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Theorem ovoliunlem2 24572
Description: Lemma for ovoliun 24574. (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 3107 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
3 iunss 4971 . . . 4 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
42, 3sylibr 233 . . 3 (𝜑 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
5 ovolcl 24547 . . 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 480 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
9 ovoliun.j . . . . . . . . . . . 12 (𝜑𝐽:ℕ–1-1-onto→(ℕ × ℕ))
10 f1of 6700 . . . . . . . . . . . 12 (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:ℕ⟶(ℕ × ℕ))
119, 10syl 17 . . . . . . . . . . 11 (𝜑𝐽:ℕ⟶(ℕ × ℕ))
1211ffvelrnda 6943 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝐽𝑘) ∈ (ℕ × ℕ))
13 xp1st 7836 . . . . . . . . . 10 ((𝐽𝑘) ∈ (ℕ × ℕ) → (1st ‘(𝐽𝑘)) ∈ ℕ)
1412, 13syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐽𝑘)) ∈ ℕ)
158, 14ffvelrnd 6944 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
16 elovolmlem 24543 . . . . . . . 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 7837 . . . . . . . 8 ((𝐽𝑘) ∈ (ℕ × ℕ) → (2nd ‘(𝐽𝑘)) ∈ ℕ)
1912, 18syl 17 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐽𝑘)) ∈ ℕ)
2017, 19ffvelrnd 6944 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))) ∈ ( ≤ ∩ (ℝ × ℝ)))
21 ovoliun.h . . . . . 6 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))
2220, 21fmptd 6970 . . . . 5 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
23 eqid 2738 . . . . . 6 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
24 ovoliun.u . . . . . 6 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
2523, 24ovolsf 24541 . . . . 5 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
26 frn 6591 . . . . 5 (𝑈:ℕ⟶(0[,)+∞) → ran 𝑈 ⊆ (0[,)+∞))
2722, 25, 263syl 18 . . . 4 (𝜑 → ran 𝑈 ⊆ (0[,)+∞))
28 icossxr 13093 . . . 4 (0[,)+∞) ⊆ ℝ*
2927, 28sstrdi 3929 . . 3 (𝜑 → ran 𝑈 ⊆ ℝ*)
30 supxrcl 12978 . . 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 12701 . . . 4 (𝜑𝐵 ∈ ℝ)
3532, 34readdcld 10935 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ)
3635rexrd 10956 . 2 (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ*)
37 eliun 4925 . . . . . 6 (𝑧 𝑛 ∈ ℕ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑧𝐴)
38 ovoliun.x1 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
39383adant3 1130 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
4013adant3 1130 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝐴 ⊆ ℝ)
417ffvelrnda 6943 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
42 elovolmlem 24543 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4341, 42sylib 217 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
44433adant3 1130 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
45 ovolfioo 24536 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ (𝐹𝑛)) ↔ ∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
4640, 44, 45syl2anc 583 . . . . . . . . . 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 1136 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝑧𝐴)
49 rsp 3129 . . . . . . . . 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 1189 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝜑)
52 f1ocnv 6712 . . . . . . . . . . . 12 (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:(ℕ × ℕ)–1-1-onto→ℕ)
53 f1of 6700 . . . . . . . . . . . 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 1190 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝑛 ∈ ℕ)
56 simpr 484 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
5754, 55, 56fovrnd 7422 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝑛𝐽𝑗) ∈ ℕ)
58 2fveq3 6761 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑛𝐽𝑗) → (1st ‘(𝐽𝑘)) = (1st ‘(𝐽‘(𝑛𝐽𝑗))))
5958fveq2d 6760 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛𝐽𝑗) → (𝐹‘(1st ‘(𝐽𝑘))) = (𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗)))))
60 2fveq3 6761 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛𝐽𝑗) → (2nd ‘(𝐽𝑘)) = (2nd ‘(𝐽‘(𝑛𝐽𝑗))))
6159, 60fveq12d 6763 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛𝐽𝑗) → ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))) = ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))))
