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Theorem ovoliunlem2 23078
 Description: Lemma for ovoliun 23080. (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 2949 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
3 iunss 4497 . . . 4 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
42, 3sylibr 223 . . 3 (𝜑 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
5 ovolcl 23053 . . 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 6050 . . . . . . . . . . . 12 (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:ℕ⟶(ℕ × ℕ))
119, 10syl 17 . . . . . . . . . . 11 (𝜑𝐽:ℕ⟶(ℕ × ℕ))
1211ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝐽𝑘) ∈ (ℕ × ℕ))
13 xp1st 7089 . . . . . . . . . 10 ((𝐽𝑘) ∈ (ℕ × ℕ) → (1st ‘(𝐽𝑘)) ∈ ℕ)
1412, 13syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐽𝑘)) ∈ ℕ)
158, 14ffvelrnd 6268 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
16 reex 9906 . . . . . . . . . . 11 ℝ ∈ V
1716, 16xpex 6860 . . . . . . . . . 10 (ℝ × ℝ) ∈ V
1817inex2 4728 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ∈ V
19 nnex 10903 . . . . . . . . 9 ℕ ∈ V
2018, 19elmap 7772 . . . . . . . 8 ((𝐹‘(1st ‘(𝐽𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ (𝐹‘(1st ‘(𝐽𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2115, 20sylib 207 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
22 xp2nd 7090 . . . . . . . 8 ((𝐽𝑘) ∈ (ℕ × ℕ) → (2nd ‘(𝐽𝑘)) ∈ ℕ)
2312, 22syl 17 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐽𝑘)) ∈ ℕ)
2421, 23ffvelrnd 6268 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))) ∈ ( ≤ ∩ (ℝ × ℝ)))
25 ovoliun.h . . . . . 6 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))
2624, 25fmptd 6292 . . . . 5 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27 eqid 2610 . . . . . 6 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
28 ovoliun.u . . . . . 6 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
2927, 28ovolsf 23048 . . . . 5 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
30 frn 5966 . . . . 5 (𝑈:ℕ⟶(0[,)+∞) → ran 𝑈 ⊆ (0[,)+∞))
3126, 29, 303syl 18 . . . 4 (𝜑 → ran 𝑈 ⊆ (0[,)+∞))
32 icossxr 12129 . . . 4 (0[,)+∞) ⊆ ℝ*
3331, 32syl6ss 3580 . . 3 (𝜑 → ran 𝑈 ⊆ ℝ*)
34 supxrcl 12017 . . 3 (ran 𝑈 ⊆ ℝ* → sup(ran 𝑈, ℝ*, < ) ∈ ℝ*)
3533, 34syl 17 . 2 (𝜑 → sup(ran 𝑈, ℝ*, < ) ∈ ℝ*)
36 ovoliun.r . . . 4 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
37 ovoliun.b . . . . 5 (𝜑𝐵 ∈ ℝ+)
3837rpred 11748 . . . 4 (𝜑𝐵 ∈ ℝ)
3936, 38readdcld 9948 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ)
4039rexrd 9968 . 2 (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ*)
41 eliun 4460 . . . . . 6 (𝑧 𝑛 ∈ ℕ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑧𝐴)
42 ovoliun.x1 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
43423adant3 1074 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
4413adant3 1074 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝐴 ⊆ ℝ)
457ffvelrnda 6267 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
4618, 19elmap 7772 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4745, 46sylib 207 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
48473adant3 1074 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
49 ovolfioo 23043 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ (𝐹𝑛)) ↔ ∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
5044, 48, 49syl2anc 691 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → (𝐴 ran ((,) ∘ (𝐹𝑛)) ↔ ∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
5143, 50mpbid 221 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → ∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))))
52 simp3 1056 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝑧𝐴)
53 rsp 2913 . . . . . . . . 9 (∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → (𝑧𝐴 → ∃𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
5451, 52, 53sylc 63 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → ∃𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))))
55 simpl1 1057 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝜑)
56 f1ocnv 6062 . . . . . . . . . . . 12 (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:(ℕ × ℕ)–1-1-onto→ℕ)
57 f1of 6050 . . . . . . . . . . . 12 (𝐽:(ℕ × ℕ)–1-1-onto→ℕ → 𝐽:(ℕ × ℕ)⟶ℕ)
5855, 9, 56, 574syl 19 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝐽:(ℕ × ℕ)⟶ℕ)
59 simpl2 1058 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝑛 ∈ ℕ)
60 simpr 476 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
6158, 59, 60fovrnd 6704 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝑛𝐽𝑗) ∈ ℕ)
62 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑛𝐽𝑗) → (𝐽𝑘) = (𝐽‘(𝑛𝐽𝑗)))
6362fveq2d 6107 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑛𝐽𝑗) → (1st ‘(𝐽𝑘)) = (1st ‘(𝐽‘(𝑛𝐽𝑗))))
6463fveq2d 6107 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛𝐽𝑗) → (𝐹‘(1st ‘(𝐽𝑘))) = (𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗)))))
6562fveq2d 6107 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛𝐽𝑗) → (2nd ‘(𝐽𝑘)) = (2nd ‘(𝐽‘(𝑛𝐽𝑗))))
