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Theorem ioombl1 23376
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 11988 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 ioossre 12273 . . . . 5 (𝐴(,)+∞) ⊆ ℝ
32a1i 11 . . . 4 (𝐴 ∈ ℝ → (𝐴(,)+∞) ⊆ ℝ)
4 elpwi 4201 . . . . . 6 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
5 simplrl 817 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → 𝑥 ⊆ ℝ)
6 simplrr 818 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → (vol*‘𝑥) ∈ ℝ)
7 simpr 476 . . . . . . . . . . 11 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
8 eqid 2651 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
98ovolgelb 23294 . . . . . . . . . . 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 1366 . . . . . . . . . 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 2651 . . . . . . . . . . 11 (𝐴(,)+∞) = (𝐴(,)+∞)
12 simplll 813 . . . . . . . . . . 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 807 . . . . . . . . . . 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 2651 . . . . . . . . . . 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 2651 . . . . . . . . . . 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 809 . . . . . . . . . . . 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 reex 10065 . . . . . . . . . . . . . . 15 ℝ ∈ V
2019, 19xpex 7004 . . . . . . . . . . . . . 14 (ℝ × ℝ) ∈ V
2120inex2 4833 . . . . . . . . . . . . 13 ( ≤ ∩ (ℝ × ℝ)) ∈ V
22 nnex 11064 . . . . . . . . . . . . 13 ℕ ∈ V
2321, 22elmap 7928 . . . . . . . . . . . 12 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2418, 23sylib 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)))
25 simprrl 821 . . . . . . . . . . 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))
26 simprrr 822 . . . . . . . . . . 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))
27 eqid 2651 . . . . . . . . . . 11 (1st ‘(𝑓𝑛)) = (1st ‘(𝑓𝑛))
28 eqid 2651 . . . . . . . . . . 11 (2nd ‘(𝑓𝑛)) = (2nd ‘(𝑓𝑛))
29 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → (𝑓𝑚) = (𝑓𝑛))
3029fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1st ‘(𝑓𝑚)) = (1st ‘(𝑓𝑛)))
3130breq1d 4695 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((1st ‘(𝑓𝑚)) ≤ 𝐴 ↔ (1st ‘(𝑓𝑛)) ≤ 𝐴))
3231, 30ifbieq2d 4144 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) = if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))))
3329fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (2nd ‘(𝑓𝑚)) = (2nd ‘(𝑓𝑛)))
3432, 33breq12d 4698 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (if((1st ‘(𝑓𝑚)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑚))) ≤ (2nd ‘(𝑓𝑚)) ↔ if((1st ‘(𝑓𝑛)) ≤ 𝐴, 𝐴, (1st ‘(𝑓𝑛))) ≤ (2nd ‘(𝑓𝑛))))
3534, 32, 33ifbieq12d 4146 . . . . . . . . . . . . 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))))
3635, 33opeq12d 4441 . . . . . . . . . . . 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))⟩)
3736cbvmptv 4783 . . . . . . . . . . 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))⟩)
3830, 35opeq12d 4441 . . . . . . . . . . . 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)))⟩)
3938cbvmptv 4783 . . . . . . . . . . 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)))⟩)
4011, 12, 13, 14, 15, 8, 16, 17, 24, 25, 26, 27, 28, 37, 39ioombl1lem4 23375 . . . . . . . . . 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))
4110, 40rexlimddv 3064 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ+) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
4241ralrimiva 2995 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ∀𝑦 ∈ ℝ+ ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol*‘𝑥) + 𝑦))
43 inss1 3866 . . . . . . . . . . . 12 (𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥
44 ovolsscl 23300 . . . . . . . . . . . 12 (((𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
4543, 44mp3an1 1451 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
4645adantl 481 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
47 difss 3770 . . . . . . . . . . . 12 (𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥
48 ovolsscl 23300 . . . . . . . . . . . 12 (((𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
4947, 48mp3an1 1451 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
5049adantl 481 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
5146, 50readdcld 10107 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ∈ ℝ)
52 simprr 811 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘𝑥) ∈ ℝ)
53 alrple 12075 . . . . . . . . 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)))
5451, 52, 53syl2anc 694 . . . . . . . 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)))
5542, 54mpbird 247 . . . . . . 7 ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥))
5655expr 642 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
574, 56sylan2 490 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
5857ralrimiva 2995 . . . 4 (𝐴 ∈ ℝ → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥)))
59 ismbl2 23341 . . . 4 ((𝐴(,)+∞) ∈ dom vol ↔ ((𝐴(,)+∞) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol*‘𝑥))))
603, 58, 59sylanbrc 699 . . 3 (𝐴 ∈ ℝ → (𝐴(,)+∞) ∈ dom vol)
61 oveq1 6697 . . . . 5 (𝐴 = +∞ → (𝐴(,)+∞) = (+∞(,)+∞))
62 iooid 12241 . . . . 5 (+∞(,)+∞) = ∅
6361, 62syl6eq 2701 . . . 4 (𝐴 = +∞ → (𝐴(,)+∞) = ∅)
64 0mbl 23353 . . . 4 ∅ ∈ dom vol
6563, 64syl6eqel 2738 . . 3 (𝐴 = +∞ → (𝐴(,)+∞) ∈ dom vol)
66 oveq1 6697 . . . . 5 (𝐴 = -∞ → (𝐴(,)+∞) = (-∞(,)+∞))
67 ioomax 12286 . . . . 5 (-∞(,)+∞) = ℝ
6866, 67syl6eq 2701 . . . 4 (𝐴 = -∞ → (𝐴(,)+∞) = ℝ)
69 rembl 23354 . . . 4 ℝ ∈ dom vol
7068, 69syl6eqel 2738 . . 3 (𝐴 = -∞ → (𝐴(,)+∞) ∈ dom vol)
7160, 65, 703jaoi 1431 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴(,)+∞) ∈ dom vol)
721, 71sylbi 207 1 (𝐴 ∈ ℝ* → (𝐴(,)+∞) ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3o 1053   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cdif 3604  cin 3606  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191  cop 4216   cuni 4468   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  supcsup 8387  cr 9973  1c1 9975   + caddc 9977  +∞cpnf 10109  -∞cmnf 10110  *cxr 10111   < clt 10112  cle 10113  cmin 10304  cn 11058  +crp 11870  (,)cioo 12213  seqcseq 12841  abscabs 14018  vol*covol 23277  volcvol 23278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279  df-vol 23280
This theorem is referenced by:  icombl1  23377
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