ovolfioo - Metamath Proof Explorer

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Theorem ovolfioo 25459

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 13398	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		inss2 4173	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		rexpssxrxp 11188	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
4	2, 3	sstri 3931	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
5		fss 6678	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝐹:ℕ⟶(ℝ^ × ℝ^*))
6	4, 5	mpan2 697	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
7		fco 6686	. . . . . 6 ⊢ (((,):(ℝ^* × ℝ^)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
8	1, 6, 7	sylancr 593	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
9		ffn 6662	. . . . 5 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
10		fniunfv 7198	. . . . 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 3954	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
13	12	adantl 482	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
14		dfss3 3911	. . 3 ⊢ (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
15		ssel2 3917	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
16		eliun 4932	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ (((,) ∘ 𝐹)‘𝑛))
17		rexr 11189	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ^*)
18	17	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℝ^*)
19		fvco3 6934	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
20		ffvelcdm 7029	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	20	elin2d 4141	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ × ℝ))
22		1st2nd2 7977	. . . . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
25		df-ov 7366	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
26	24, 25	eqtr4di 2793	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
27	19, 26	eqtrd 2775	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
28	27	eleq2d 2826	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ 𝑧 ∈ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛)))))
29		ovolfcl 25458	. . . . . . . . . . . 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 11189	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝐹‘𝑛)) ∈ ℝ → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^*)
31		rexr 11189	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(𝐹‘𝑛)) ∈ ℝ → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*)
32		elioo1 13336	. . . . . . . . . . . . . . 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 602	. . . . . . . . . . . . . 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 1100	. . . . . . . . . . . . . 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 288	. . . . . . . . . . . . 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 1138	. . . . . . . . . . . 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 280	. . . . . . . . . 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 720	. . . . . . . . 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 713	. . . . . . . 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 3162	. . . . . . 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 284	. . . . . 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 586	. . . . 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 658	. . . 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 3161	. . 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 284	. 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 282	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 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3054 ∃wrex 3064 ∩ cin 3889 ⊆ wss 3890 𝒫 cpw 4536 ⟨cop 4568 ∪ cuni 4845 ∪ ciun 4928 class class class wbr 5079 × cxp 5623 ran crn 5626 ∘ ccom 5629 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 1^st c1st 7936 2^nd c2nd 7937 ℝcr 11035 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 ℕcn 12172 (,)cioo 13296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-pre-lttri 11110 ax-pre-lttrn 11111

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-po 5533 df-so 5534 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-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-ioo 13300

This theorem is referenced by: ovollb2lem 25480 ovolunlem1 25489 ovoliunlem2 25495 ovolshftlem1 25501 ovolscalem1 25505 ioombl1lem4 25553

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