ovolfioo - Metamath Proof Explorer

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Theorem ovolfioo 25396

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 13349	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		inss2 4187	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		rexpssxrxp 11164	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
4	2, 3	sstri 3940	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
5		fss 6672	. . . . . . 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 6680	. . . . . 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 6656	. . . . 5 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
10		fniunfv 7187	. . . . 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 3963	. . 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 3919	. . 3 ⊢ (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
15		ssel2 3925	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
16		eliun 4945	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ (((,) ∘ 𝐹)‘𝑛))
17		rexr 11165	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ^*)
18	17	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℝ^*)
19		fvco3 6927	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
20		ffvelcdm 7020	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	20	elin2d 4154	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ × ℝ))
22		1st2nd2 7966	. . . . . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
25		df-ov 7355	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
26	24, 25	eqtr4di 2786	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
27	19, 26	eqtrd 2768	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
28	27	eleq2d 2819	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ 𝑧 ∈ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛)))))
29		ovolfcl 25395	. . . . . . . . . . . 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 11165	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝐹‘𝑛)) ∈ ℝ → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^*)
31		rexr 11165	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(𝐹‘𝑛)) ∈ ℝ → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*)
32		elioo1 13287	. . . . . . . . . . . . . . 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 3155	. . . . . . 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 3154	. . 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 2113 ∀wral 3048 ∃wrex 3057 ∩ cin 3897 ⊆ wss 3898 𝒫 cpw 4549 ⟨cop 4581 ∪ cuni 4858 ∪ ciun 4941 class class class wbr 5093 × cxp 5617 ran crn 5620 ∘ ccom 5623 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 1^st c1st 7925 2^nd c2nd 7926 ℝcr 11012 ℝ^*cxr 11152 < clt 11153 ≤ cle 11154 ℕcn 12132 (,)cioo 13247

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-pre-lttri 11087 ax-pre-lttrn 11088

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-ioo 13251

This theorem is referenced by: ovollb2lem 25417 ovolunlem1 25426 ovoliunlem2 25432 ovolshftlem1 25438 ovolscalem1 25442 ioombl1lem4 25490

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