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Theorem ioombl1 25542

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 13061	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		ioossre 13354	. . . . 5 ⊢ (𝐴(,)+∞) ⊆ ℝ
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ⊆ ℝ)
4		elpwi 4549	. . . . . 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 2737	. . . . . . . . . . . 12 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
9	8	ovolgelb 25460	. . . . . . . . . . 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 2737	. . . . . . . . . . 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 2737	. . . . . . . . . . 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 2737	. . . . . . . . . . 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 25454	. . . . . . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (1^st ‘(𝑓‘𝑛)) = (1^st ‘(𝑓‘𝑛))
24		eqid 2737	. . . . . . . . . . 11 ⊢ (2^nd ‘(𝑓‘𝑛)) = (2^nd ‘(𝑓‘𝑛))
25		2fveq3 6840	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑓‘𝑚)) = (1^st ‘(𝑓‘𝑛)))
26	25	breq1d 5096	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑓‘𝑚)) ≤ 𝐴 ↔ (1^st ‘(𝑓‘𝑛)) ≤ 𝐴))
27	26, 25	ifbieq2d 4494	. . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑓‘𝑚)) = (2^nd ‘(𝑓‘𝑛)))
29	27, 28	breq12d 5099	. . . . . . . . . . . . . 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 4496	. . . . . . . . . . . . 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 4825	. . . . . . . . . . . 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 5190	. . . . . . . . . . 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 4825	. . . . . . . . . . . 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 5190	. . . . . . . . . . 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 25541	. . . . . . . . . 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 3145	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ⁺) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol‘𝑥) + 𝑦))
37	36	ralrimiva 3130	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ∀𝑦 ∈ ℝ⁺ ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol‘𝑥) + 𝑦))
38		inss1 4178	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥
39		ovolsscl 25466	. . . . . . . . . . . 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 4077	. . . . . . . . . . . 12 ⊢ (𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥
43		ovolsscl 25466	. . . . . . . . . . . 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 11168	. . . . . . . . 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 13152	. . . . . . . . 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 3130	. . . 4 ⊢ (𝐴 ∈ ℝ → ∀𝑥 ∈ 𝒫 ℝ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
54		ismbl2 25507	. . . 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 7368	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) = (+∞(,)+∞))
57		iooid 13320	. . . . 5 ⊢ (+∞(,)+∞) = ∅
58	56, 57	eqtrdi 2788	. . . 4 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) = ∅)
59		0mbl 25519	. . . 4 ⊢ ∅ ∈ dom vol
60	58, 59	eqeltrdi 2845	. . 3 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) ∈ dom vol)
61		oveq1 7368	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) = (-∞(,)+∞))
62		ioomax 13369	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
63	61, 62	eqtrdi 2788	. . . 4 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) = ℝ)
64		rembl 25520	. . . 4 ⊢ ℝ ∈ dom vol
65	63, 64	eqeltrdi 2845	. . 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 3052 ∃wrex 3062 ∖ cdif 3887 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 ifcif 4467 𝒫 cpw 4542 ⟨cop 4574 ∪ cuni 4851 class class class wbr 5086 ↦ cmpt 5167 × cxp 5623 dom cdm 5625 ran crn 5626 ∘ ccom 5629 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 1^st c1st 7934 2^nd c2nd 7935 ↑_m cmap 8767 supcsup 9347 ℝcr 11031 1c1 11033 + caddc 11035 +∞cpnf 11170 -∞cmnf 11171 ℝ^*cxr 11172 < clt 11173 ≤ cle 11174 − cmin 11371 ℕcn 12168 ℝ⁺crp 12936 (,)cioo 13292 seqcseq 13957 abscabs 15190 vol*covol 25442 volcvol 25443

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 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-n0 12432 df-z 12519 df-uz 12783 df-q 12893 df-rp 12937 df-xadd 13058 df-ioo 13296 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-seq 13958 df-exp 14018 df-hash 14287 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-clim 15444 df-rlim 15445 df-sum 15643 df-xmet 21340 df-met 21341 df-ovol 25444 df-vol 25445

This theorem is referenced by: icombl1 25543

W3C validator