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Theorem ioombl1 25412

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.

Step	Hyp	Ref	Expression
1		elxr 13103	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		ioossre 13392	. . . . 5 ⊢ (𝐴(,)+∞) ⊆ ℝ
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ⊆ ℝ)
4		elpwi 4609	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
5		simplrl 774	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ⁺) → 𝑥 ⊆ ℝ)
6		simplrr 775	. . . . . . . . . . 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 2731	. . . . . . . . . . . 12 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
9	8	ovolgelb 25330	. . . . . . . . . . 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)))
10	5, 6, 7, 9	syl3anc 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 2731	. . . . . . . . . . 11 ⊢ (𝐴(,)+∞) = (𝐴(,)+∞)
12		simplll 772	. . . . . . . . . . 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)
13	5	adantr 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)
14	6	adantr 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 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)))) → 𝑦 ∈ ℝ⁺)
16		eqid 2731	. . . . . . . . . . 11 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨if(if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) ≤ (2^nd ‘(𝑓‘𝑚)), if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚))⟩))) = seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨if(if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) ≤ (2^nd ‘(𝑓‘𝑚)), if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚))⟩)))
17		eqid 2731	. . . . . . . . . . 11 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨(1^st ‘(𝑓‘𝑚)), if(if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) ≤ (2^nd ‘(𝑓‘𝑚)), if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚)))⟩))) = seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨(1^st ‘(𝑓‘𝑚)), if(if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) ≤ (2^nd ‘(𝑓‘𝑚)), if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚)))⟩)))
18		simprl 768	. . . . . . . . . . . 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 25324	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
20	18, 19	sylib 217	. . . . . . . . . . 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 778	. . . . . . . . . . 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 779	. . . . . . . . . . 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 2731	. . . . . . . . . . 11 ⊢ (1^st ‘(𝑓‘𝑛)) = (1^st ‘(𝑓‘𝑛))
24		eqid 2731	. . . . . . . . . . 11 ⊢ (2^nd ‘(𝑓‘𝑛)) = (2^nd ‘(𝑓‘𝑛))
25		2fveq3 6896	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑓‘𝑚)) = (1^st ‘(𝑓‘𝑛)))
26	25	breq1d 5158	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑓‘𝑚)) ≤ 𝐴 ↔ (1^st ‘(𝑓‘𝑛)) ≤ 𝐴))
27	26, 25	ifbieq2d 4554	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) = if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))))
28		2fveq3 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑓‘𝑚)) = (2^nd ‘(𝑓‘𝑛)))
29	27, 28	breq12d 5161	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) ≤ (2^nd ‘(𝑓‘𝑚)) ↔ if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))) ≤ (2^nd ‘(𝑓‘𝑛))))
30	29, 27, 28	ifbieq12d 4556	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → if(if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) ≤ (2^nd ‘(𝑓‘𝑚)), if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚))) = if(if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))) ≤ (2^nd ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))), (2^nd ‘(𝑓‘𝑛))))
31	30, 28	opeq12d 4881	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ⟨if(if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) ≤ (2^nd ‘(𝑓‘𝑚)), if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚))⟩ = ⟨if(if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))) ≤ (2^nd ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))), (2^nd ‘(𝑓‘𝑛))), (2^nd ‘(𝑓‘𝑛))⟩)
32	31	cbvmptv 5261	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ ↦ ⟨if(if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) ≤ (2^nd ‘(𝑓‘𝑚)), if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚))⟩) = (𝑛 ∈ ℕ ↦ ⟨if(if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))) ≤ (2^nd ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))), (2^nd ‘(𝑓‘𝑛))), (2^nd ‘(𝑓‘𝑛))⟩)
33	25, 30	opeq12d 4881	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ⟨(1^st ‘(𝑓‘𝑚)), if(if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) ≤ (2^nd ‘(𝑓‘𝑚)), if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚)))⟩ = ⟨(1^st ‘(𝑓‘𝑛)), if(if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))) ≤ (2^nd ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))), (2^nd ‘(𝑓‘𝑛)))⟩)
