ovolficc - Metamath Proof Explorer

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Theorem ovolficc 25608

Description: Unpack the interval covering property using closed intervals. (Contributed by Mario Carneiro, 16-Mar-2014.)

**Assertion**
Ref	Expression
ovolficc	⊢ ((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 ovolficc**
Step	Hyp	Ref	Expression
1		iccf 13476	. . . . . 6 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
2		inss2 4191	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		rexpssxrxp 11255	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
4	2, 3	sstri 3947	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
5		fss 6724	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝐹:ℕ⟶(ℝ^ × ℝ^*))
6	4, 5	mpan2 703	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
7		fco 6732	. . . . . 6 ⊢ (([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ ∧ 𝐹:ℕ⟶(ℝ^* × ℝ^)) → ([,] ∘ 𝐹):ℕ⟶𝒫 ℝ^)
8	1, 6, 7	sylancr 598	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] ∘ 𝐹):ℕ⟶𝒫 ℝ^*)
9		ffn 6707	. . . . 5 ⊢ (([,] ∘ 𝐹):ℕ⟶𝒫 ℝ^* → ([,] ∘ 𝐹) Fn ℕ)
10		fniunfv 7247	. . . . 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 3970	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)))
13	12	adantl 486	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)))
14		dfss3 3927	. . 3 ⊢ (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛))
15		ssel2 3933	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
16		eliun 4961	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ (([,] ∘ 𝐹)‘𝑛))
17		simpll 778	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℝ)
18		fvco3 6983	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑛) = ([,]‘(𝐹‘𝑛)))
19		ffvelcdm 7078	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
20	19	elin2d 4159	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ × ℝ))
21		1st2nd2 8026	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) ∈ (ℝ × ℝ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
22	20, 21	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
23	22	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ([,]‘(𝐹‘𝑛)) = ([,]‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
24		df-ov 7415	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛))) = ([,]‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
25	23, 24	eqtr4di 2816	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ([,]‘(𝐹‘𝑛)) = ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛))))
26	18, 25	eqtrd 2798	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑛) = ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛))))
27	26	eleq2d 2849	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ 𝑧 ∈ ((1^st ‘(𝐹‘𝑛))[,](2^nd ‘(𝐹‘𝑛)))))
28		ovolfcl 25606	. . . . . . . . . . . 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))))
29		elicc2 13439	. . . . . . . . . . . . . 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)))))
30		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)))))
31	29, 30	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))))))
32	31	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))))))
33	28, 32	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))))))
34	27, 33	bitrd 282	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ ∧ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛))))))
35	34	adantll 726	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ ∧ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛))))))
36	17, 35	mpbirand 719	. . . . . . . 8 ⊢ (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
37	36	rexbidva 3187	. . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∃𝑛 ∈ ℕ 𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1^st ‘(𝐹‘𝑛)) ≤ 𝑧 ∧ 𝑧 ≤ (2^nd ‘(𝐹‘𝑛)))))
38	16, 37	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)))))
39	15, 38	sylan 591	. . . . 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)))))
40	39	an32s 664	. . . 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)))))
41	40	ralbidva 3186	. . 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)))))
42	14, 41	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)))))
43	13, 42	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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ∩ cin 3905 ⊆ wss 3906 𝒫 cpw 4563 ⟨cop 4596 ∪ cuni 4873 ∪ ciun 4957 class class class wbr 5110 × cxp 5661 ran crn 5664 ∘ ccom 5667 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 1^st c1st 7985 2^nd c2nd 7986 ℝcr 11100 ℝ^*cxr 11243 ≤ cle 11245 ℕcn 12234 [,]cicc 13376

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-pre-lttri 11175 ax-pre-lttrn 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 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 7415 df-oprab 7416 df-mpo 7417 df-1st 7987 df-2nd 7988 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-icc 13380

This theorem is referenced by: ovollb2lem 25628 ovolctb 25630 ovolicc1 25656 ioombl1lem4 25701

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