Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolficc Structured version   Visualization version   GIF version

Theorem ovolficc 24113
 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) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
Distinct variable groups:   𝑧,𝑛,𝐴   𝑛,𝐹,𝑧

Proof of Theorem ovolficc
StepHypRef Expression
1 iccf 12846 . . . . . 6 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
2 inss2 4159 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3 rexpssxrxp 10693 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
42, 3sstri 3926 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
5 fss 6509 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
64, 5mpan2 690 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
7 fco 6513 . . . . . 6 (([,]:(ℝ* × ℝ*)⟶𝒫 ℝ*𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹):ℕ⟶𝒫 ℝ*)
81, 6, 7sylancr 590 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ([,] ∘ 𝐹):ℕ⟶𝒫 ℝ*)
9 ffn 6495 . . . . 5 (([,] ∘ 𝐹):ℕ⟶𝒫 ℝ* → ([,] ∘ 𝐹) Fn ℕ)
10 fniunfv 6994 . . . . 5 (([,] ∘ 𝐹) Fn ℕ → 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) = ran ([,] ∘ 𝐹))
118, 9, 103syl 18 . . . 4 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) = ran ([,] ∘ 𝐹))
1211sseq2d 3949 . . 3 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝐴 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ 𝐴 ran ([,] ∘ 𝐹)))
1312adantl 485 . 2 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ 𝐴 ran ([,] ∘ 𝐹)))
14 dfss3 3905 . . 3 (𝐴 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴 𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛))
15 ssel2 3912 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑧𝐴) → 𝑧 ∈ ℝ)
16 eliun 4889 . . . . . . 7 (𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ (([,] ∘ 𝐹)‘𝑛))
17 simpll 766 . . . . . . . . 9 (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℝ)
18 fvco3 6747 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑛) = ([,]‘(𝐹𝑛)))
19 ffvelrn 6836 . . . . . . . . . . . . . . . . 17 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
2019elin2d 4129 . . . . . . . . . . . . . . . 16 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (ℝ × ℝ))
21 1st2nd2 7723 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ (ℝ × ℝ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
2220, 21syl 17 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
2322fveq2d 6659 . . . . . . . . . . . . . 14 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ([,]‘(𝐹𝑛)) = ([,]‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
24 df-ov 7148 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) = ([,]‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
2523, 24eqtr4di 2851 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ([,]‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))))
2618, 25eqtrd 2833 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))))
2726eleq2d 2875 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ 𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛)))))
28 ovolfcl 24111 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
29 elicc2 12810 . . . . . . . . . . . . . 14 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
30 3anass 1092 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3129, 30syl6bb 290 . . . . . . . . . . . . 13 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
32313adant3 1129 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → (𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
3328, 32syl 17 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ ((1st ‘(𝐹𝑛))[,](2nd ‘(𝐹𝑛))) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
3427, 33bitrd 282 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
3534adantll 713 . . . . . . . . 9 (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ (𝑧 ∈ ℝ ∧ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛))))))
3617, 35mpbirand 706 . . . . . . . 8 (((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑛 ∈ ℕ) → (𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3736rexbidva 3256 . . . . . . 7 ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∃𝑛 ∈ ℕ 𝑧 ∈ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3816, 37syl5bb 286 . . . . . 6 ((𝑧 ∈ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
3915, 38sylan 583 . . . . 5 (((𝐴 ⊆ ℝ ∧ 𝑧𝐴) ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
4039an32s 651 . . . 4 (((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) ∧ 𝑧𝐴) → (𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
4140ralbidva 3161 . . 3 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∀𝑧𝐴 𝑧 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
4214, 41syl5bb 286 . 2 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 𝑛 ∈ ℕ (([,] ∘ 𝐹)‘𝑛) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
4313, 42bitr3d 284 1 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ([,] ∘ 𝐹) ↔ ∀𝑧𝐴𝑛 ∈ ℕ ((1st ‘(𝐹𝑛)) ≤ 𝑧𝑧 ≤ (2nd ‘(𝐹𝑛)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4500  ⟨cop 4534  ∪ cuni 4804  ∪ ciun 4885   class class class wbr 5034   × cxp 5521  ran crn 5524   ∘ ccom 5527   Fn wfn 6327  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  1st c1st 7682  2nd c2nd 7683  ℝcr 10543  ℝ*cxr 10681   ≤ cle 10683  ℕcn 11643  [,]cicc 12749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-pre-lttri 10618  ax-pre-lttrn 10619 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7684  df-2nd 7685  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-icc 12753 This theorem is referenced by:  ovollb2lem  24133  ovolctb  24135  ovolicc1  24161  ioombl1lem4  24206
 Copyright terms: Public domain W3C validator