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Theorem ioombl1lem3 25577
Description: Lemma for ioombl1 25579. (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 25483 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
42, 3sylan 578 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
54simp2d 1140 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
61, 5eqeltrid 2830 . . . 4 ((𝜑𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
76recnd 11283 . . 3 ((𝜑𝑛 ∈ ℕ) → 𝑄 ∈ ℂ)
8 ioombl1.a . . . . . . 7 (𝜑𝐴 ∈ ℝ)
98adantr 479 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
10 ioombl1.p . . . . . . 7 𝑃 = (1st ‘(𝐹𝑛))
114simp1d 1139 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
1210, 11eqeltrid 2830 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
139, 12ifcld 4569 . . . . 5 ((𝜑𝑛 ∈ ℕ) → if(𝑃𝐴, 𝐴, 𝑃) ∈ ℝ)
1413, 6ifcld 4569 . . . 4 ((𝜑𝑛 ∈ ℕ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
1514recnd 11283 . . 3 ((𝜑𝑛 ∈ ℕ) → if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) ∈ ℂ)
1612recnd 11283 . . 3 ((𝜑𝑛 ∈ ℕ) → 𝑃 ∈ ℂ)
177, 15, 16npncand 11636 . 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 25575 . . . . . 6 (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
3029simpld 493 . . . . 5 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31 eqid 2726 . . . . . 6 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
3231ovolfsval 25487 . . . . 5 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
3330, 32sylan 578 . . . 4 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))))
34 simpr 483 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
35 opex 5462 . . . . . . . 8 ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ V
3627fvmpt2 7012 . . . . . . . 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 584 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = ⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
3837fveq2d 6897 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
39 op2ndg 8008 . . . . . . 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 582 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (2nd ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = 𝑄)
4138, 40eqtrd 2766 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = 𝑄)
4237fveq2d 6897 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = (1st ‘⟨if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
43 op1stg 8007 . . . . . . 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 582 . . . . . 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 2766 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
4641, 45oveq12d 7434 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) − (1st ‘(𝐺𝑛))) = (𝑄 − if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)))
4733, 46eqtrd 2766 . . 3 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (𝑄 − if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)))
4829simprd 494 . . . . 5 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
49 eqid 2726 . . . . . 6 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
5049ovolfsval 25487 . . . . 5 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = ((2nd ‘(𝐻𝑛)) − (1st ‘(𝐻𝑛))))
5148, 50sylan 578 . . . 4 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = ((2nd ‘(𝐻𝑛)) − (1st ‘(𝐻𝑛))))
52 opex 5462 . . . . . . . 8 𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ V
5328fvmpt2 7012 . . . . . . . 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 584 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐻𝑛) = ⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩)
5554fveq2d 6897 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐻𝑛)) = (2nd ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩))
56 op2ndg 8008 . . . . . . 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 582 . . . . . 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 2766 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐻𝑛)) = if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄))
5954fveq2d 6897 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩))
60 op1stg 8007 . . . . . . 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 582 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (1st ‘⟨𝑃, if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄)⟩) = 𝑃)
6259, 61eqtrd 2766 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐻𝑛)) = 𝑃)
6358, 62oveq12d 7434 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐻𝑛)) − (1st ‘(𝐻𝑛))) = (if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) − 𝑃))
6451, 63eqtrd 2766 . . 3 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = (if(if(𝑃𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃𝐴, 𝐴, 𝑃), 𝑄) − 𝑃))
6547, 64oveq12d 7434 . 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 2726 . . . . 5 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
6766ovolfsval 25487 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
682, 67sylan 578 . . 3 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛))))
691, 10oveq12i 7428 . . 3 (𝑄𝑃) = ((2nd ‘(𝐹𝑛)) − (1st ‘(𝐹𝑛)))
7068, 69eqtr4di 2784 . 2 ((𝜑𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = (𝑄𝑃))
7117, 65, 703eqtr4d 2776 1 ((𝜑𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  Vcvv 3462  cin 3945  wss 3946  ifcif 4523  cop 4629   cuni 4905   class class class wbr 5145  cmpt 5228   × cxp 5672  ran crn 5675  ccom 5678  wf 6542  cfv 6546  (class class class)co 7416  1st c1st 7993  2nd c2nd 7994  supcsup 9476  cr 11148  1c1 11150   + caddc 11152  +∞cpnf 11286  *cxr 11288   < clt 11289  cle 11290  cmin 11485  cn 12258  +crp 13022  (,)cioo 13372  seqcseq 14015  abscabs 15234  vol*covol 25479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-sup 9478  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-div 11913  df-nn 12259  df-2 12321  df-3 12322  df-n0 12519  df-z 12605  df-uz 12869  df-rp 13023  df-seq 14016  df-exp 14076  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236
This theorem is referenced by:  ioombl1lem4  25578
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