovoliun 24869. (Contributed by Mario Carneiro, 12-Jun-2014.)
Hypotheses
Ref Expression
ovoliun.t 𝑇 = seq1( + , 𝐺)
ovoliun.g 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
ovoliun.a ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
ovoliun.r (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
ovoliun.b (𝜑𝐵 ∈ ℝ+)
ovoliun.s 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛)))
ovoliun.u 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ovoliun.h 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))
ovoliun.j (𝜑𝐽:ℕ–1-1-onto→(ℕ × ℕ))
ovoliun.f (𝜑𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
ovoliun.x1 ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
ovoliun.x2 ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
Assertion
Ref Expression
ovoliunlem2 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Distinct variable groups:   𝐴,𝑘   𝑘,𝑛,𝐵   𝑘,𝐹,𝑛   𝑘,𝐽,𝑛   𝑛,𝐻   𝜑,𝑘,𝑛   𝑆,𝑘   𝑘,𝐺   𝑇,𝑘   𝑛,𝐺   𝑇,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝑆(𝑛)   𝑈(𝑘,𝑛)   𝐻(𝑘)

Proof of Theorem ovoliunlem2
Dummy variables 𝑗 𝑚 𝑥 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovoliun.a . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
21ralrimiva 3143 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
3 iunss 5005 . . . 4 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
42, 3sylibr 233 . . 3 (𝜑 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
5 ovolcl 24842 . . 3 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
64, 5syl 17 . 2 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
7 ovoliun.f . . . . . . . . . 10 (𝜑𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
87adantr 481 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
9 ovoliun.j . . . . . . . . . . . 12 (𝜑𝐽:ℕ–1-1-onto→(ℕ × ℕ))
10 f1of 6784 . . . . . . . . . . . 12 (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:ℕ⟶(ℕ × ℕ))
119, 10syl 17 . . . . . . . . . . 11 (𝜑𝐽:ℕ⟶(ℕ × ℕ))
1211ffvelcdmda 7035 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝐽𝑘) ∈ (ℕ × ℕ))
13 xp1st 7953 . . . . . . . . . 10 ((𝐽𝑘) ∈ (ℕ × ℕ) → (1st ‘(𝐽𝑘)) ∈ ℕ)
1412, 13syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐽𝑘)) ∈ ℕ)
158, 14ffvelcdmd 7036 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
16 elovolmlem 24838 . . . . . . . 8 ((𝐹‘(1st ‘(𝐽𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ (𝐹‘(1st ‘(𝐽𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1715, 16sylib 217 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18 xp2nd 7954 . . . . . . . 8 ((𝐽𝑘) ∈ (ℕ × ℕ) → (2nd ‘(𝐽𝑘)) ∈ ℕ)
1912, 18syl 17 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐽𝑘)) ∈ ℕ)
2017, 19ffvelcdmd 7036 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))) ∈ ( ≤ ∩ (ℝ × ℝ)))
21 ovoliun.h . . . . . 6 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))
2220, 21fmptd 7062 . . . . 5 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
23 eqid 2736 . . . . . 6 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
24 ovoliun.u . . . . . 6 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
2523, 24ovolsf 24836 . . . . 5 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
26 frn 6675 . . . . 5 (𝑈:ℕ⟶(0[,)+∞) → ran 𝑈 ⊆ (0[,)+∞))
2722, 25, 263syl 18 . . . 4 (𝜑 → ran 𝑈 ⊆ (0[,)+∞))
28 icossxr 13349 . . . 4 (0[,)+∞) ⊆ ℝ*
2927, 28sstrdi 3956 . . 3 (𝜑 → ran 𝑈 ⊆ ℝ*)
30 supxrcl 13234 . . 3 (ran 𝑈 ⊆ ℝ* → sup(ran 𝑈, ℝ*, < ) ∈ ℝ*)
3129, 30syl 17 . 2 (𝜑 → sup(ran 𝑈, ℝ*, < ) ∈ ℝ*)
32 ovoliun.r . . . 4 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
33 ovoliun.b . . . . 5 (𝜑𝐵 ∈ ℝ+)
3433rpred 12957 . . . 4 (𝜑𝐵 ∈ ℝ)
3532, 34readdcld 11184 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ)
3635rexrd 11205 . 2 (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ*)
37 eliun 4958 . . . . . 6 (𝑧 𝑛 ∈ ℕ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑧𝐴)
38 ovoliun.x1 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
39383adant3 1132 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
4013adant3 1132 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝐴 ⊆ ℝ)
417ffvelcdmda 7035 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
