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Theorem uniiccvol 23568
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 23542.) (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 . . 3 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2 ssid 3773 . . 3 ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)
3 uniioombl.3 . . . 4 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
43ovollb2 23477 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)) → (vol*‘ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
51, 2, 4sylancl 574 . 2 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
6 uniioombl.2 . . . 4 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
71, 6, 3uniioovol 23567 . . 3 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
8 ioossicc 12464 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ⊆ ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥)))
9 df-ov 6796 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
10 df-ov 6796 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
118, 9, 103sstr3i 3792 . . . . . . . . . . 11 ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩) ⊆ ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1211a1i 11 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩) ⊆ ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
13 inss2 3982 . . . . . . . . . . . . 13 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
14 ffvelrn 6500 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1513, 14sseldi 3750 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ (ℝ × ℝ))
16 1st2nd2 7354 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ (ℝ × ℝ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1715, 16syl 17 . . . . . . . . . . 11 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
1817fveq2d 6336 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
1917fveq2d 6336 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
2012, 18, 193sstr4d 3797 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) ⊆ ([,]‘(𝐹𝑥)))
21 fvco3 6417 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
22 fvco3 6417 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
2320, 21, 223sstr4d 3797 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
241, 23sylan 569 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
2524ralrimiva 3115 . . . . . 6 (𝜑 → ∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
26 ss2iun 4670 . . . . . 6 (∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥) → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥))
2725, 26syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥))
28 ioof 12477 . . . . . . . 8 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
29 ffn 6185 . . . . . . . 8 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
3028, 29ax-mp 5 . . . . . . 7 (,) Fn (ℝ* × ℝ*)
31 rexpssxrxp 10286 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
3213, 31sstri 3761 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
33 fss 6196 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
341, 32, 33sylancl 574 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
35 fnfco 6209 . . . . . . 7 (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
3630, 34, 35sylancr 575 . . . . . 6 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
37 fniunfv 6648 . . . . . 6 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
3836, 37syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
39 iccf 12478 . . . . . . . 8 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
40 ffn 6185 . . . . . . . 8 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
4139, 40ax-mp 5 . . . . . . 7 [,] Fn (ℝ* × ℝ*)
42 fnfco 6209 . . . . . . 7 (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
4341, 34, 42sylancr 575 . . . . . 6 (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
44 fniunfv 6648 . . . . . 6 (([,] ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4543, 44syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4627, 38, 453sstr3d 3796 . . . 4 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
47 ovolficcss 23457 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
481, 47syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
49 ovolss 23473 . . . 4 (( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ ran ([,] ∘ 𝐹) ⊆ ℝ) → (vol*‘ ran ((,) ∘ 𝐹)) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
5046, 48, 49syl2anc 573 . . 3 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
517, 50eqbrtrrd 4810 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
52 ovolcl 23466 . . . 4 ( ran ([,] ∘ 𝐹) ⊆ ℝ → (vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ*)
5348, 52syl 17 . . 3 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ*)
54 eqid 2771 . . . . . . . 8 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
5554, 3ovolsf 23460 . . . . . . 7 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
561, 55syl 17 . . . . . 6 (𝜑𝑆:ℕ⟶(0[,)+∞))
57 frn 6193 . . . . . 6 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
5856, 57syl 17 . . . . 5 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
59 icossxr 12463 . . . . 5 (0[,)+∞) ⊆ ℝ*
6058, 59syl6ss 3764 . . . 4 (𝜑 → ran 𝑆 ⊆ ℝ*)
61 supxrcl 12350 . . . 4 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
6260, 61syl 17 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
63 xrletri3 12190 . . 3 (((vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ* ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ*) → ((vol*‘ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ) ↔ ((vol*‘ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ) ∧ sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ([,] ∘ 𝐹)))))
6453, 62, 63syl2anc 573 . 2 (𝜑 → ((vol*‘ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ) ↔ ((vol*‘ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ) ∧ sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ([,] ∘ 𝐹)))))
655, 51, 64mpbir2and 692 1 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  cin 3722  wss 3723  𝒫 cpw 4297  cop 4322   cuni 4574   ciun 4654  Disj wdisj 4754   class class class wbr 4786   × cxp 5247  ran crn 5250  ccom 5253   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  supcsup 8502  cr 10137  0cc0 10138  1c1 10139   + caddc 10141  +∞cpnf 10273  *cxr 10275   < clt 10276  cle 10277  cmin 10468  cn 11222  (,)cioo 12380  [,)cico 12382  [,]cicc 12383  seqcseq 13008  abscabs 14182  vol*covol 23450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-disj 4755  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-rlim 14428  df-sum 14625  df-rest 16291  df-topgen 16312  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-top 20919  df-topon 20936  df-bases 20971  df-cmp 21411  df-ovol 23452  df-vol 23453
This theorem is referenced by:  mblfinlem2  33780
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