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Theorem uniiccvol 23249
 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 23224.) (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 3608 . . 3 ran ([,] ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹)
3 uniioombl.3 . . . 4 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
43ovollb2 23159 . . 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 23248 . . 3 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
8 ioossicc 12198 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) ⊆ ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥)))
9 df-ov 6608 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
10 df-ov 6608 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑥))[,](2nd ‘(𝐹𝑥))) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
118, 9, 103sstr3i 3627 . . . . . . . . . . 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 3817 . . . . . . . . . . . . 13 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
14 ffvelrn 6314 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
1513, 14sseldi 3586 . . . . . . . . . . . 12 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹𝑥) ∈ (ℝ × ℝ))
16 1st2nd2 7153 . . . . . . . . . . . 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 6154 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) = ((,)‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
1917fveq2d 6154 . . . . . . . . . 10 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹𝑥)) = ([,]‘⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
2012, 18, 193sstr4d 3632 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹𝑥)) ⊆ ([,]‘(𝐹𝑥)))
21 fvco3 6233 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹𝑥)))
22 fvco3 6233 . . . . . . . . 9 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹𝑥)))
2320, 21, 223sstr4d 3632 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
241, 23sylan 488 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
2524ralrimiva 2965 . . . . . 6 (𝜑 → ∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
26 ss2iun 4507 . . . . . 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 12210 . . . . . . . 8 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
29 ffn 6004 . . . . . . . 8 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
3028, 29ax-mp 5 . . . . . . 7 (,) Fn (ℝ* × ℝ*)
31 rexpssxrxp 10029 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
3213, 31sstri 3597 . . . . . . . 8 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
33 fss 6015 . . . . . . . 8 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
341, 32, 33sylancl 693 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
35 fnfco 6028 . . . . . . 7 (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
3630, 34, 35sylancr 694 . . . . . 6 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
37 fniunfv 6460 . . . . . 6 (((,) ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
3836, 37syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ran ((,) ∘ 𝐹))
39 iccf 12211 . . . . . . . 8 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
40 ffn 6004 . . . . . . . 8 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
4139, 40ax-mp 5 . . . . . . 7 [,] Fn (ℝ* × ℝ*)
42 fnfco 6028 . . . . . . 7 (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
4341, 34, 42sylancr 694 . . . . . 6 (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
44 fniunfv 6460 . . . . . 6 (([,] ∘ 𝐹) Fn ℕ → 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4543, 44syl 17 . . . . 5 (𝜑 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ran ([,] ∘ 𝐹))
4627, 38, 453sstr3d 3631 . . . 4 (𝜑 ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹))
47 ovolficcss 23140 . . . . 5 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ran ([,] ∘ 𝐹) ⊆ ℝ)
481, 47syl 17 . . . 4 (𝜑 ran ([,] ∘ 𝐹) ⊆ ℝ)
49 ovolss 23155 . . . 4 (( ran ((,) ∘ 𝐹) ⊆ ran ([,] ∘ 𝐹) ∧ ran ([,] ∘ 𝐹) ⊆ ℝ) → (vol*‘ ran ((,) ∘ 𝐹)) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
5046, 48, 49syl2anc 692 . . 3 (𝜑 → (vol*‘ ran ((,) ∘ 𝐹)) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
517, 50eqbrtrrd 4642 . 2 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran ([,] ∘ 𝐹)))
52 ovolcl 23148 . . . 4 ( ran ([,] ∘ 𝐹) ⊆ ℝ → (vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ*)
5348, 52syl 17 . . 3 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) ∈ ℝ*)
54 eqid 2626 . . . . . . . 8 ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
5554, 3ovolsf 23143 . . . . . . 7 (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
561, 55syl 17 . . . . . 6 (𝜑𝑆:ℕ⟶(0[,)+∞))
57 frn 6012 . . . . . 6 (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
5856, 57syl 17 . . . . 5 (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
59 icossxr 12197 . . . . 5 (0[,)+∞) ⊆ ℝ*
6058, 59syl6ss 3600 . . . 4 (𝜑 → ran 𝑆 ⊆ ℝ*)
61 supxrcl 12085 . . . 4 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
6260, 61syl 17 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
63 xrletri3 11929 . . 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 692 . 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 956 1 (𝜑 → (vol*‘ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1992  ∀wral 2912   ∩ cin 3559   ⊆ wss 3560  𝒫 cpw 4135  ⟨cop 4159  ∪ cuni 4407  ∪ ciun 4490  Disj wdisj 4588   class class class wbr 4618   × cxp 5077  ran crn 5080   ∘ ccom 5083   Fn wfn 5845  ⟶wf 5846  ‘cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  supcsup 8291  ℝcr 9880  0cc0 9881  1c1 9882   + caddc 9884  +∞cpnf 10016  ℝ*cxr 10018   < clt 10019   ≤ cle 10020   − cmin 10211  ℕcn 10965  (,)cioo 12114  [,)cico 12116  [,]cicc 12117  seqcseq 12738  abscabs 13903  vol*covol 23133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-fi 8262  df-sup 8293  df-inf 8294  df-oi 8360  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-z 11323  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12118  df-ico 12120  df-icc 12121  df-fz 12266  df-fzo 12404  df-fl 12530  df-seq 12739  df-exp 12798  df-hash 13055  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-clim 14148  df-rlim 14149  df-sum 14346  df-rest 15999  df-topgen 16020  df-psmet 19652  df-xmet 19653  df-met 19654  df-bl 19655  df-mopn 19656  df-top 20616  df-bases 20617  df-topon 20618  df-cmp 21095  df-ovol 23135  df-vol 23136 This theorem is referenced by:  mblfinlem2  33065
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