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Theorem ioombl1 23550
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 12155 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 ioossre 12440 . . . . 5 (𝐴(,)+∞) ⊆ ℝ
32a1i 11 . . . 4 (𝐴 ∈ ℝ → (𝐴(,)+∞) ⊆ ℝ)
4 elpwi 4307 . . . . . 6 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
5 simplrl 762 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → 𝑥 ⊆ ℝ)
6 simplrr 763 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → (vol*‘𝑥) ∈ ℝ)
7 simpr 471 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
8 eqid 2771 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
98ovolgelb 23468 . . . . . . . . . . 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 1476 . . . . . . . . . 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 2771 . . . . . . . . . . 11 (𝐴(,)+∞) = (𝐴(,)+∞)
12 simplll 758 . . . . . . . . . . 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 466 . . . . . . . . . . 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 466 . . . . . . . . . . 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 752 . . . . . . . . . . 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 2771 . . . . . . . . . . 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 2771 . . . . . . . . . . 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 754 . . . . . . . . . . . 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 23462 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2018, 19sylib 208 . . . . . . . . . . 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 766 . . . . . . . . . . 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 767 . . . . . . . . . . 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 2771 . . . . . . . . . . 11 (1st ‘(𝑓𝑛)) = (1st ‘(𝑓𝑛))
24 eqid 2771 . . . . . . . . . . 11 (2nd ‘(𝑓𝑛)) = (2nd ‘(𝑓𝑛))
25 fveq2 6332 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → (𝑓𝑚) = (𝑓𝑛))
2625fveq2d 6336 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1st ‘(𝑓𝑚)) = (1st ‘(𝑓𝑛)))
2726breq1d 4796 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((1st ‘(𝑓𝑚)) ≤ 𝐴 ↔ (1st ‘(𝑓𝑛)) ≤ 𝐴))
2827, 26ifbieq2d 4250 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) = if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))))
2925fveq2d 6336 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (2nd ‘(𝑓𝑚)) = (2nd ‘(𝑓𝑛)))
3028, 29breq12d 4799 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)) ↔ if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))) ≤ (2nd ‘(𝑓𝑛))))
3130, 28, 29ifbieq12d 4252 . . . . . . . . . . . . 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))))
3231, 29opeq12d 4547 . . . . . . . . . . . 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))⟩)
3332cbvmptv 4884 . . . . . . . . . . 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))⟩)
3426, 31opeq12d 4547 . . . . . . . . . . . 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)))⟩)
3534cbvmptv 4884 . . . . . . . . . . 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)))⟩)
3611, 12, 13, 14, 15, 8, 16, 17, 20, 21, 22, 23, 24, 33, 35ioombl1lem4 23549 . . . . . . . . . 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))
3710, 36rexlimddv 3183 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
3837ralrimiva 3115 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ∀𝑦 ∈ ℝ+ ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
39 inss1 3981 . . . . . . . . . . . 12 (𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥
40 ovolsscl 23474 . . . . . . . . . . . 12 (((𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
4139, 40mp3an1 1559 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
4241adantl 467 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
43 difss 3888 . . . . . . . . . . . 12 (𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥
44 ovolsscl 23474 . . . . . . . . . . . 12 (((𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4543, 44mp3an1 1559 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4645adantl 467 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4742, 46readdcld 10271 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ∈ ℝ)
48 simprr 756 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
49 alrple 12242 . . . . . . . . 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)))
5047, 48, 49syl2anc 573 . . . . . . . 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)))
5138, 50mpbird 247 . . . . . . 7 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥))
5251expr 444 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
534, 52sylan2 580 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
5453ralrimiva 3115 . . . 4 (𝐴 ∈ ℝ → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
55 ismbl2 23515 . . . 4 ((𝐴(,)+∞) ∈ dom vol ↔ ((𝐴(,)+∞) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥))))
563, 54, 55sylanbrc 572 . . 3 (𝐴 ∈ ℝ → (𝐴(,)+∞) ∈ dom vol)
57 oveq1 6800 . . . . 5 (𝐴 = +∞ → (𝐴(,)+∞) = (+∞(,)+∞))
58 iooid 12408 . . . . 5 (+∞(,)+∞) = ∅
5957, 58syl6eq 2821 . . . 4 (𝐴 = +∞ → (𝐴(,)+∞) = ∅)
60 0mbl 23527 . . . 4 ∅ ∈ dom vol
6159, 60syl6eqel 2858 . . 3 (𝐴 = +∞ → (𝐴(,)+∞) ∈ dom vol)
62 oveq1 6800 . . . . 5 (𝐴 = -∞ → (𝐴(,)+∞) = (-∞(,)+∞))
63 ioomax 12453 . . . . 5 (-∞(,)+∞) = ℝ
6462, 63syl6eq 2821 . . . 4 (𝐴 = -∞ → (𝐴(,)+∞) = ℝ)
65 rembl 23528 . . . 4 ℝ ∈ dom vol
6664, 65syl6eqel 2858 . . 3 (𝐴 = -∞ → (𝐴(,)+∞) ∈ dom vol)
6756, 61, 663jaoi 1539 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴(,)+∞) ∈ dom vol)
681, 67sylbi 207 1 (𝐴 ∈ ℝ* → (𝐴(,)+∞) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3o 1070   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cdif 3720  cin 3722  wss 3723  c0 4063  ifcif 4225  𝒫 cpw 4297  cop 4322   cuni 4574   class class class wbr 4786  cmpt 4863   × cxp 5247  dom cdm 5249  ran crn 5250  ccom 5253  wf 6027  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  𝑚 cmap 8009  supcsup 8502  cr 10137  1c1 10139   + caddc 10141  +∞cpnf 10273  -∞cmnf 10274  *cxr 10275   < clt 10276  cle 10277  cmin 10468  cn 11222  +crp 12035  (,)cioo 12380  seqcseq 13008  abscabs 14182  vol*covol 23450  volcvol 23451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xadd 12152  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-rlim 14428  df-sum 14625  df-xmet 19954  df-met 19955  df-ovol 23452  df-vol 23453
This theorem is referenced by:  icombl1  23551
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