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Theorem ovoliunlem2 23317
Description: Lemma for ovoliun 23319. (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 2995 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
3 iunss 4593 . . . 4 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
42, 3sylibr 224 . . 3 (𝜑 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
5 ovolcl 23292 . . 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 6175 . . . . . . . . . . . 12 (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:ℕ⟶(ℕ × ℕ))
119, 10syl 17 . . . . . . . . . . 11 (𝜑𝐽:ℕ⟶(ℕ × ℕ))
1211ffvelrnda 6399 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝐽𝑘) ∈ (ℕ × ℕ))
13 xp1st 7242 . . . . . . . . . 10 ((𝐽𝑘) ∈ (ℕ × ℕ) → (1st ‘(𝐽𝑘)) ∈ ℕ)
1412, 13syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (1st ‘(𝐽𝑘)) ∈ ℕ)
158, 14ffvelrnd 6400 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
16 reex 10065 . . . . . . . . . . 11 ℝ ∈ V
1716, 16xpex 7004 . . . . . . . . . 10 (ℝ × ℝ) ∈ V
1817inex2 4833 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ∈ V
19 nnex 11064 . . . . . . . . 9 ℕ ∈ V
2018, 19elmap 7928 . . . . . . . 8 ((𝐹‘(1st ‘(𝐽𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ (𝐹‘(1st ‘(𝐽𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2115, 20sylib 208 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
22 xp2nd 7243 . . . . . . . 8 ((𝐽𝑘) ∈ (ℕ × ℕ) → (2nd ‘(𝐽𝑘)) ∈ ℕ)
2312, 22syl 17 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (2nd ‘(𝐽𝑘)) ∈ ℕ)
2421, 23ffvelrnd 6400 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))) ∈ ( ≤ ∩ (ℝ × ℝ)))
25 ovoliun.h . . . . . 6 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))))
2624, 25fmptd 6425 . . . . 5 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27 eqid 2651 . . . . . 6 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
28 ovoliun.u . . . . . 6 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
2927, 28ovolsf 23287 . . . . 5 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
30 frn 6091 . . . . 5 (𝑈:ℕ⟶(0[,)+∞) → ran 𝑈 ⊆ (0[,)+∞))
3126, 29, 303syl 18 . . . 4 (𝜑 → ran 𝑈 ⊆ (0[,)+∞))
32 icossxr 12296 . . . 4 (0[,)+∞) ⊆ ℝ*
3331, 32syl6ss 3648 . . 3 (𝜑 → ran 𝑈 ⊆ ℝ*)
34 supxrcl 12183 . . 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 11910 . . . 4 (𝜑𝐵 ∈ ℝ)
3936, 38readdcld 10107 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ)
4039rexrd 10127 . 2 (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ*)
41 eliun 4556 . . . . . 6 (𝑧 𝑛 ∈ ℕ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑧𝐴)
42 ovoliun.x1 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
43423adant3 1101 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝐴 ran ((,) ∘ (𝐹𝑛)))
4413adant3 1101 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝐴 ⊆ ℝ)
457ffvelrnda 6399 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
4618, 19elmap 7928 . . . . . . . . . . . . 13 ((𝐹𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4745, 46sylib 208 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
48473adant3 1101 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
49 ovolfioo 23282 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ (𝐹𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ (𝐹𝑛)) ↔ ∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
5044, 48, 49syl2anc 694 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → (𝐴 ran ((,) ∘ (𝐹𝑛)) ↔ ∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
5143, 50mpbid 222 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → ∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))))
52 simp3 1083 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → 𝑧𝐴)
53 rsp 2958 . . . . . . . . 9 (∀𝑧𝐴𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → (𝑧𝐴 → ∃𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
5451, 52, 53sylc 65 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) → ∃𝑗 ∈ ℕ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))))
55 simpl1 1084 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝜑)
56 f1ocnv 6187 . . . . . . . . . . . 12 (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:(ℕ × ℕ)–1-1-onto→ℕ)
57 f1of 6175 . . . . . . . . . . . 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 1085 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝑛 ∈ ℕ)
60 simpr 476 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
6158, 59, 60fovrnd 6848 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝑛𝐽𝑗) ∈ ℕ)
62 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑛𝐽𝑗) → (𝐽𝑘) = (𝐽‘(𝑛𝐽𝑗)))
6362fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑛𝐽𝑗) → (1st ‘(𝐽𝑘)) = (1st ‘(𝐽‘(𝑛𝐽𝑗))))
