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Theorem uniiccvol 23154
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 23129.) (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 3587 . . 3 ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)
3 uniioombl.3 . . . 4 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
43ovollb2 23064 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)) → (vol*‘ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
51, 2, 4sylancl 693 . 2 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
6 uniioombl.2 . . . 4 (𝜑Disj 𝑥 ∈ ℕ ((,)‘(𝐹𝑥)))
71, 6, 3uniioovol 23153 . . 3 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
8 ioossicc 12130 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ⊆ ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥)))
9 df-ov 6552 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
10 df-ov 6552 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
118, 9, 103sstr3i 3606 . . . . . . . . . . 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 3796 . . . . . . . . . . . . 13 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
14 ffvelrn 6265 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1513, 14sseldi 3566 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ (ℝ × ℝ))
16 1st2nd2 7096 . . . . . . . . . . . 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 6107 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
1917fveq2d 6107 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
2012, 18, 193sstr4d 3611 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) ⊆ ([,]‘(𝐹𝑥)))
21 fvco3 6185 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
22 fvco3 6185 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
2320, 21, 223sstr4d 3611 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
241, 23sylan 487 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
2524ralrimiva 2949 . . . . . 6 (𝜑 → ∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
26 ss2iun 4472 . . . . . 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 12142 . . . . . . . 8 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
29 ffn 5958 . . . . . . . 8 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
3028, 29ax-mp 5 . . . . . . 7 (,) Fn (ℝ* × ℝ*)
31 rexpssxrxp 9963 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
3213, 31sstri 3577 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
33 fss 5969 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
341, 32, 33sylancl 693 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
35 fnfco 5982 . . . . . . 7 (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
3630, 34, 35sylancr 694 . . . . . 6 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
37 fniunfv 6409 . . . . . 6 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
3836, 37syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
39 iccf 12143 . . . . . . . 8 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
40 ffn 5958 . . . . . . . 8 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
4139, 40ax-mp 5 . . . . . . 7 [,] Fn (ℝ* × ℝ*)
42 fnfco 5982 . . . . . . 7 (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
4341, 34, 42sylancr 694 . . . . . 6 (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
44 fniunfv 6409 . . . . . 6 (([,] ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4543, 44syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4627, 38, 453sstr3d 3610 . . . 4 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
47 ovolficcss 23045 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
481, 47syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
49 ovolss 23060 . . . 4 (( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ ran ([,] ∘ 𝐹) ⊆ ℝ) → (vol*‘ ran ((,) ∘ 𝐹)) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
5046, 48, 49syl2anc 691 . . 3 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
517, 50eqbrtrrd 4607 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
52 ovolcl 23053 . . . 4 ( ran ([,] ∘ 𝐹) ⊆ ℝ → (vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ*)
5348, 52syl 17 . . 3 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ*)
54 eqid 2610 . . . . . . . 8 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
5554, 3ovolsf 23048 . . . . . . 7 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
561, 55syl 17 . . . . . 6 (𝜑𝑆:ℕ⟶(0[,)+∞))
57 frn 5966 . . . . . 6 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
5856, 57syl 17 . . . . 5 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
59 icossxr 12129 . . . . 5 (0[,)+∞) ⊆ ℝ*
6058, 59syl6ss 3580 . . . 4 (𝜑 → ran 𝑆 ⊆ ℝ*)
61 supxrcl 12017 . . . 4 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
6260, 61syl 17 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
63 xrletri3 11861 . . 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 691 . 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 959 1 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  cin 3539  wss 3540  𝒫 cpw 4108  cop 4131   cuni 4372   ciun 4455  Disj wdisj 4553   class class class wbr 4583   × cxp 5036  ran crn 5039  ccom 5042   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  supcsup 8229  cr 9814  0cc0 9815  1c1 9816   + caddc 9818  +∞cpnf 9950  *cxr 9952   < clt 9953  cle 9954  cmin 10145  cn 10897  (,)cioo 12046  [,)cico 12048  [,]cicc 12049  seqcseq 12663  abscabs 13822  vol*covol 23038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041
This theorem is referenced by:  mblfinlem2  32617
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