ovolfioo - Metamath Proof Explorer

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Theorem ovolfioo 25793

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 13581	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		inss2 4134	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		rexpssxrxp 11357	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
4	2, 3	sstri 3917	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
5		fss 6726	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝐹:ℕ⟶(ℝ^ × ℝ^*))
6	4, 5	mpan2 704	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
7		fco 6734	. . . . . 6 ⊢ (((,):(ℝ^* × ℝ^)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
8	1, 6, 7	sylancr 599	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
9		ffn 6709	. . . . 5 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
10		fniunfv 7251	. . . . 5 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ ran ((,) ∘ 𝐹))
11	8, 9, 10	3syl 19	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ ran ((,) ∘ 𝐹))
12	11	sseq2d 3940	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
13	12	adantl 487	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹)))
14		dfss3 3897	. . 3 ⊢ (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
15		ssel2 3903	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
16		eliun 4954	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ (((,) ∘ 𝐹)‘𝑛))
17		rexr 11358	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ^*)
18	17	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℝ^*)
19		fvco3 6985	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
20		ffvelcdm 7081	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	20	elin2d 4102	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ × ℝ))
22		1st2nd2 8040	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
23	21, 22	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
24	23	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
25		df-ov 7423	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
26	24, 25	eqtr4di 2813	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
27	19, 26	eqtrd 2795	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
28	27	eleq2d 2846	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (((,) ∘ 𝐹)‘𝑛) ↔ 𝑧 ∈ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛)))))
29		ovolfcl 25792	. . . . . . . . . . . 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 11358	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(𝐹‘𝑛)) ∈ ℝ → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^*)
31		rexr 11358	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(𝐹‘𝑛)) ∈ ℝ → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*)
32		elioo1 13519	. . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . 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 1111	. . . . . . . . . . . . . 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 290	. . . . . . . . . . . . 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 1150	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 282	. . . . . . . . . 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 727	. . . . . . . . 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 720	. . . . . . . 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 3184	. . . . . . 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 286	. . . . . 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 592	. . . . 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 665	. . . 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 3183	. . 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 286	. 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 284	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 ⊆ wss 3895 ∩ cin 4089 𝒫 cpw 4556 ⟨cop 4834 ∪ cuni 4866 ∪ ciun 4950 class class class wbr 5102 × cxp 5648 ran crn 5651 ∘ ccom 5654 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 ℝcr 11202 ℝ^*cxr 11345 < clt 11346 ≤ cle 11347 ℕcn 12338 (,)cioo 13479

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-pre-lttri 11277 ax-pre-lttrn 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-po 5558 df-so 5559 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-er 8720 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-ioo 13483

This theorem is used by: ovollb2lem 25814 ovolunlem1 25823 ovoliunlem2 25829 ovolshftlem1 25835 ovolscalem1 25839 ioombl1lem4 25887

W3C validator