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

Theorem uniiccvol 25628
Description: An almost-disjoint union of closed intervals (disjoint interiors) has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 25602.) (Contributed by Mario Carneiro, 25-Mar-2015.)
Hypotheses
Ref Expression
uniioombl.1 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.2 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
uniioombl.3 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
Assertion
Ref Expression
uniiccvol (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Distinct variable groups:   𝑥,𝐹   𝜑,𝑥
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem uniiccvol
StepHypRef Expression
1 uniioombl.1 . . . 4 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2 ovolficcss 25517 . . . 4 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
31, 2syl 17 . . 3 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
4 ovolcl 25526 . . 3 ( ran ([,] ∘ 𝐹) ⊆ ℝ → (vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ*)
53, 4syl 17 . 2 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ*)
6 eqid 2734 . . . . . . 7 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
7 uniioombl.3 . . . . . . 7 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
86, 7ovolsf 25520 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
91, 8syl 17 . . . . 5 (𝜑𝑆:ℕ⟶(0[,)+∞))
109frnd 6744 . . . 4 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
11 icossxr 13468 . . . 4 (0[,)+∞) ⊆ ℝ*
1210, 11sstrdi 4007 . . 3 (𝜑 → ran 𝑆 ⊆ ℝ*)
13 supxrcl 13353 . . 3 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
1412, 13syl 17 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
15 ssid 4017 . . 3 ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)
167ovollb2 25537 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)) → (vol*‘ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
171, 15, 16sylancl 586 . 2 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
18 uniioombl.2 . . . 4 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
191, 18, 7uniioovol 25627 . . 3 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
20 ioossicc 13469 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ⊆ ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥)))
21 df-ov 7433 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
22 df-ov 7433 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
2320, 21, 223sstr3i 4037 . . . . . . . . . . 11 ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩) ⊆ ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
2423a1i 11 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩) ⊆ ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
25 ffvelcdm 7100 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
2625elin2d 4214 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ (ℝ × ℝ))
27 1st2nd2 8051 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
2826, 27syl 17 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
2928fveq2d 6910 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
3028fveq2d 6910 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
3124, 29, 303sstr4d 4042 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) ⊆ ([,]‘(𝐹𝑥)))
32 fvco3 7007 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
33 fvco3 7007 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
3431, 32, 333sstr4d 4042 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
351, 34sylan 580 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
3635ralrimiva 3143 . . . . . 6 (𝜑 → ∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
37 ss2iun 5014 . . . . . 6 (∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥) → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥))
3836, 37syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥))
39 ioof 13483 . . . . . . . 8 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
40 ffn 6736 . . . . . . . 8 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
4139, 40ax-mp 5 . . . . . . 7 (,) Fn (ℝ* × ℝ*)
42 inss2 4245 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
43 rexpssxrxp 11303 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4442, 43sstri 4004 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
45 fss 6752 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
461, 44, 45sylancl 586 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
47 fnfco 6773 . . . . . . 7 (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
4841, 46, 47sylancr 587 . . . . . 6 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
49 fniunfv 7266 . . . . . 6 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
5048, 49syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
51 iccf 13484 . . . . . . . 8 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
52 ffn 6736 . . . . . . . 8 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
5351, 52ax-mp 5 . . . . . . 7 [,] Fn (ℝ* × ℝ*)
54 fnfco 6773 . . . . . . 7 (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
5553, 46, 54sylancr 587 . . . . . 6 (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
56 fniunfv 7266 . . . . . 6 (([,] ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
5755, 56syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
5838, 50, 573sstr3d 4041 . . . 4 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
59 ovolss 25533 . . . 4 (( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ ran ([,] ∘ 𝐹) ⊆ ℝ) → (vol*‘ ran ((,) ∘ 𝐹)) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
6058, 3, 59syl2anc 584 . . 3 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
6119, 60eqbrtrrd 5171 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
625, 14, 17, 61xrletrid 13193 1 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  cin 3961  wss 3962  𝒫 cpw 4604  cop 4636   cuni 4911   ciun 4995  Disj wdisj 5114   class class class wbr 5147   × cxp 5686  ran crn 5689  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  supcsup 9477  cr 11151  0cc0 11152  1c1 11153   + caddc 11155  +∞cpnf 11289  *cxr 11291   < clt 11292  cle 11293  cmin 11489  cn 12263  (,)cioo 13383  [,)cico 13385  [,]cicc 13386  seqcseq 14038  abscabs 15269  vol*covol 25510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-rest 17468  df-topgen 17489  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-top 22915  df-topon 22932  df-bases 22968  df-cmp 23410  df-ovol 25512  df-vol 25513
This theorem is referenced by:  mblfinlem2  37644
  Copyright terms: Public domain W3C validator