6463fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛𝐽𝑗) → (𝐹‘(1st ‘(𝐽𝑘))) = (𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗)))))
6562fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑛𝐽𝑗) → (2nd ‘(𝐽𝑘)) = (2nd ‘(𝐽‘(𝑛𝐽𝑗))))
6664, 65fveq12d 6235 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑛𝐽𝑗) → ((𝐹‘(1st ‘(𝐽𝑘)))‘(2nd ‘(𝐽𝑘))) = ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))))
67 fvex 6239 . . . . . . . . . . . . . . . . 17 ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))) ∈ V
6866, 25, 67fvmpt 6321 . . . . . . . . . . . . . . . 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 6693 . . . . . . . . . . . . . . . . . . . . 21 (𝑛𝐽𝑗) = (𝐽‘⟨𝑛, 𝑗⟩)
7170fveq2i 6232 . . . . . . . . . . . . . . . . . . . 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 5182 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ ℕ ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
7459, 60, 73syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
75 f1ocnvfv2 6573 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽:ℕ–1-1-onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → (𝐽‘(𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
7672, 74, 75syl2anc 694 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
7771, 76syl5eq 2697 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(𝑛𝐽𝑗)) = ⟨𝑛, 𝑗⟩)
7877fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐽‘(𝑛𝐽𝑗))) = (1st ‘⟨𝑛, 𝑗⟩))
79 vex 3234 . . . . . . . . . . . . . . . . . . 19 𝑛 ∈ V
80 vex 3234 . . . . . . . . . . . . . . . . . . 19 𝑗 ∈ V
8179, 80op1st 7218 . . . . . . . . . . . . . . . . . 18 (1st ‘⟨𝑛, 𝑗⟩) = 𝑛
8278, 81syl6eq 2701 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐽‘(𝑛𝐽𝑗))) = 𝑛)
8382fveq2d 6233 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗)))) = (𝐹𝑛))
8477fveq2d 6233 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐽‘(𝑛𝐽𝑗))) = (2nd ‘⟨𝑛, 𝑗⟩))
8579, 80op2nd 7219 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨𝑛, 𝑗⟩) = 𝑗
8684, 85syl6eq 2701 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐽‘(𝑛𝐽𝑗))) = 𝑗)
8783, 86fveq12d 6235 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽‘(𝑛𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛𝐽𝑗)))) = ((𝐹𝑛)‘𝑗))
8869, 87eqtrd 2685 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑛𝐽𝑗)) = ((𝐹𝑛)‘𝑗))
8988fveq2d 6233 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐻‘(𝑛𝐽𝑗))) = (1st ‘((𝐹𝑛)‘𝑗)))
9089breq1d 4695 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧 ↔ (1st ‘((𝐹𝑛)‘𝑗)) < 𝑧))
9188fveq2d 6233 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐻‘(𝑛𝐽𝑗))) = (2nd ‘((𝐹𝑛)‘𝑗)))
9291breq2d 4697 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))) ↔ 𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))))
9390, 92anbi12d 747 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗)))) ↔ ((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗)))))
9493biimprd 238 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))))
95 fveq2 6229 . . . . . . . . . . . . . 14 (𝑚 = (𝑛𝐽𝑗) → (𝐻𝑚) = (𝐻‘(𝑛𝐽𝑗)))
9695fveq2d 6233 . . . . . . . . . . . . 13 (𝑚 = (𝑛𝐽𝑗) → (1st ‘(𝐻𝑚)) = (1st ‘(𝐻‘(𝑛𝐽𝑗))))
9796breq1d 4695 . . . . . . . . . . . 12 (𝑚 = (𝑛𝐽𝑗) → ((1st ‘(𝐻𝑚)) < 𝑧 ↔ (1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧))
9895fveq2d 6233 . . . . . . . . . . . . 13 (𝑚 = (𝑛𝐽𝑗) → (2nd ‘(𝐻𝑚)) = (2nd ‘(𝐻‘(𝑛𝐽𝑗))))
9998breq2d 4697 . . . . . . . . . . . 12 (𝑚 = (𝑛𝐽𝑗) → (𝑧 < (2nd ‘(𝐻𝑚)) ↔ 𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗)))))
10097, 99anbi12d 747 . . . . . . . . . . 11 (𝑚 = (𝑛𝐽𝑗) → (((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))) ↔ ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))))
101100rspcev 3340 . . . . . . . . . 10 (((𝑛𝐽𝑗) ∈ ℕ ∧ ((1st ‘(𝐻‘(𝑛𝐽𝑗))) < 𝑧𝑧 < (2nd ‘(𝐻‘(𝑛𝐽𝑗))))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))))
10261, 94, 101syl6an 567 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑧𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘((𝐹𝑛)‘𝑗)) < 𝑧𝑧 < (2nd ‘((𝐹𝑛)‘𝑗))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
103102rexlimdva 3060 . . . . . . . 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 3062 . . . . . 6 (𝜑 → (∃𝑛 ∈ ℕ 𝑧𝐴 → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
10641, 105syl5bi 232 . . . . 5 (𝜑 → (𝑧 𝑛 ∈ ℕ 𝐴 → ∃𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
107106ralrimiv 2994 . . . 4 (𝜑 → ∀𝑧 𝑛 ∈ ℕ 𝐴𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚))))
108 ovolfioo 23282 . . . . 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 694 . . . 4 (𝜑 → ( 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧 𝑛 ∈ ℕ 𝐴𝑚 ∈ ℕ ((1st ‘(𝐻𝑚)) < 𝑧𝑧 < (2nd ‘(𝐻𝑚)))))