42 elovolmlem 24838 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4341, 42sylib 217 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
44433adant3 1132 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
45 ovolfioo 24831 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ (𝐹𝑛)) ↔ ∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
4640, 44, 45syl2anc 584 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → (𝐴 ran ((,) ∘ (𝐹𝑛)) ↔ ∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
4739, 46mpbid 231 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → ∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))))
48 simp3 1138 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝑧𝐴)
49 rsp 3230 . . . . . . . . 9 (∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → (𝑧𝐴 → ∃𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
5047, 48, 49sylc 65 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → ∃𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))))
51 simpl1 1191 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝜑)
52 f1ocnv 6796 . . . . . . . . . . . 12 (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:(ℕ × ℕ)–1-1-onto→ℕ)
53 f1of 6784 . . . . . . . . . . . 12 (𝐽:(ℕ × ℕ)–1-1-onto→ℕ → 𝐽:(ℕ × ℕ)⟶ℕ)
5451, 9, 52, 534syl 19 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝐽:(ℕ × ℕ)⟶ℕ)
55 simpl2 1192 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝑛 ∈ ℕ)
56 simpr 485 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
5754, 55, 56fovcdmd 7526 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝑛𝐽𝑗) ∈ ℕ)
58 2fveq3 6847 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑛𝐽𝑗) → (1st ‘(𝐽𝑘)) = (1st ‘(𝐽‘(𝑛𝐽𝑗))))
5958fveq2d 6846 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛𝐽𝑗) → (𝐹‘(1st ‘(𝐽𝑘))) = (𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗)))))
60 2fveq3 6847 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛𝐽𝑗) → (2nd ‘(𝐽𝑘)) = (2nd ‘(𝐽‘(𝑛𝐽𝑗))))
6159, 60fveq12d 6849 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛𝐽𝑗) → ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))) = ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))))
62 fvex 6855 . . . . . . . . . . . . . . . . 17 ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))) ∈ V
6361, 21, 62fvmpt 6948 . . . . . . . . . . . . . . . 16 ((𝑛𝐽𝑗) ∈ ℕ → (𝐻‘(𝑛𝐽𝑗)) = ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))))
6457, 63syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑛𝐽𝑗)) = ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))))
65 df-ov 7360 . . . . . . . . . . . . . . . . . . . . 21 (𝑛𝐽𝑗) = (𝐽‘⟨𝑛, 𝑗⟩)
6665fveq2i 6845 . . . . . . . . . . . . . . . . . . . 20 (𝐽‘(𝑛𝐽𝑗)) = (𝐽‘(𝐽‘⟨𝑛, 𝑗⟩))
6751, 9syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
6855, 56opelxpd 5671 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
69 f1ocnvfv2 7223 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽:ℕ–1-1-onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → (𝐽‘(𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
7067, 68, 69syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
7166, 70eqtrid 2788 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(𝑛𝐽𝑗)) = ⟨𝑛, 𝑗⟩)
7271fveq2d 6846 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐽‘(𝑛𝐽𝑗))) = (1st ‘⟨𝑛, 𝑗⟩))
73 vex 3449 . . . . . . . . . . . . . . . . . . 19 𝑛 ∈ V
74 vex 3449 . . . . . . . . . . . . . . . . . . 19 𝑗 ∈ V
7573, 74op1st 7929 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨𝑛, 𝑗⟩) = 𝑛
7672, 75eqtrdi 2792 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐽‘(𝑛𝐽𝑗))) = 𝑛)
7776fveq2d 6846 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗)))) = (𝐹𝑛))
7871fveq2d 6846 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐽‘(𝑛𝐽𝑗))) = (2nd ‘⟨𝑛, 𝑗⟩))
7973, 74op2nd 7930 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨𝑛, 𝑗⟩) = 𝑗
8078, 79eqtrdi 2792 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐽‘(𝑛𝐽𝑗))) = 𝑗)
8177, 80fveq12d 6849 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))) = ((𝐹𝑛)‘𝑗))
8264, 81eqtrd 2776 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑛𝐽𝑗)) = ((𝐹𝑛)‘𝑗))