62 fvex 6769 . . . . . . . . . . . . . . . . 17 ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))) ∈ V
6361, 21, 62fvmpt 6857 . . . . . . . . . . . . . . . 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 7258 . . . . . . . . . . . . . . . . . . . . 21 (𝑛𝐽𝑗) = (𝐽‘⟨𝑛, 𝑗⟩)
6665fveq2i 6759 . . . . . . . . . . . . . . . . . . . 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 5618 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
69 f1ocnvfv2 7130 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽:ℕ–1-1-onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → (𝐽‘(𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
7067, 68, 69syl2anc 583 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
7166, 70syl5eq 2791 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(𝑛𝐽𝑗)) = ⟨𝑛, 𝑗⟩)
7271fveq2d 6760 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐽‘(𝑛𝐽𝑗))) = (1st ‘⟨𝑛, 𝑗⟩))
73 vex 3426 . . . . . . . . . . . . . . . . . . 19 𝑛 ∈ V
74 vex 3426 . . . . . . . . . . . . . . . . . . 19 𝑗 ∈ V
7573, 74op1st 7812 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨𝑛, 𝑗⟩) = 𝑛
7672, 75eqtrdi 2795 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐽‘(𝑛𝐽𝑗))) = 𝑛)
7776fveq2d 6760 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗)))) = (𝐹𝑛))
7871fveq2d 6760 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐽‘(𝑛𝐽𝑗))) = (2nd ‘⟨𝑛, 𝑗⟩))
7973, 74op2nd 7813 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨𝑛, 𝑗⟩) = 𝑗
8078, 79eqtrdi 2795 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐽‘(𝑛𝐽𝑗))) = 𝑗)
8177, 80fveq12d 6763 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))) = ((𝐹𝑛)‘𝑗))
8264, 81eqtrd 2778 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑛𝐽𝑗)) = ((𝐹𝑛)‘𝑗))
8382fveq2d 6760 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐻‘(𝑛𝐽𝑗))) = (1st ‘((𝐹𝑛)‘𝑗)))
8483breq1d 5080 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧 ↔ (1st ‘((𝐹𝑛)‘𝑗)) < 𝑧))
8582fveq2d 6760 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐻‘(𝑛𝐽𝑗))) = (2nd ‘((𝐹𝑛)‘𝑗)))
8685breq2d 5082 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))) ↔ 𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))))
8784, 86anbi12d 630 . . . . . . . . . . 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 6761 . . . . . . . . . . . . 13 (𝑚 = (𝑛𝐽𝑗) → (1st ‘(𝐻𝑚)) = (1st ‘(𝐻‘(𝑛𝐽𝑗))))
9089breq1d 5080 . . . . . . . . . . . 12 (𝑚 = (𝑛𝐽𝑗) → ((1st ‘(𝐻𝑚)) < 𝑧 ↔ (1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧))
91 2fveq3 6761 . . . . . . . . . . . . 13 (𝑚 = (𝑛𝐽𝑗) → (2nd ‘(𝐻𝑚)) = (2nd ‘(𝐻‘(𝑛𝐽𝑗))))
9291breq2d 5082 . . . . . . . . . . . 12 (𝑚 = (𝑛𝐽𝑗) → (𝑧 < (2nd ‘(𝐻𝑚)) ↔ 𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗)))))
9390, 92anbi12d 630 . . . . . . . . . . 11 (𝑚 = (𝑛𝐽𝑗) → (((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))) ↔ ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))))
9493rspcev 3552 . . . . . . . . . 10 (((𝑛𝐽𝑗) ∈ ℕ ∧ ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))))
9557, 88, 94syl6an 680 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
9695rexlimdva 3212 . . . . . . . 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 3214 . . . . . 6 (𝜑 → (∃𝑛 ∈ ℕ 𝑧𝐴 → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
9937, 98syl5bi 241 . . . . 5 (𝜑 → (𝑧 𝑛 ∈ ℕ 𝐴 → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
10099ralrimiv 3106 . . . 4 (𝜑 → ∀𝑧 𝑛 ∈ ℕ 𝐴𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))))
101 ovolfioo 24536 . . . . 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 583 . . . 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 24548 . . 3 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻)) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ*, < ))
10522, 103, 104syl2anc 583 . 2 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ*, < ))
106 fzfi 13620 . . . . . . 7 (1...𝑗) ∈ Fin
107 elfznn 13214 . . . . . . . . . 10 (𝑤 ∈ (1...𝑗) → 𝑤 ∈ ℕ)
108 ffvelrn 6941 . . . . . . . . . . 11 ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (𝐽𝑤) ∈ (ℕ × ℕ))
109 xp1st 7836 . . . . . . . . . . 11 ((𝐽𝑤) ∈ (ℕ × ℕ) → (1st ‘(𝐽𝑤)) ∈ ℕ)