6664, 65fveq12d 6109 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛𝐽𝑗) → ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))) = ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))))
67 fvex 6113 . . . . . . . . . . . . . . . . 17 ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))) ∈ V
6866, 25, 67fvmpt 6191 . . . . . . . . . . . . . . . 16 ((𝑛𝐽𝑗) ∈ ℕ → (𝐻‘(𝑛𝐽𝑗)) = ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))))
6961, 68syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑛𝐽𝑗)) = ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))))
70 df-ov 6552 . . . . . . . . . . . . . . . . . . . . 21 (𝑛𝐽𝑗) = (𝐽‘⟨𝑛, 𝑗⟩)
7170fveq2i 6106 . . . . . . . . . . . . . . . . . . . 20 (𝐽‘(𝑛𝐽𝑗)) = (𝐽‘(𝐽‘⟨𝑛, 𝑗⟩))
7255, 9syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
73 opelxpi 5072 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
7459, 60, 73syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
75 f1ocnvfv2 6433 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽:ℕ–1-1-onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → (𝐽‘(𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
7672, 74, 75syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
7771, 76syl5eq 2656 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(𝑛𝐽𝑗)) = ⟨𝑛, 𝑗⟩)
7877fveq2d 6107 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐽‘(𝑛𝐽𝑗))) = (1st ‘⟨𝑛, 𝑗⟩))
79 vex 3176 . . . . . . . . . . . . . . . . . . 19 𝑛 ∈ V
80 vex 3176 . . . . . . . . . . . . . . . . . . 19 𝑗 ∈ V
8179, 80op1st 7067 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨𝑛, 𝑗⟩) = 𝑛
8278, 81syl6eq 2660 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐽‘(𝑛𝐽𝑗))) = 𝑛)
8382fveq2d 6107 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗)))) = (𝐹𝑛))
8477fveq2d 6107 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐽‘(𝑛𝐽𝑗))) = (2nd ‘⟨𝑛, 𝑗⟩))
8579, 80op2nd 7068 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨𝑛, 𝑗⟩) = 𝑗
8684, 85syl6eq 2660 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐽‘(𝑛𝐽𝑗))) = 𝑗)
8783, 86fveq12d 6109 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))) = ((𝐹𝑛)‘𝑗))
8869, 87eqtrd 2644 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑛𝐽𝑗)) = ((𝐹𝑛)‘𝑗))
8988fveq2d 6107 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐻‘(𝑛𝐽𝑗))) = (1st ‘((𝐹𝑛)‘𝑗)))
9089breq1d 4593 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧 ↔ (1st ‘((𝐹𝑛)‘𝑗)) < 𝑧))
9188fveq2d 6107 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐻‘(𝑛𝐽𝑗))) = (2nd ‘((𝐹𝑛)‘𝑗)))
9291breq2d 4595 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))) ↔ 𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))))
9390, 92anbi12d 743 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗)))) ↔ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
9493biimprd 237 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))))
95 fveq2 6103 . . . . . . . . . . . . . 14 (𝑚 = (𝑛𝐽𝑗) → (𝐻𝑚) = (𝐻‘(𝑛𝐽𝑗)))
9695fveq2d 6107 . . . . . . . . . . . . 13 (𝑚 = (𝑛𝐽𝑗) → (1st ‘(𝐻𝑚)) = (1st ‘(𝐻‘(𝑛𝐽𝑗))))
9796breq1d 4593 . . . . . . . . . . . 12 (𝑚 = (𝑛𝐽𝑗) → ((1st ‘(𝐻𝑚)) < 𝑧 ↔ (1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧))
9895fveq2d 6107 . . . . . . . . . . . . 13 (𝑚 = (𝑛𝐽𝑗) → (2nd ‘(𝐻𝑚)) = (2nd ‘(𝐻‘(𝑛𝐽𝑗))))
9998breq2d 4595 . . . . . . . . . . . 12 (𝑚 = (𝑛𝐽𝑗) → (𝑧 < (2nd ‘(𝐻𝑚)) ↔ 𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗)))))
10097, 99anbi12d 743 . . . . . . . . . . 11 (𝑚 = (𝑛𝐽𝑗) → (((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))) ↔ ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))))
101100rspcev 3282 . . . . . . . . . 10 (((𝑛𝐽𝑗) ∈ ℕ ∧ ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))))
10261, 94, 101syl6an 566 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
103102rexlimdva 3013 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → (∃𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
10454, 103mpd 15 . . . . . . 7 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))))
105104rexlimdv3a 3015 . . . . . 6 (𝜑 → (∃𝑛 ∈ ℕ 𝑧𝐴 → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
10641, 105syl5bi 231 . . . . 5 (𝜑 → (𝑧 𝑛 ∈ ℕ 𝐴 → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
107106ralrimiv 2948 . . . 4 (𝜑 → ∀𝑧 𝑛 ∈ ℕ 𝐴𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))))
108 ovolfioo 23043 . . . . 5 (( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ( 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧 𝑛 ∈ ℕ 𝐴𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
1094, 26, 108syl2anc 691 . . . 4 (𝜑 → ( 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧 𝑛 ∈ ℕ 𝐴𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
110107, 109mpbird 246 . . 3 (𝜑 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻))
11128ovollb 23054 . . 3 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻)) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ*, < ))
11226, 110, 111syl2anc 691 . 2 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ*, < ))
113 fzfi 12633 . . . . . . 7 (1...𝑗) ∈ Fin
114 elfznn 12241 . . . . . . . . . 10 (𝑤 ∈ (1...𝑗) → 𝑤 ∈ ℕ)
115 ffvelrn 6265 . . . . . . . . . . 11 ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (𝐽𝑤) ∈ (ℕ × ℕ))