34	33	cbvmptv 5261	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ ↦ ⟨(1^st ‘(𝑓‘𝑚)), if(if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))) ≤ (2^nd ‘(𝑓‘𝑚)), if((1^st ‘(𝑓‘𝑚)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑚))), (2^nd ‘(𝑓‘𝑚)))⟩) = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝑓‘𝑛)), if(if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))) ≤ (2^nd ‘(𝑓‘𝑛)), if((1^st ‘(𝑓‘𝑛)) ≤ 𝐴, 𝐴, (1^st ‘(𝑓‘𝑛))), (2^nd ‘(𝑓‘𝑛)))⟩)
35	11, 12, 13, 14, 15, 8, 16, 17, 20, 21, 22, 23, 24, 32, 34	ioombl1lem4 25411	. . . . . . . . . 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))
36	10, 35	rexlimddv 3160	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ⁺) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol‘𝑥) + 𝑦))
37	36	ralrimiva 3145	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ∀𝑦 ∈ ℝ⁺ ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol‘𝑥) + 𝑦))
38		inss1 4228	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥
39		ovolsscl 25336	. . . . . . . . . . . 12 ⊢ (((𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
40	38, 39	mp3an1 1447	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
41	40	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → (vol‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
42		difss 4131	. . . . . . . . . . . 12 ⊢ (𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥
43		ovolsscl 25336	. . . . . . . . . . . 12 ⊢ (((𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
44	42, 43	mp3an1 1447	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
45	44	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → (vol‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
46	41, 45	readdcld 11250	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ∈ ℝ)
47		simprr 770	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → (vol‘𝑥) ∈ ℝ)
48		alrple 13192	. . . . . . . . 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)))
49	46, 47, 48	syl2anc 583	. . . . . . . 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)))
50	37, 49	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥))
51	50	expr 456	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ⊆ ℝ) → ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
52	4, 51	sylan2 592	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
53	52	ralrimiva 3145	. . . 4 ⊢ (𝐴 ∈ ℝ → ∀𝑥 ∈ 𝒫 ℝ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
54		ismbl2 25377	. . . 4 ⊢ ((𝐴(,)+∞) ∈ dom vol ↔ ((𝐴(,)+∞) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥))))
55	3, 53, 54	sylanbrc 582	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ∈ dom vol)
56		oveq1 7419	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) = (+∞(,)+∞))
57		iooid 13359	. . . . 5 ⊢ (+∞(,)+∞) = ∅
58	56, 57	eqtrdi 2787	. . . 4 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) = ∅)
59		0mbl 25389	. . . 4 ⊢ ∅ ∈ dom vol
60	58, 59	eqeltrdi 2840	. . 3 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) ∈ dom vol)
61		oveq1 7419	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) = (-∞(,)+∞))
62		ioomax 13406	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
63	61, 62	eqtrdi 2787	. . . 4 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) = ℝ)
64		rembl 25390	. . . 4 ⊢ ℝ ∈ dom vol
65	63, 64	eqeltrdi 2840	. . 3 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) ∈ dom vol)
66	55, 60, 65	3jaoi 1426	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴(,)+∞) ∈ dom vol)
67	1, 66	sylbi 216	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,)+∞) ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ w3o 1085 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ifcif 4528 𝒫 cpw 4602 ⟨cop 4634 ∪ cuni 4908 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 dom cdm 5676 ran crn 5677 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 1^st c1st 7977 2^nd c2nd 7978 ↑_m cmap 8826 supcsup 9441 ℝcr 11115 1c1 11117 + caddc 11119 +∞cpnf 11252 -∞cmnf 11253 ℝ^*cxr 11254 < clt 11255 ≤ cle 11256 − cmin 11451 ℕcn 12219 ℝ⁺crp 12981 (,)cioo 13331 seqcseq 13973 abscabs 15188 vol*covol 25312 volcvol 25313

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-xadd 13100 df-ioo 13335 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-sum 15640 df-xmet 21227 df-met 21228 df-ovol 25314 df-vol 25315

This theorem is referenced by: icombl1 25413

W3C validator