8382fveq2d 6846 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐻‘(𝑛𝐽𝑗))) = (1st ‘((𝐹𝑛)‘𝑗)))
8483breq1d 5115 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧 ↔ (1st ‘((𝐹𝑛)‘𝑗)) < 𝑧))
8582fveq2d 6846 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐻‘(𝑛𝐽𝑗))) = (2nd ‘((𝐹𝑛)‘𝑗)))
8685breq2d 5117 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))) ↔ 𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))))
8784, 86anbi12d 631 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗)))) ↔ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
8887biimprd 247 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))))
89 2fveq3 6847 . . . . . . . . . . . . 13 (𝑚 = (𝑛𝐽𝑗) → (1st ‘(𝐻𝑚)) = (1st ‘(𝐻‘(𝑛𝐽𝑗))))
9089breq1d 5115 . . . . . . . . . . . 12 (𝑚 = (𝑛𝐽𝑗) → ((1st ‘(𝐻𝑚)) < 𝑧 ↔ (1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧))
91 2fveq3 6847 . . . . . . . . . . . . 13 (𝑚 = (𝑛𝐽𝑗) → (2nd ‘(𝐻𝑚)) = (2nd ‘(𝐻‘(𝑛𝐽𝑗))))
9291breq2d 5117 . . . . . . . . . . . 12 (𝑚 = (𝑛𝐽𝑗) → (𝑧 < (2nd ‘(𝐻𝑚)) ↔ 𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗)))))
9390, 92anbi12d 631 . . . . . . . . . . 11 (𝑚 = (𝑛𝐽𝑗) → (((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))) ↔ ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))))
9493rspcev 3581 . . . . . . . . . 10 (((𝑛𝐽𝑗) ∈ ℕ ∧ ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))))
9557, 88, 94syl6an 682 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
9695rexlimdva 3152 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → (∃𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
9750, 96mpd 15 . . . . . . 7 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))))
9897rexlimdv3a 3156 . . . . . 6 (𝜑 → (∃𝑛 ∈ ℕ 𝑧𝐴 → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
9937, 98biimtrid 241 . . . . 5 (𝜑 → (𝑧 𝑛 ∈ ℕ 𝐴 → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
10099ralrimiv 3142 . . . 4 (𝜑 → ∀𝑧 𝑛 ∈ ℕ 𝐴𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))))
101 ovolfioo 24831 . . . . 5 (( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ( 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧 𝑛 ∈ ℕ 𝐴𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
1024, 22, 101syl2anc 584 . . . 4 (𝜑 → ( 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧 𝑛 ∈ ℕ 𝐴𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
103100, 102mpbird 256 . . 3 (𝜑 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻))
10424ovollb 24843 . . 3 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻)) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ*, < ))
10522, 103, 104syl2anc 584 . 2 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ*, < ))
106 fzfi 13877 . . . . . . 7 (1...𝑗) ∈ Fin
107 elfznn 13470 . . . . . . . . . 10 (𝑤 ∈ (1...𝑗) → 𝑤 ∈ ℕ)
108 ffvelcdm 7032 . . . . . . . . . . 11 ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (𝐽𝑤) ∈ (ℕ × ℕ))
109 xp1st 7953 . . . . . . . . . . 11 ((𝐽𝑤) ∈ (ℕ × ℕ) → (1st ‘(𝐽𝑤)) ∈ ℕ)
110 nnre 12160 . . . . . . . . . . 11 ((1st ‘(𝐽𝑤)) ∈ ℕ → (1st ‘(𝐽𝑤)) ∈ ℝ)
111108, 109, 1103syl 18 . . . . . . . . . 10 ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (1st ‘(𝐽𝑤)) ∈ ℝ)
11211, 107, 111syl2an 596 . . . . . . . . 9 ((𝜑𝑤 ∈ (1...𝑗)) → (1st ‘(𝐽𝑤)) ∈ ℝ)
113112ralrimiva 3143 . . . . . . . 8 (𝜑 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ)
114113adantr 481 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ)
115 fimaxre3 12101 . . . . . . 7 (((1...𝑗) ∈ Fin ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥)
116106, 114, 115sylancr 587 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥)
117 fllep1 13706 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
118117ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ≤ ((⌊‘𝑥) + 1))
119112adantlr 713 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (1st ‘(𝐽𝑤)) ∈ ℝ)
120 simplr 767 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ∈ ℝ)
121 flcl 13700 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℤ)
122121peano2zd 12610 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℤ)
123122zred 12607 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ)
