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Theorem ioombl1 23137

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 11826	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		ioossre 12106	. . . . 5 ⊢ (𝐴(,)+∞) ⊆ ℝ
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ⊆ ℝ)
4		elpwi 4117	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
5		simplrl 796	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ⁺) → 𝑥 ⊆ ℝ)
6		simplrr 797	. . . . . . . . . . 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 2610	. . . . . . . . . . . 12 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
9	8	ovolgelb 23055	. . . . . . . . . . 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 1318	. . . . . . . . . 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 2610	. . . . . . . . . . 11 ⊢ (𝐴(,)+∞) = (𝐴(,)+∞)
12		simplll 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)))) → 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 788	. . . . . . . . . . 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 2610	. . . . . . . . . . 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 2610	. . . . . . . . . . 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 790	. . . . . . . . . . . 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 9906	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
20	19, 19	xpex 6860	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) ∈ V
21	20	inex2 4728	. . . . . . . . . . . . 13 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
22		nnex 10903	. . . . . . . . . . . . 13 ⊢ ℕ ∈ V
23	21, 22	elmap 7772	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24	18, 23	sylib 207	. . . . . . . . . . 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 800	. . . . . . . . . . 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 801	. . . . . . . . . . 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 2610	. . . . . . . . . . 11 ⊢ (1^st ‘(𝑓‘𝑛)) = (1^st ‘(𝑓‘𝑛))
28		eqid 2610	. . . . . . . . . . 11 ⊢ (2^nd ‘(𝑓‘𝑛)) = (2^nd ‘(𝑓‘𝑛))
29		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (𝑓‘𝑚) = (𝑓‘𝑛))
30	29	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝑓‘𝑚)) = (1^st ‘(𝑓‘𝑛)))
31	30	breq1d 4593	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝑓‘𝑚)) ≤ 𝐴 ↔ (1^st ‘(𝑓‘𝑛)) ≤ 𝐴))
32	31, 30	ifbieq2d 4061	. . . . . . . . . . . . . . 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))))
33	29	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝑓‘𝑚)) = (2^nd ‘(𝑓‘𝑛)))
34	32, 33	breq12d 4596	. . . . . . . . . . . . . 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))))
35	34, 32, 33	ifbieq12d 4063	. . . . . . . . . . . . 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))))
36	35, 33	opeq12d 4348	. . . . . . . . . . . 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))⟩)
37	36	cbvmptv 4678	. . . . . . . . . . 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))⟩)
38	30, 35	opeq12d 4348	. . . . . . . . . . . 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)))⟩)
39	38	cbvmptv 4678	. . . . . . . . . . 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)))⟩)
40	11, 12, 13, 14, 15, 8, 16, 17, 24, 25, 26, 27, 28, 37, 39	ioombl1lem4 23136	. . . . . . . . . 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))
41	10, 40	rexlimddv 3017	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) ∧ 𝑦 ∈ ℝ⁺) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol‘𝑥) + 𝑦))
42	41	ralrimiva 2949	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ∀𝑦 ∈ ℝ⁺ ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ ((vol‘𝑥) + 𝑦))
43		inss1 3795	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥
44		ovolsscl 23061	. . . . . . . . . . . 12 ⊢ (((𝑥 ∩ (𝐴(,)+∞)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
45	43, 44	mp3an1 1403	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
46	45	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → (vol‘(𝑥 ∩ (𝐴(,)+∞))) ∈ ℝ)
47		difss 3699	. . . . . . . . . . . 12 ⊢ (𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥
48		ovolsscl 23061	. . . . . . . . . . . 12 ⊢ (((𝑥 ∖ (𝐴(,)+∞)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
49	47, 48	mp3an1 1403	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
50	49	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → (vol‘(𝑥 ∖ (𝐴(,)+∞))) ∈ ℝ)
51	46, 50	readdcld 9948	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol*‘(𝑥 ∖ (𝐴(,)+∞)))) ∈ ℝ)
52		simprr 792	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → (vol‘𝑥) ∈ ℝ)
53		alrple 11911	. . . . . . . . 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)))
54	51, 52, 53	syl2anc 691	. . . . . . . 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)))
55	42, 54	mpbird 246	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ)) → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥))
56	55	expr 641	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ⊆ ℝ) → ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
57	4, 56	sylan2 490	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
58	57	ralrimiva 2949	. . . 4 ⊢ (𝐴 ∈ ℝ → ∀𝑥 ∈ 𝒫 ℝ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥)))
59		ismbl2 23102	. . . 4 ⊢ ((𝐴(,)+∞) ∈ dom vol ↔ ((𝐴(,)+∞) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol‘𝑥) ∈ ℝ → ((vol‘(𝑥 ∩ (𝐴(,)+∞))) + (vol‘(𝑥 ∖ (𝐴(,)+∞)))) ≤ (vol‘𝑥))))
60	3, 58, 59	sylanbrc 695	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ∈ dom vol)
61		oveq1 6556	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) = (+∞(,)+∞))
62		iooid 12074	. . . . 5 ⊢ (+∞(,)+∞) = ∅
63	61, 62	syl6eq 2660	. . . 4 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) = ∅)
64		0mbl 23114	. . . 4 ⊢ ∅ ∈ dom vol
65	63, 64	syl6eqel 2696	. . 3 ⊢ (𝐴 = +∞ → (𝐴(,)+∞) ∈ dom vol)
66		oveq1 6556	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) = (-∞(,)+∞))
67		ioomax 12119	. . . . 5 ⊢ (-∞(,)+∞) = ℝ
68	66, 67	syl6eq 2660	. . . 4 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) = ℝ)
69		rembl 23115	. . . 4 ⊢ ℝ ∈ dom vol
70	68, 69	syl6eqel 2696	. . 3 ⊢ (𝐴 = -∞ → (𝐴(,)+∞) ∈ dom vol)
71	60, 65, 70	3jaoi 1383	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴(,)+∞) ∈ dom vol)
72	1, 71	sylbi 206	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴(,)+∞) ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 ⟨cop 4131 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ran crn 5039 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 ↑_𝑚 cmap 7744 supcsup 8229 ℝcr 9814 1c1 9816 + caddc 9818 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℝ⁺crp 11708 (,)cioo 12046 seqcseq 12663 abscabs 13822 vol*covol 23038 volcvol 23039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041

This theorem is referenced by: icombl1 23138

W3C validator