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Theorem ioombl1lem3 25077
Description: Lemma for ioombl1 25079. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
ioombl1.b 𝐵 = (𝐴(,)+∞)
ioombl1.a (𝜑𝐴 ∈ ℝ)
ioombl1.e (𝜑𝐸 ⊆ ℝ)
ioombl1.v (𝜑 → (vol*‘𝐸) ∈ ℝ)
ioombl1.c (𝜑𝐶 ∈ ℝ+)
ioombl1.s 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ioombl1.t 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
ioombl1.u 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ioombl1.f1 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ioombl1.f2 (𝜑𝐸 ran ((,) ∘ 𝐹))
ioombl1.f3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
ioombl1.p 𝑃 = (1st ‘(𝐹𝑛))
ioombl1.q 𝑄 = (2nd ‘(𝐹𝑛))
ioombl1.g 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
ioombl1.h 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
Assertion
Ref Expression
ioombl1lem3 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
Distinct variable groups:   𝐵,𝑛   𝐶,𝑛   𝑛,𝐸   𝑛,𝐹   𝑛,𝐺   𝑛,𝐻   𝜑,𝑛   𝑆,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝑃(𝑛)   𝑄(𝑛)   𝑇(𝑛)   𝑈(𝑛)

Proof of Theorem ioombl1lem3
StepHypRef Expression
1 ioombl1.q . . . . 5 𝑄 = (2nd ‘(𝐹𝑛))
2 ioombl1.f1 . . . . . . 7 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3 ovolfcl 24983 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
42, 3sylan 581 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
54simp2d 1144 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
61, 5eqeltrid 2838 . . . 4 ((𝜑𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
76recnd 11242 . . 3 ((𝜑𝑛 ∈ ℕ) → 𝑄 ∈ ℂ)
8 ioombl1.a . . . . . . 7 (𝜑𝐴 ∈ ℝ)
98adantr 482 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
10 ioombl1.p . . . . . . 7 𝑃 = (1st ‘(𝐹𝑛))
114simp1d 1143 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
1210, 11eqeltrid 2838 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
139, 12ifcld 4575 . . . . 5 ((𝜑𝑛 ∈ ℕ) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
1413, 6ifcld 4575 . . . 4 ((𝜑𝑛 ∈ ℕ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
1514recnd 11242 . . 3 ((𝜑𝑛 ∈ ℕ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℂ)
1612recnd 11242 . . 3 ((𝜑𝑛 ∈ ℕ) → 𝑃 ∈ ℂ)
177, 15, 16npncand 11595 . 2 ((𝜑𝑛 ∈ ℕ) → ((𝑄 − if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)) + (if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) − 𝑃)) = (𝑄𝑃))
18 ioombl1.b . . . . . . 7 𝐵 = (𝐴(,)+∞)
19 ioombl1.e . . . . . . 7 (𝜑𝐸 ⊆ ℝ)
20 ioombl1.v . . . . . . 7 (𝜑 → (vol*‘𝐸) ∈ ℝ)
21 ioombl1.c . . . . . . 7 (𝜑𝐶 ∈ ℝ+)
22 ioombl1.s . . . . . . 7 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
23 ioombl1.t . . . . . . 7 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
24 ioombl1.u . . . . . . 7 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
25 ioombl1.f2 . . . . . . 7 (𝜑𝐸 ran ((,) ∘ 𝐹))
26 ioombl1.f3 . . . . . . 7 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
27 ioombl1.g . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
28 ioombl1.h . . . . . . 7 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
2918, 8, 19, 20, 21, 22, 23, 24, 2, 25, 26, 10, 1, 27, 28ioombl1lem1 25075 . . . . . 6 (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
3029simpld 496 . . . . 5 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31 eqid 2733 . . . . . 6 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
3231ovolfsval 24987 . . . . 5 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
3330, 32sylan 581 . . . 4 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
34 simpr 486 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
35 opex 5465 . . . . . . . 8 ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ V
3627fvmpt2 7010 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ V) → (𝐺𝑛) = ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
3734, 35, 36sylancl 587 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
3837fveq2d 6896 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
39 op2ndg 7988 . . . . . . 7 ((if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = 𝑄)
4014, 6, 39syl2anc 585 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = 𝑄)
4138, 40eqtrd 2773 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = 𝑄)
4237fveq2d 6896 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = (1st ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
43 op1stg 7987 . . . . . . 7 ((if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (1st ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
4414, 6, 43syl2anc 585 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (1st ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
4542, 44eqtrd 2773 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
4641, 45oveq12d 7427 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (𝑄 − if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)))
4733, 46eqtrd 2773 . . 3 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (𝑄 − if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)))
4829simprd 497 . . . . 5 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
49 eqid 2733 . . . . . 6 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
5049ovolfsval 24987 . . . . 5 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = ((2nd ‘(𝐻𝑛)) − (1st ‘(𝐻𝑛))))
5148, 50sylan 581 . . . 4 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = ((2nd ‘(𝐻𝑛)) − (1st ‘(𝐻𝑛))))
52 opex 5465 . . . . . . . 8 𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ V
5328fvmpt2 7010 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ V) → (𝐻𝑛) = ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
5434, 52, 53sylancl 587 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐻𝑛) = ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
5554fveq2d 6896 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐻𝑛)) = (2nd ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩))
56 op2ndg 7988 . . . . . . 7 ((𝑃 ∈ ℝ ∧ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) → (2nd ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
5712, 14, 56syl2anc 585 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (2nd ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
5855, 57eqtrd 2773 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐻𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
5954fveq2d 6896 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩))
60 op1stg 7987 . . . . . . 7 ((𝑃 ∈ ℝ ∧ if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) → (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = 𝑃)
6112, 14, 60syl2anc 585 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = 𝑃)
6259, 61eqtrd 2773 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = 𝑃)
6358, 62oveq12d 7427 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐻𝑛)) − (1st ‘(𝐻𝑛))) = (if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) − 𝑃))
6451, 63eqtrd 2773 . . 3 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = (if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) − 𝑃))
6547, 64oveq12d 7427 . 2 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = ((𝑄 − if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)) + (if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) − 𝑃)))
66 eqid 2733 . . . . 5 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
6766ovolfsval 24987 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
682, 67sylan 581 . . 3 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
691, 10oveq12i 7421 . . 3 (𝑄𝑃) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛)))
7068, 69eqtr4di 2791 . 2 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = (𝑄𝑃))
7117, 65, 703eqtr4d 2783 1 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  cin 3948  wss 3949  ifcif 4529  cop 4635   cuni 4909   class class class wbr 5149  cmpt 5232   × cxp 5675  ran crn 5678  ccom 5681  wf 6540  cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  supcsup 9435  cr 11109  1c1 11111   + caddc 11113  +∞cpnf 11245  *cxr 11247   < clt 11248  cle 11249  cmin 11444  cn 12212  +crp 12974  (,)cioo 13324  seqcseq 13966  abscabs 15181  vol*covol 24979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183
This theorem is referenced by:  ioombl1lem4  25078
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