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Theorem ioombl1 25597
Description: An open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
Assertion
Ref Expression
ioombl1 (𝐴 ∈ ℝ* → (𝐴(,)+∞) ∈ dom vol)

Proof of Theorem ioombl1
Dummy variables 𝑓 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxr 13158 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 ioossre 13448 . . . . 5 (𝐴(,)+∞) ⊆ ℝ
32a1i 11 . . . 4 (𝐴 ∈ ℝ → (𝐴(,)+∞) ⊆ ℝ)
4 elpwi 4607 . . . . . 6 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
5 simplrl 777 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → 𝑥 ⊆ ℝ)
6 simplrr 778 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → (vol*‘𝑥) ∈ ℝ)
7 simpr 484 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
8 eqid 2737 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
98ovolgelb 25515 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ ∧ 𝑦 ∈ ℝ+) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))
105, 6, 7, 9syl3anc 1373 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))
11 eqid 2737 . . . . . . . . . . 11 (𝐴(,)+∞) = (𝐴(,)+∞)
12 simplll 775 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))) → 𝐴 ∈ ℝ)
135adantr 480 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))) → 𝑥 ⊆ ℝ)
146adantr 480 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))) → (vol*‘𝑥) ∈ ℝ)
15 simplr 769 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))) → 𝑦 ∈ ℝ+)
16 eqid 2737 . . . . . . . . . . 11 seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨if(if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)), if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚))⟩))) = seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨if(if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)), if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚))⟩)))
17 eqid 2737 . . . . . . . . . . 11 seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨(1st ‘(𝑓𝑚)), if(if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)), if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚)))⟩))) = seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨(1st ‘(𝑓𝑚)), if(if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)), if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚)))⟩)))
18 simprl 771 . . . . . . . . . . . 12 ((((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))) → 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
19 elovolmlem 25509 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2018, 19sylib 218 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
21 simprrl 781 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))) → 𝑥 ran ((,) ∘ 𝑓))
22 simprrr 782 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦))
23 eqid 2737 . . . . . . . . . . 11 (1st ‘(𝑓𝑛)) = (1st ‘(𝑓𝑛))
24 eqid 2737 . . . . . . . . . . 11 (2nd ‘(𝑓𝑛)) = (2nd ‘(𝑓𝑛))
25 2fveq3 6911 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1st ‘(𝑓𝑚)) = (1st ‘(𝑓𝑛)))
2625breq1d 5153 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((1st ‘(𝑓𝑚)) ≤ 𝐴 ↔ (1st ‘(𝑓𝑛)) ≤ 𝐴))
2726, 25ifbieq2d 4552 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) = if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))))
28 2fveq3 6911 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (2nd ‘(𝑓𝑚)) = (2nd ‘(𝑓𝑛)))
2927, 28breq12d 5156 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)) ↔ if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))) ≤ (2nd ‘(𝑓𝑛))))
3029, 27, 28ifbieq12d 4554 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → if(if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)), if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚))) = if(if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))) ≤ (2nd ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))), (2nd ‘(𝑓𝑛))))
3130, 28opeq12d 4881 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ⟨if(if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)), if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚))⟩ = ⟨if(if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))) ≤ (2nd ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))), (2nd ‘(𝑓𝑛))), (2nd ‘(𝑓𝑛))⟩)
3231cbvmptv 5255 . . . . . . . . . . 11 (𝑚 ∈ ℕ ↦ ⟨if(if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)), if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚))⟩) = (𝑛 ∈ ℕ ↦ ⟨if(if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))) ≤ (2nd ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))), (2nd ‘(𝑓𝑛))), (2nd ‘(𝑓𝑛))⟩)