110 nnre 11910 . . . . . . . . . . 11 ((1st ‘(𝐽𝑤)) ∈ ℕ → (1st ‘(𝐽𝑤)) ∈ ℝ)
111108, 109, 1103syl 18 . . . . . . . . . 10 ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (1st ‘(𝐽𝑤)) ∈ ℝ)
11211, 107, 111syl2an 595 . . . . . . . . 9 ((𝜑𝑤 ∈ (1...𝑗)) → (1st ‘(𝐽𝑤)) ∈ ℝ)
113112ralrimiva 3107 . . . . . . . 8 (𝜑 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ)
114113adantr 480 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ)
115 fimaxre3 11851 . . . . . . 7 (((1...𝑗) ∈ Fin ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥)
116106, 114, 115sylancr 586 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥)
117 fllep1 13449 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
118117ad2antlr 723 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ≤ ((⌊‘𝑥) + 1))
119112adantlr 711 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (1st ‘(𝐽𝑤)) ∈ ℝ)
120 simplr 765 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ∈ ℝ)
121 flcl 13443 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℤ)
122121peano2zd 12358 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℤ)
123122zred 12355 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ)
124123ad2antlr 723 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((⌊‘𝑥) + 1) ∈ ℝ)
125 letr 10999 . . . . . . . . . . . 12 (((1st ‘(𝐽𝑤)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ((⌊‘𝑥) + 1) ∈ ℝ) → (((1st ‘(𝐽𝑤)) ≤ 𝑥𝑥 ≤ ((⌊‘𝑥) + 1)) → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
126119, 120, 124, 125syl3anc 1369 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (((1st ‘(𝐽𝑤)) ≤ 𝑥𝑥 ≤ ((⌊‘𝑥) + 1)) → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
127118, 126mpan2d 690 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((1st ‘(𝐽𝑤)) ≤ 𝑥 → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
128127ralimdva 3102 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
129128adantlr 711 . . . . . . . 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 763 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝜑)
133132, 1sylan 579 . . . . . . . . . 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 579 . . . . . . . . . 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 579 . . . . . . . . . 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 579 . . . . . . . . . 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 765 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝑗 ∈ ℕ)
145122ad2antrl 724 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → ((⌊‘𝑥) + 1) ∈ ℤ)
146 simprr 769 . . . . . . . . . 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 24571 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
148147expr 456 . . . . . . . 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 3212 . . . . . 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 3107 . . . 4 (𝜑 → ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
153 ffn 6584 . . . . 5 (𝑈:ℕ⟶(0[,)+∞) → 𝑈 Fn ℕ)
154 breq1 5073 . . . . . 6 (𝑧 = (𝑈𝑗) → (𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
155154ralrn 6946 . . . . 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 12989 . . . 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 583 . . 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 12825 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1539  wcel 2108  wral 3063  wrex 3064  cin 3882  wss 3883  cop 4564   cuni 4836   ciun 4921   class class class wbr 5070  cmpt 5153   × cxp 5578  ccnv 5579  ran crn 5581  ccom 5584   Fn wfn 6413  wf 6414  1-1-ontowf1o 6417  cfv 6418  (class class class)co 7255  1st c1st 7802  2nd c2nd 7803  m cmap 8573  Fincfn 8691  supcsup 9129  cr 10801  0cc0 10802  1c1 10803   + caddc 10805  +∞cpnf 10937  *cxr 10939   < clt 10940  cle 10941  cmin 11135   / cdiv 11562  cn 11903  2c2 11958  cz 12249  +crp 12659  (,)cioo 13008  [,)cico 13010  ...cfz 13168  cfl 13438  seqcseq 13649  cexp 13710  abscabs 14873  vol*covol 24531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-ioo 13012  df-ico 13014  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-sum 15326  df-ovol 24533
This theorem is referenced by:  ovoliunlem3  24573
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