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Theorem ioombl1 24207
 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 12519 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 ioossre 12806 . . . . 5 (𝐴(,)+∞) ⊆ ℝ
32a1i 11 . . . 4 (𝐴 ∈ ℝ → (𝐴(,)+∞) ⊆ ℝ)
4 elpwi 4509 . . . . . 6 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
5 simplrl 776 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → 𝑥 ⊆ ℝ)
6 simplrr 777 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → (vol*‘𝑥) ∈ ℝ)
7 simpr 488 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
8 eqid 2798 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
98ovolgelb 24125 . . . . . . . . . . 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 1368 . . . . . . . . . 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 2798 . . . . . . . . . . 11 (𝐴(,)+∞) = (𝐴(,)+∞)
12 simplll 774 . . . . . . . . . . 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 484 . . . . . . . . . . 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 484 . . . . . . . . . . 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 768 . . . . . . . . . . 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 2798 . . . . . . . . . . 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 2798 . . . . . . . . . . 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 770 . . . . . . . . . . . 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 24119 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2018, 19sylib 221 . . . . . . . . . . 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 780 . . . . . . . . . . 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 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)))) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝑥) + 𝑦))
23 eqid 2798 . . . . . . . . . . 11 (1st ‘(𝑓𝑛)) = (1st ‘(𝑓𝑛))
24 eqid 2798 . . . . . . . . . . 11 (2nd ‘(𝑓𝑛)) = (2nd ‘(𝑓𝑛))
25 2fveq3 6660 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1st ‘(𝑓𝑚)) = (1st ‘(𝑓𝑛)))
2625breq1d 5044 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((1st ‘(𝑓𝑚)) ≤ 𝐴 ↔ (1st ‘(𝑓𝑛)) ≤ 𝐴))
2726, 25ifbieq2d 4453 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) = if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))))
28 2fveq3 6660 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (2nd ‘(𝑓𝑚)) = (2nd ‘(𝑓𝑛)))
2927, 28breq12d 5047 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)) ↔ if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))) ≤ (2nd ‘(𝑓𝑛))))
3029, 27, 28ifbieq12d 4455 . . . . . . . . . . . . 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 4777 . . . . . . . . . . . 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 5137 . . . . . . . . . . 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 4777 . . . . . . . . . . . 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 5137 . . . . . . . . . . 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 24206 . . . . . . . . . 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 3251 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
3736ralrimiva 3149 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ∀𝑦 ∈ ℝ+ ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
38 inss1 4158 . . . . . . . . . . . 12 (𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥
39 ovolsscl 24131 . . . . . . . . . . . 12 (((𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
4038, 39mp3an1 1445 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
4140adantl 485 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
42 difss 4062 . . . . . . . . . . . 12 (𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥
43 ovolsscl 24131 . . . . . . . . . . . 12 (((𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4442, 43mp3an1 1445 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4544adantl 485 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4641, 45readdcld 10677 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ∈ ℝ)
47 simprr 772 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
48 alrple 12607 . . . . . . . . 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 587 . . . . . . . 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 260 . . . . . . 7 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥))
5150expr 460 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
524, 51sylan2 595 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
5352ralrimiva 3149 . . . 4 (𝐴 ∈ ℝ → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
54 ismbl2 24172 . . . 4 ((𝐴(,)+∞) ∈ dom vol ↔ ((𝐴(,)+∞) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥))))
553, 53, 54sylanbrc 586 . . 3 (𝐴 ∈ ℝ → (𝐴(,)+∞) ∈ dom vol)
56 oveq1 7152 . . . . 5 (𝐴 = +∞ → (𝐴(,)+∞) = (+∞(,)+∞))
57 iooid 12774 . . . . 5 (+∞(,)+∞) = ∅
5856, 57eqtrdi 2849 . . . 4 (𝐴 = +∞ → (𝐴(,)+∞) = ∅)
59 0mbl 24184 . . . 4 ∅ ∈ dom vol
6058, 59eqeltrdi 2898 . . 3 (𝐴 = +∞ → (𝐴(,)+∞) ∈ dom vol)
61 oveq1 7152 . . . . 5 (𝐴 = -∞ → (𝐴(,)+∞) = (-∞(,)+∞))
62 ioomax 12820 . . . . 5 (-∞(,)+∞) = ℝ
6361, 62eqtrdi 2849 . . . 4 (𝐴 = -∞ → (𝐴(,)+∞) = ℝ)
64 rembl 24185 . . . 4 ℝ ∈ dom vol
6563, 64eqeltrdi 2898 . . 3 (𝐴 = -∞ → (𝐴(,)+∞) ∈ dom vol)
6655, 60, 653jaoi 1424 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴(,)+∞) ∈ dom vol)
671, 66sylbi 220 1 (𝐴 ∈ ℝ* → (𝐴(,)+∞) ∈ dom vol)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4246  ifcif 4428  𝒫 cpw 4500  ⟨cop 4534  ∪ cuni 4804   class class class wbr 5034   ↦ cmpt 5114   × cxp 5521  dom cdm 5523  ran crn 5524   ∘ ccom 5527  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  1st c1st 7682  2nd c2nd 7683   ↑m cmap 8407  supcsup 8906  ℝcr 10543  1c1 10545   + caddc 10547  +∞cpnf 10679  -∞cmnf 10680  ℝ*cxr 10681   < clt 10682   ≤ cle 10683   − cmin 10877  ℕcn 11643  ℝ+crp 12397  (,)cioo 12746  seqcseq 13384  abscabs 14605  vol*covol 24107  volcvol 24108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-inf2 9106  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-pre-sup 10622 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7400  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-oadd 8107  df-er 8290  df-map 8409  df-pm 8410  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-sup 8908  df-inf 8909  df-oi 8976  df-dju 9332  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-div 11305  df-nn 11644  df-2 11706  df-3 11707  df-n0 11904  df-z 11990  df-uz 12252  df-q 12357  df-rp 12398  df-xadd 12516  df-ioo 12750  df-ico 12752  df-icc 12753  df-fz 12906  df-fzo 13049  df-fl 13177  df-seq 13385  df-exp 13446  df-hash 13707  df-cj 14470  df-re 14471  df-im 14472  df-sqrt 14606  df-abs 14607  df-clim 14857  df-rlim 14858  df-sum 15055  df-xmet 20105  df-met 20106  df-ovol 24109  df-vol 24110 This theorem is referenced by:  icombl1  24208
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