110107, 109mpbird 247 . . 3 (𝜑 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻))
11128ovollb 23293 . . 3 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ 𝐴 ran ((,) ∘ 𝐻)) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ*, < ))
11226, 110, 111syl2anc 694 . 2 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ*, < ))
113 fzfi 12811 . . . . . . 7 (1...𝑗) ∈ Fin
114 elfznn 12408 . . . . . . . . . 10 (𝑤 ∈ (1...𝑗) → 𝑤 ∈ ℕ)
115 ffvelrn 6397 . . . . . . . . . . 11 ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (𝐽𝑤) ∈ (ℕ × ℕ))
116 xp1st 7242 . . . . . . . . . . 11 ((𝐽𝑤) ∈ (ℕ × ℕ) → (1st ‘(𝐽𝑤)) ∈ ℕ)
117 nnre 11065 . . . . . . . . . . 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 2995 . . . . . . . 8 (𝜑 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ)
121120adantr 480 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ)
122 fimaxre3 11008 . . . . . . 7 (((1...𝑗) ∈ Fin ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥)
123113, 121, 122sylancr 696 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥)
124 fllep1 12642 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
125124ad2antlr 763 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ≤ ((⌊‘𝑥) + 1))
126119adantlr 751 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (1st ‘(𝐽𝑤)) ∈ ℝ)
127 simplr 807 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ∈ ℝ)
128 flcl 12636 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℤ)
129128peano2zd 11523 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℤ)
130129zred 11520 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ)
131130ad2antlr 763 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((⌊‘𝑥) + 1) ∈ ℝ)
132 letr 10169 . . . . . . . . . . . 12 (((1st ‘(𝐽𝑤)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ((⌊‘𝑥) + 1) ∈ ℝ) → (((1st ‘(𝐽𝑤)) ≤ 𝑥𝑥 ≤ ((⌊‘𝑥) + 1)) → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
133126, 127, 131, 132syl3anc 1366 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (((1st ‘(𝐽𝑤)) ≤ 𝑥𝑥 ≤ ((⌊‘𝑥) + 1)) → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
134125, 133mpan2d 710 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((1st ‘(𝐽𝑤)) ≤ 𝑥 → (1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
135134ralimdva 2991 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1)))
136135adantlr 751 . . . . . . . 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 805 . . . . . . . . . . 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 807 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝑗 ∈ ℕ)
152129ad2antrl 764 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → ((⌊‘𝑥) + 1) ∈ ℤ)
153 simprr 811 . . . . . . . . . 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 23316 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1))) → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
155154expr 642 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ ((⌊‘𝑥) + 1) → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
156136, 155syld 47 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽𝑤)) ≤ 𝑥 → (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
157156rexlimdva 3060 . . . . . 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 2995 . . . 4 (𝜑 → ∀𝑗 ∈ ℕ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
160 ffn 6083 . . . . 5 (𝑈:ℕ⟶(0[,)+∞) → 𝑈 Fn ℕ)
161 breq1 4688 . . . . . 6 (𝑧 = (𝑈𝑗) → (𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ (𝑈𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
162161ralrn 6402 . . . . 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 247 . . 3 (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
165 supxrleub 12194 . . . 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 694 . . 3 (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
167164, 166mpbird 247 . 2 (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
1686, 35, 40, 112, 167xrletrd 12031 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cin 3606  wss 3607  cop 4216   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762   × cxp 5141  ccnv 5142  ran crn 5144  ccom 5147   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Fincfn 7997  supcsup 8387  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  cz 11415  +crp 11870  (,)cioo 12213  [,)cico 12215  ...cfz 12364  cfl 12631  seqcseq 12841  cexp 12900  abscabs 14018  vol*covol 23277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ioo 12217  df-ico 12219  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-ovol 23279
This theorem is referenced by:  ovoliunlem3  23318
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