ovolfioo - Metamath Proof Explorer

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Theorem ovolfioo 25393

Description: Unpack the interval covering property of the outer measure definition. (Contributed by Mario Carneiro, 16-Mar-2014.)

**Assertion**
Ref	Expression
ovolfioo	⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))

Distinct variable groups: 𝑧,𝑛,𝐴 𝑛,𝐹,𝑧

**Proof of Theorem ovolfioo**
Step	Hyp	Ref	Expression
1		ioof 13344	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		inss2 4188	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		rexpssxrxp 11154	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
4	2, 3	sstri 3944	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
5		fss 6667	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝐹:ℕ⟶(ℝ^ × ℝ^*))
6	4, 5	mpan2 691	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
7		fco 6675	. . . . . 6 ⊢ (((,):(ℝ^* × ℝ^)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
8	1, 6, 7	sylancr 587	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
9		ffn 6651	. . . . 5 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
10		fniunfv 7181	. . . . 5 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ ran ((,) ∘ 𝐹))
11	8, 9, 10	3syl 18	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ ran ((,) ∘ 𝐹))
12	11	sseq2d 3967	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
13	12	adantl 481	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
14		dfss3 3923	. . 3 ⊢ (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
15		ssel2 3929	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
16		eliun 4945	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ (((,) ∘ 𝐹)‘𝑛))
17		rexr 11155	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ^*)
18	17	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℝ^*)
19		fvco3 6921	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
20		ffvelcdm 7014	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	20	elin2d 4155	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ × ℝ))
22		1st2nd2 7960	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
23	21, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
24	23	fveq2d 6826	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
25		df-ov 7349	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
26	24, 25	eqtr4di 2784	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
27	19, 26	eqtrd 2766	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
28	27	eleq2d 2817	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ 𝑧 ∈ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛)))))
29		ovolfcl 25392	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1^st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))))
30		rexr 11155	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝐹‘𝑛)) ∈ ℝ → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^*)
31		rexr 11155	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(𝐹‘𝑛)) ∈ ℝ → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*)
32		elioo1 13282	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(𝐹‘𝑛)) ∈ ℝ^* ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^) → (𝑧 ∈ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ^ ∧ (1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))
33	30, 31, 32	syl2an 596	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ) → (𝑧 ∈ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ^* ∧ (1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))
34		3anass 1094	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℝ^* ∧ (1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ^* ∧ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))
35	33, 34	bitrdi 287	. . . . . . . . . . . . 13 ⊢ (((1^st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ) → (𝑧 ∈ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ^* ∧ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛))))))
36	35	3adant3 1132	. . . . . . . . . . . 12 ⊢ (((1^st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → (𝑧 ∈ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ^* ∧ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛))))))
37	29, 36	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) ↔ (𝑧 ∈ ℝ^* ∧ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛))))))
38	28, 37	bitrd 279	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ^* ∧ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛))))))
39	38	adantll 714	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ^* ∧ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛))))))
40	18, 39	mpbirand 707	. . . . . . . 8 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))
41	40	rexbidva 3154	. . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∃𝑛 ∈ ℕ 𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))
42	16, 41	bitrid 283	. . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))
43	15, 42	sylan 580	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))
44	43	an32s 652	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))
45	44	ralbidva 3153	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))
46	14, 45	bitrid 283	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))
47	13, 46	bitr3d 281	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐹‘𝑛)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 ∩ cin 3901 ⊆ wss 3902 𝒫 cpw 4550 ⟨cop 4582 ∪ cuni 4859 ∪ ciun 4941 class class class wbr 5091 × cxp 5614 ran crn 5617 ∘ ccom 5620 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 ℝcr 11002 ℝ^*cxr 11142 < clt 11143 ≤ cle 11144 ℕcn 12122 (,)cioo 13242

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-pre-lttri 11077 ax-pre-lttrn 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-po 5524 df-so 5525 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-ioo 13246

This theorem is referenced by: ovollb2lem 25414 ovolunlem1 25423 ovoliunlem2 25429 ovolshftlem1 25435 ovolscalem1 25439 ioombl1lem4 25487

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