116 xp1st 7089 . . . . . . . . . . 11 ((𝐽𝑤) ∈ (ℕ × ℕ) → (1st ‘(𝐽𝑤)) ∈ ℕ)
117 nnre 10904 . . . . . . . . . . 11 ((1st ‘(𝐽𝑤)) ∈ ℕ → (1st ‘(𝐽𝑤)) ∈ ℝ)
118115, 116, 1173syl 18 . . . . . . . . . 10 ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (1st ‘(𝐽𝑤)) ∈ ℝ)
11911, 114, 118syl2an 493 . . . . . . . . 9 ((𝜑𝑤 ∈ (1...𝑗)) → (1st ‘(𝐽𝑤)) ∈ ℝ)
120119ralrimiva 2949 . . . . . . . 8 (𝜑 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ)
121120adantr 480 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ)
122 fimaxre3 10849 . . . . . . 7 (((1...𝑗) ∈ Fin ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥)
123113, 121, 122sylancr 694 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥)
124 fllep1 12464 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
125124ad2antlr 759 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ≤ ((⌊‘𝑥) + 1))
126119adantlr 747 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (1st ‘(𝐽𝑤)) ∈ ℝ)
127 simplr 788 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ∈ ℝ)
128 flcl 12458 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℤ)
129128peano2zd 11361 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℤ)
130129zred 11358 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ)
131130ad2antlr 759 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((⌊‘𝑥) + 1) ∈ ℝ)
132 letr 10010 . . . . . . . . . . . 12 (((1st ‘(𝐽𝑤)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ((⌊‘𝑥) + 1) ∈ ℝ) → (((1st ‘(𝐽𝑤)) ≤ 𝑥𝑥 ≤ ((⌊‘𝑥) + 1)) → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
133126, 127, 131, 132syl3anc 1318 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (((1st ‘(𝐽𝑤)) ≤ 𝑥𝑥 ≤ ((⌊‘𝑥) + 1)) → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
134125, 133mpan2d 706 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((1st ‘(𝐽𝑤)) ≤ 𝑥 → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
135134ralimdva 2945 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
136135adantlr 747 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
137 ovoliun.t . . . . . . . . . 10 𝑇 = seq1( + , 𝐺)
138 ovoliun.g . . . . . . . . . 10 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
139 simpll 786 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝜑)
140139, 1sylan 487 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
141 ovoliun.v . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
142139, 141sylan 487 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
143139, 36syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
144139, 37syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐵 ∈ ℝ+)
145 ovoliun.s . . . . . . . . . 10 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹𝑛)))
146139, 9syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
147139, 7syl 17 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
148139, 42sylan 487 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
149 ovoliun.x2 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
150139, 149sylan 487 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
151 simplr 788 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝑗 ∈ ℕ)
152129ad2antrl 760 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → ((⌊‘𝑥) + 1) ∈ ℤ)
153 simprr 792 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))
154137, 138, 140, 142, 143, 144, 145, 28, 25, 146, 147, 148, 150, 151, 152, 153ovoliunlem1 23077 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
155154expr 641 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1) → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
156136, 155syld 46 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
157156rexlimdva 3013 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
158123, 157mpd 15 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
159158ralrimiva 2949 . . . 4 (𝜑 → ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
160 ffn 5958 . . . . 5 (𝑈:ℕ⟶(0[,)+∞) → 𝑈 Fn ℕ)
161 breq1 4586 . . . . . 6 (𝑧 = (𝑈𝑗) → (𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
162161ralrn 6270 . . . . 5 (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
16326, 29, 160, 1624syl 19 . . . 4 (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
164159, 163mpbird 246 . . 3 (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
165 supxrleub 12028 . . . 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)))
16633, 40, 165syl2anc 691 . . 3 (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
167164, 166mpbird 246 . 2 (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
1686, 35, 40, 112, 167xrletrd 11869 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  ran crn 5039   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  Fincfn 7841  supcsup 8229  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  2c2 10947  ℤcz 11254  ℝ+crp 11708  (,)cioo 12046  [,)cico 12048  ...cfz 12197  ⌊cfl 12453  seqcseq 12663  ↑cexp 12722  abscabs 13822  vol*covol 23038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ioo 12050  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-ovol 23040 This theorem is referenced by:  ovoliunlem3  23079
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