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Theorem ioombl1 25529

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 13067	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		ioossre 13360	. . . . 5 ⊢ (𝐴(,)+∞) ⊆ ℝ
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ⊆ ℝ)
4		elpwi 4548	. . . . . 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 2736	. . . . . . . . . . . 12 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
9	8	ovolgelb 25447	. . . . . . . . . . 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 1374	. . . . . . . . . 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 2736	. . . . . . . . . . 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)
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 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 2736	. . . . . . . . . . 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 2736	. . . . . . . . . . 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 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 25441	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
20	18, 19	sylib 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 2736	. . . . . . . . . . 11 ⊢ (1^st ‘(𝑓‘𝑛)) = (1^st ‘(𝑓‘𝑛))
24		eqid 2736	. . . . . . . . . . 11 ⊢ (2^nd ‘(𝑓‘𝑛)) = (2^nd ‘(𝑓‘𝑛))
25		2fveq3 6845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑓‘𝑚)) = (1^st ‘(𝑓‘𝑛)))
26	25	breq1d 5095	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑓‘𝑚)) ≤ 𝐴 ↔ (1^st ‘(𝑓‘𝑛)) ≤ 𝐴))
27	26, 25	ifbieq2d 4493	. . . . . . . . . . . . . . 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 6845	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑓‘𝑚)) = (2^nd ‘(𝑓‘𝑛)))
29	27, 28	breq12d 5098	. . . . . . . . . . . . . 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 4495	. . . . . . . . . . . . 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 4824	. . . . . . . . . . . 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 5189	. . . . . . . . . . 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 4824	. . . . . . . . . . . 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 5189	. . . . . . . . . . 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 25528	. . . . . . . . . 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 3144	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ⁺) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol‘𝑥) + 𝑦))
37	36	ralrimiva 3129	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ∀𝑦 ∈ ℝ⁺ ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol‘𝑥) + 𝑦))
38		inss1 4177	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥
39		ovolsscl 25453	. . . . . . . . . . . 12 ⊢ (((𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
40	38, 39	mp3an1 1451	. . . . . . . . . . 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 4076	. . . . . . . . . . . 12 ⊢ (𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥
43		ovolsscl 25453	. . . . . . . . . . . 12 ⊢ (((𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
44	42, 43	mp3an1 1451	. . . . . . . . . . 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 11174	. . . . . . . . 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 13158	. . . . . . . . 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 585	. . . . . . . 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 594	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
53	52	ralrimiva 3129	. . . 4 ⊢ (𝐴 ∈ ℝ → ∀𝑥 ∈ 𝒫 ℝ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
54		ismbl2 25494	. . . 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 584	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ∈ dom vol)
56		oveq1 7374	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) = (+∞(,)+∞))
57		iooid 13326	. . . . 5 ⊢ (+∞(,)+∞) = ∅
58	56, 57	eqtrdi 2787	. . . 4 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) = ∅)
59		0mbl 25506	. . . 4 ⊢ ∅ ∈ dom vol
60	58, 59	eqeltrdi 2844	. . 3 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) ∈ dom vol)
61		oveq1 7374	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) = (-∞(,)+∞))
62		ioomax 13375	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
63	61, 62	eqtrdi 2787	. . . 4 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) = ℝ)
64		rembl 25507	. . . 4 ⊢ ℝ ∈ dom vol
65	63, 64	eqeltrdi 2844	. . 3 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) ∈ dom vol)
66	55, 60, 65	3jaoi 1431	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴(,)+∞) ∈ dom vol)
67	1, 66	sylbi 217	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,)+∞) ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1086 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 ∖ cdif 3886 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 ifcif 4466 𝒫 cpw 4541 ⟨cop 4573 ∪ cuni 4850 class class class wbr 5085 ↦ cmpt 5166 × cxp 5629 dom cdm 5631 ran crn 5632 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 ↑_m cmap 8773 supcsup 9353 ℝcr 11037 1c1 11039 + caddc 11041 +∞cpnf 11176 -∞cmnf 11177 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 − cmin 11377 ℕcn 12174 ℝ⁺crp 12942 (,)cioo 13298 seqcseq 13963 abscabs 15196 vol*covol 25429 volcvol 25430

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xadd 13064 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-xmet 21345 df-met 21346 df-ovol 25431 df-vol 25432

This theorem is referenced by: icombl1 25530

W3C validator