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Theorem uniiccvol 24180
 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 24154.) (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 24069 . . . 4 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
31, 2syl 17 . . 3 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
4 ovolcl 24078 . . 3 ( ran ([,] ∘ 𝐹) ⊆ ℝ → (vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ*)
53, 4syl 17 . 2 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ*)
6 eqid 2821 . . . . . . 7 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
7 uniioombl.3 . . . . . . 7 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
86, 7ovolsf 24072 . . . . . 6 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
91, 8syl 17 . . . . 5 (𝜑𝑆:ℕ⟶(0[,)+∞))
109frnd 6520 . . . 4 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
11 icossxr 12820 . . . 4 (0[,)+∞) ⊆ ℝ*
1210, 11sstrdi 3978 . . 3 (𝜑 → ran 𝑆 ⊆ ℝ*)
13 supxrcl 12707 . . 3 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
1412, 13syl 17 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
15 ssid 3988 . . 3 ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)
167ovollb2 24089 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)) → (vol*‘ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
171, 15, 16sylancl 588 . 2 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
18 uniioombl.2 . . . 4 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
191, 18, 7uniioovol 24179 . . 3 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
20 ioossicc 12821 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ⊆ ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥)))
21 df-ov 7158 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
22 df-ov 7158 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
2320, 21, 223sstr3i 4008 . . . . . . . . . . 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 ffvelrn 6848 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
2625elin2d 4175 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ (ℝ × ℝ))
27 1st2nd2 7727 . . . . . . . . . . . 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 6673 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
3028fveq2d 6673 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
3124, 29, 303sstr4d 4013 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) ⊆ ([,]‘(𝐹𝑥)))
32 fvco3 6759 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
33 fvco3 6759 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
3431, 32, 333sstr4d 4013 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
351, 34sylan 582 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
3635ralrimiva 3182 . . . . . 6 (𝜑 → ∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
37 ss2iun 4936 . . . . . 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 12834 . . . . . . . 8 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
40 ffn 6513 . . . . . . . 8 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
4139, 40ax-mp 5 . . . . . . 7 (,) Fn (ℝ* × ℝ*)
42 inss2 4205 . . . . . . . . 9 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
43 rexpssxrxp 10685 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4442, 43sstri 3975 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
45 fss 6526 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
461, 44, 45sylancl 588 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
47 fnfco 6542 . . . . . . 7 (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
4841, 46, 47sylancr 589 . . . . . 6 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
49 fniunfv 7005 . . . . . 6 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
5048, 49syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
51 iccf 12835 . . . . . . . 8 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
52 ffn 6513 . . . . . . . 8 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
5351, 52ax-mp 5 . . . . . . 7 [,] Fn (ℝ* × ℝ*)
54 fnfco 6542 . . . . . . 7 (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
5553, 46, 54sylancr 589 . . . . . 6 (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
56 fniunfv 7005 . . . . . 6 (([,] ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
5755, 56syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
5838, 50, 573sstr3d 4012 . . . 4 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
59 ovolss 24085 . . . 4 (( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ ran ([,] ∘ 𝐹) ⊆ ℝ) → (vol*‘ ran ((,) ∘ 𝐹)) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
6058, 3, 59syl2anc 586 . . 3 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
6119, 60eqbrtrrd 5089 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
625, 14, 17, 61xrletrid 12547 1 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ∩ cin 3934   ⊆ wss 3935  𝒫 cpw 4538  ⟨cop 4572  ∪ cuni 4837  ∪ ciun 4918  Disj wdisj 5030   class class class wbr 5065   × cxp 5552  ran crn 5555   ∘ ccom 5558   Fn wfn 6349  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  1st c1st 7686  2nd c2nd 7687  supcsup 8903  ℝcr 10535  0cc0 10536  1c1 10537   + caddc 10539  +∞cpnf 10671  ℝ*cxr 10673   < clt 10674   ≤ cle 10675   − cmin 10869  ℕcn 11637  (,)cioo 12737  [,)cico 12739  [,]cicc 12740  seqcseq 13368  abscabs 14592  vol*covol 24062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-disj 5031  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-oadd 8105  df-er 8288  df-map 8407  df-pm 8408  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-fi 8874  df-sup 8905  df-inf 8906  df-oi 8973  df-dju 9329  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-q 12348  df-rp 12389  df-xneg 12506  df-xadd 12507  df-xmul 12508  df-ioo 12741  df-ico 12743  df-icc 12744  df-fz 12892  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13429  df-hash 13690  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-clim 14844  df-rlim 14845  df-sum 15042  df-rest 16695  df-topgen 16716  df-psmet 20536  df-xmet 20537  df-met 20538  df-bl 20539  df-mopn 20540  df-top 21501  df-topon 21518  df-bases 21553  df-cmp 21994  df-ovol 24064  df-vol 24065 This theorem is referenced by:  mblfinlem2  34929
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