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Theorem ioombl1 25876

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 13238	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		ioossre 13531	. . . . 5 ⊢ (𝐴(,)+∞) ⊆ ℝ
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ⊆ ℝ)
4		elpwi 4564	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
5		simplrl 789	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ⁺) → 𝑥 ⊆ ℝ)
6		simplrr 790	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ⁺) → (vol‘𝑥) ∈ ℝ)
7		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
8		eqid 2761	. . . . . . . . . . . 12 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
9	8	ovolgelb 25794	. . . . . . . . . . 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 1398	. . . . . . . . . 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 2761	. . . . . . . . . . 11 ⊢ (𝐴(,)+∞) = (𝐴(,)+∞)
12		simplll 787	. . . . . . . . . . 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 486	. . . . . . . . . . 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 486	. . . . . . . . . . 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 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)))) → 𝑦 ∈ ℝ⁺)
16		eqid 2761	. . . . . . . . . . 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 2761	. . . . . . . . . . 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 783	. . . . . . . . . . . 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 25788	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
20	18, 19	sylib 221	. . . . . . . . . . 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 793	. . . . . . . . . . 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 794	. . . . . . . . . . 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 2761	. . . . . . . . . . 11 ⊢ (1^st ‘(𝑓‘𝑛)) = (1^st ‘(𝑓‘𝑛))
24		eqid 2761	. . . . . . . . . . 11 ⊢ (2^nd ‘(𝑓‘𝑛)) = (2^nd ‘(𝑓‘𝑛))
25		2fveq3 6888	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑓‘𝑚)) = (1^st ‘(𝑓‘𝑛)))
26	25	breq1d 5113	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑓‘𝑚)) ≤ 𝐴 ↔ (1^st ‘(𝑓‘𝑛)) ≤ 𝐴))
27	26, 25	ifbieq2d 4509	. . . . . . . . . . . . . . 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 6888	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑓‘𝑚)) = (2^nd ‘(𝑓‘𝑛)))
29	27, 28	breq12d 5116	. . . . . . . . . . . . . 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 4511	. . . . . . . . . . . . 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 4841	. . . . . . . . . . . 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 5209	. . . . . . . . . . 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 4841	. . . . . . . . . . . 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 5209	. . . . . . . . . . 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 25875	. . . . . . . . . 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 3170	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ⁺) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol‘𝑥) + 𝑦))
37	36	ralrimiva 3155	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ∀𝑦 ∈ ℝ⁺ ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol‘𝑥) + 𝑦))
38		inss1 4182	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥
39		ovolsscl 25800	. . . . . . . . . . . 12 ⊢ (((𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
40	38, 39	mp3an1 1477	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
41	40	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → (vol‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
42		difss 4083	. . . . . . . . . . . 12 ⊢ (𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥
43		ovolsscl 25800	. . . . . . . . . . . 12 ⊢ (((𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
44	42, 43	mp3an1 1477	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
45	44	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → (vol‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
46	41, 45	readdcld 11331	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ∈ ℝ)
47		simprr 785	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → (vol‘𝑥) ∈ ℝ)
48		alrple 13329	. . . . . . . . 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 596	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥))
51	50	expr 462	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ⊆ ℝ) → ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
52	4, 51	sylan2 605	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
53	52	ralrimiva 3155	. . . 4 ⊢ (𝐴 ∈ ℝ → ∀𝑥 ∈ 𝒫 ℝ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
54		ismbl2 25841	. . . 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 595	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ∈ dom vol)
56		oveq1 7425	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) = (+∞(,)+∞))
57		iooid 13497	. . . . 5 ⊢ (+∞(,)+∞) = ∅
58	56, 57	eqtrdi 2812	. . . 4 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) = ∅)
59		0mbl 25853	. . . 4 ⊢ ∅ ∈ dom vol
60	58, 59	eqeltrdi 2869	. . 3 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) ∈ dom vol)
61		oveq1 7425	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) = (-∞(,)+∞))
62		ioomax 13546	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
63	61, 62	eqtrdi 2812	. . . 4 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) = ℝ)
64		rembl 25854	. . . 4 ⊢ ℝ ∈ dom vol
65	63, 64	eqeltrdi 2869	. . 3 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) ∈ dom vol)
66	55, 60, 65	3jaoi 1454	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴(,)+∞) ∈ dom vol)
67	1, 66	sylbi 220	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,)+∞) ∈ dom vol)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ∃wrex 3087 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 ifcif 4482 𝒫 cpw 4557 ⟨cop 4590 ∪ cuni 4867 class class class wbr 5103 ↦ cmpt 5186 × cxp 5649 dom cdm 5651 ran crn 5652 ∘ ccom 5655 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 1^st c1st 7997 2^nd c2nd 7998 ↑_m cmap 8840 supcsup 9425 ℝcr 11192 1c1 11194 + caddc 11196 +∞cpnf 11333 -∞cmnf 11334 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 − cmin 11534 ℕcn 12328 ℝ⁺crp 13113 (,)cioo 13469 seqcseq 14137 abscabs 15394 vol*covol 25776 volcvol 25777

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-pm 8843 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9427 df-inf 9428 df-oi 9497 df-dju 9975 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-n0 12600 df-z 12687 df-uz 12959 df-q 13069 df-rp 13114 df-xadd 13235 df-ioo 13473 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-fl 13925 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-rlim 15649 df-sum 15847 df-xmet 21664 df-met 21665 df-ovol 25778 df-vol 25779

This theorem is used by: icombl1 25877

W3C validator