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Theorem ioombl1 25611
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 13156 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 ioossre 13445 . . . . 5 (𝐴(,)+∞) ⊆ ℝ
32a1i 11 . . . 4 (𝐴 ∈ ℝ → (𝐴(,)+∞) ⊆ ℝ)
4 elpwi 4612 . . . . . 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 2735 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
98ovolgelb 25529 . . . . . . . . . . 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 1370 . . . . . . . . . 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 2735 . . . . . . . . . . 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 2735 . . . . . . . . . . 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 2735 . . . . . . . . . . 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 25523 . . . . . . . . . . . 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 2735 . . . . . . . . . . 11 (1st ‘(𝑓𝑛)) = (1st ‘(𝑓𝑛))
24 eqid 2735 . . . . . . . . . . 11 (2nd ‘(𝑓𝑛)) = (2nd ‘(𝑓𝑛))
25 2fveq3 6912 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1st ‘(𝑓𝑚)) = (1st ‘(𝑓𝑛)))
2625breq1d 5158 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((1st ‘(𝑓𝑚)) ≤ 𝐴 ↔ (1st ‘(𝑓𝑛)) ≤ 𝐴))
2726, 25ifbieq2d 4557 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) = if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))))
28 2fveq3 6912 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (2nd ‘(𝑓𝑚)) = (2nd ‘(𝑓𝑛)))
2927, 28breq12d 5161 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)) ↔ if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))) ≤ (2nd ‘(𝑓𝑛))))
3029, 27, 28ifbieq12d 4559 . . . . . . . . . . . . 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 4886 . . . . . . . . . . . 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 5261 . . . . . . . . . . 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 4886 . . . . . . . . . . . 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 5261 . . . . . . . . . . 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 25610 . . . . . . . . . 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 3159 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
3736ralrimiva 3144 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ∀𝑦 ∈ ℝ+ ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
38 inss1 4245 . . . . . . . . . . . 12 (𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥
39 ovolsscl 25535 . . . . . . . . . . . 12 (((𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
4038, 39mp3an1 1447 . . . . . . . . . . 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 4146 . . . . . . . . . . . 12 (𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥
43 ovolsscl 25535 . . . . . . . . . . . 12 (((𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4442, 43mp3an1 1447 . . . . . . . . . . 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 11288 . . . . . . . . 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 13245 . . . . . . . . 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 3144 . . . 4 (𝐴 ∈ ℝ → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
54 ismbl2 25576 . . . 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 13412 . . . . 5 (+∞(,)+∞) = ∅
5856, 57eqtrdi 2791 . . . 4 (𝐴 = +∞ → (𝐴(,)+∞) = ∅)
59 0mbl 25588 . . . 4 ∅ ∈ dom vol
6058, 59eqeltrdi 2847 . . 3 (𝐴 = +∞ → (𝐴(,)+∞) ∈ dom vol)
61 oveq1 7438 . . . . 5 (𝐴 = -∞ → (𝐴(,)+∞) = (-∞(,)+∞))
62 ioomax 13459 . . . . 5 (-∞(,)+∞) = ℝ
6361, 62eqtrdi 2791 . . . 4 (𝐴 = -∞ → (𝐴(,)+∞) = ℝ)
64 rembl 25589 . . . 4 ℝ ∈ dom vol
6563, 64eqeltrdi 2847 . . 3 (𝐴 = -∞ → (𝐴(,)+∞) ∈ dom vol)
6655, 60, 653jaoi 1427 . 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 1085   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cdif 3960  cin 3962  wss 3963  c0 4339  ifcif 4531  𝒫 cpw 4605  cop 4637   cuni 4912   class class class wbr 5148  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  m cmap 8865  supcsup 9478  cr 11152  1c1 11154   + caddc 11156  +∞cpnf 11290  -∞cmnf 11291  *cxr 11292   < clt 11293  cle 11294  cmin 11490  cn 12264  +crp 13032  (,)cioo 13384  seqcseq 14039  abscabs 15270  vol*covol 25511  volcvol 25512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xadd 13153  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-xmet 21375  df-met 21376  df-ovol 25513  df-vol 25514
This theorem is referenced by:  icombl1  25612
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