124123ad2antlr 725 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((⌊‘𝑥) + 1) ∈ ℝ)
125 letr 11249 . . . . . . . . . . . 12 (((1st ‘(𝐽𝑤)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ((⌊‘𝑥) + 1) ∈ ℝ) → (((1st ‘(𝐽𝑤)) ≤ 𝑥𝑥 ≤ ((⌊‘𝑥) + 1)) → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
126119, 120, 124, 125syl3anc 1371 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (((1st ‘(𝐽𝑤)) ≤ 𝑥𝑥 ≤ ((⌊‘𝑥) + 1)) → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
127118, 126mpan2d 692 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((1st ‘(𝐽𝑤)) ≤ 𝑥 → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
128127ralimdva 3164 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
129128adantlr 713 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
130 ovoliun.t . . . . . . . . . 10 𝑇 = seq1( + , 𝐺)
131 ovoliun.g . . . . . . . . . 10 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
132 simpll 765 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝜑)
133132, 1sylan 580 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
134 ovoliun.v . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
135132, 134sylan 580 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
136132, 32syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
137132, 33syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐵 ∈ ℝ+)
138 ovoliun.s . . . . . . . . . 10 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛)))
139132, 9syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
140132, 7syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
141132, 38sylan 580 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
142 ovoliun.x2 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
143132, 142sylan 580 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
144 simplr 767 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝑗 ∈ ℕ)
145122ad2antrl 726 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → ((⌊‘𝑥) + 1) ∈ ℤ)
146 simprr 771 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))
147130, 131, 133, 135, 136, 137, 138, 24, 21, 139, 140, 141, 143, 144, 145, 146ovoliunlem1 24866 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
148147expr 457 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1) → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
149129, 148syld 47 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
150149rexlimdva 3152 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
151116, 150mpd 15 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
152151ralrimiva 3143 . . . 4 (𝜑 → ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
153 ffn 6668 . . . . 5 (𝑈:ℕ⟶(0[,)+∞) → 𝑈 Fn ℕ)
154 breq1 5108 . . . . . 6 (𝑧 = (𝑈𝑗) → (𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
155154ralrn 7038 . . . . 5 (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
15622, 25, 153, 1554syl 19 . . . 4 (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
157152, 156mpbird 256 . . 3 (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
158 supxrleub 13245 . . . 4 ((ran 𝑈 ⊆ ℝ* ∧ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ*) → (sup(ran 𝑈, ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
15929, 36, 158syl2anc 584 . . 3 (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
160157, 159mpbird 256 . 2 (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
1616, 31, 36, 105, 160xrletrd 13081 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  cin 3909  wss 3910  cop 4592   cuni 4865   ciun 4954   class class class wbr 5105  cmpt 5188   × cxp 5631  ccnv 5632  ran crn 5634  ccom 5637   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  m cmap 8765  Fincfn 8883  supcsup 9376  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  2c2 12208  cz 12499  +crp 12915  (,)cioo 13264  [,)cico 13266  ...cfz 13424  cfl 13695  seqcseq 13906  cexp 13967  abscabs 15119  vol*covol 24826
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ioo 13268  df-ico 13270  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-ovol 24828
This theorem is referenced by:  ovoliunlem3  24868
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