3325, 30opeq12d 4881 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ⟨(1st ‘(𝑓𝑚)), if(if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)), if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚)))⟩ = ⟨(1st ‘(𝑓𝑛)), if(if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))) ≤ (2nd ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))), (2nd ‘(𝑓𝑛)))⟩)
3433cbvmptv 5255 . . . . . . . . . . 11 (𝑚 ∈ ℕ ↦ ⟨(1st ‘(𝑓𝑚)), if(if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)), if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))), (2nd ‘(𝑓𝑚)))⟩) = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝑓𝑛)), if(if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))) ≤ (2nd ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))), (2nd ‘(𝑓𝑛)))⟩)
3511, 12, 13, 14, 15, 8, 16, 17, 20, 21, 22, 23, 24, 32, 34ioombl1lem4 25596 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝑥 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦)))) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
3610, 35rexlimddv 3161 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
3736ralrimiva 3146 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ∀𝑦 ∈ ℝ+ ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
38 inss1 4237 . . . . . . . . . . . 12 (𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥
39 ovolsscl 25521 . . . . . . . . . . . 12 (((𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
4038, 39mp3an1 1450 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
4140adantl 481 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
42 difss 4136 . . . . . . . . . . . 12 (𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥
43 ovolsscl 25521 . . . . . . . . . . . 12 (((𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4442, 43mp3an1 1450 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4544adantl 481 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4641, 45readdcld 11290 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ∈ ℝ)
47 simprr 773 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
48 alrple 13248 . . . . . . . . 9 ((((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ∈ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥) ↔ ∀𝑦 ∈ ℝ+ ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦)))
4946, 47, 48syl2anc 584 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥) ↔ ∀𝑦 ∈ ℝ+ ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦)))
5037, 49mpbird 257 . . . . . . 7 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥))
5150expr 456 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
524, 51sylan2 593 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
5352ralrimiva 3146 . . . 4 (𝐴 ∈ ℝ → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
54 ismbl2 25562 . . . 4 ((𝐴(,)+∞) ∈ dom vol ↔ ((𝐴(,)+∞) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥))))
553, 53, 54sylanbrc 583 . . 3 (𝐴 ∈ ℝ → (𝐴(,)+∞) ∈ dom vol)
56 oveq1 7438 . . . . 5 (𝐴 = +∞ → (𝐴(,)+∞) = (+∞(,)+∞))
57 iooid 13415 . . . . 5 (+∞(,)+∞) = ∅
5856, 57eqtrdi 2793 . . . 4 (𝐴 = +∞ → (𝐴(,)+∞) = ∅)
59 0mbl 25574 . . . 4 ∅ ∈ dom vol
6058, 59eqeltrdi 2849 . . 3 (𝐴 = +∞ → (𝐴(,)+∞) ∈ dom vol)
61 oveq1 7438 . . . . 5 (𝐴 = -∞ → (𝐴(,)+∞) = (-∞(,)+∞))
62 ioomax 13462 . . . . 5 (-∞(,)+∞) = ℝ
6361, 62eqtrdi 2793 . . . 4 (𝐴 = -∞ → (𝐴(,)+∞) = ℝ)
64 rembl 25575 . . . 4 ℝ ∈ dom vol
6563, 64eqeltrdi 2849 . . 3 (𝐴 = -∞ → (𝐴(,)+∞) ∈ dom vol)
6655, 60, 653jaoi 1430 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴(,)+∞) ∈ dom vol)
671, 66sylbi 217 1 (𝐴 ∈ ℝ* → (𝐴(,)+∞) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3o 1086   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cdif 3948  cin 3950  wss 3951  c0 4333  ifcif 4525  𝒫 cpw 4600  cop 4632   cuni 4907   class class class wbr 5143  cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  m cmap 8866  supcsup 9480  cr 11154  1c1 11156   + caddc 11158  +∞cpnf 11292  -∞cmnf 11293  *cxr 11294   < clt 11295  cle 11296  cmin 11492  cn 12266  +crp 13034  (,)cioo 13387  seqcseq 14042  abscabs 15273  vol*covol 25497  volcvol 25498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xadd 13155  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-xmet 21357  df-met 21358  df-ovol 25499  df-vol 25500
This theorem is referenced by:  icombl1  25598
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