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Theorem ovoliun2 24114
Description: The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 24113.) (Contributed by Mario Carneiro, 12-Jun-2014.)
Hypotheses
Ref Expression
ovoliun.t 𝑇 = seq1( + , 𝐺)
ovoliun.g 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
ovoliun.a ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
ovoliun2.t (𝜑𝑇 ∈ dom ⇝ )
Assertion
Ref Expression
ovoliun2 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))
Distinct variable group:   𝜑,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝑇(𝑛)   𝐺(𝑛)

Proof of Theorem ovoliun2
Dummy variables 𝑘 𝑚 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovoliun.t . . 3 𝑇 = seq1( + , 𝐺)
2 ovoliun.g . . 3 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
3 ovoliun.a . . 3 ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
4 ovoliun.v . . 3 ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
51, 2, 3, 4ovoliun 24113 . 2 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
6 nnuz 12273 . . . . . . . 8 ℕ = (ℤ‘1)
7 1zzd 12005 . . . . . . . 8 (𝜑 → 1 ∈ ℤ)
8 fvex 6662 . . . . . . . . . . 11 (vol*‘𝑚 / 𝑛𝐴) ∈ V
9 nfcv 2958 . . . . . . . . . . . . . 14 𝑚(vol*‘𝐴)
10 nfcv 2958 . . . . . . . . . . . . . . 15 𝑛vol*
11 nfcsb1v 3855 . . . . . . . . . . . . . . 15 𝑛𝑚 / 𝑛𝐴
1210, 11nffv 6659 . . . . . . . . . . . . . 14 𝑛(vol*‘𝑚 / 𝑛𝐴)
13 csbeq1a 3845 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
1413fveq2d 6653 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘𝑚 / 𝑛𝐴))
159, 12, 14cbvmpt 5134 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
162, 15eqtri 2824 . . . . . . . . . . . 12 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
1716fvmpt2 6760 . . . . . . . . . . 11 ((𝑚 ∈ ℕ ∧ (vol*‘𝑚 / 𝑛𝐴) ∈ V) → (𝐺𝑚) = (vol*‘𝑚 / 𝑛𝐴))
188, 17mpan2 690 . . . . . . . . . 10 (𝑚 ∈ ℕ → (𝐺𝑚) = (vol*‘𝑚 / 𝑛𝐴))
1918adantl 485 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) = (vol*‘𝑚 / 𝑛𝐴))
204ralrimiva 3152 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
219nfel1 2974 . . . . . . . . . . . 12 𝑚(vol*‘𝐴) ∈ ℝ
2212nfel1 2974 . . . . . . . . . . . 12 𝑛(vol*‘𝑚 / 𝑛𝐴) ∈ ℝ
2314eleq1d 2877 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ))
2421, 22, 23cbvralw 3390 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
2520, 24sylib 221 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
2625r19.21bi 3176 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
2719, 26eqeltrd 2893 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ ℝ)
286, 7, 27serfre 13399 . . . . . . 7 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
291feq1i 6482 . . . . . . 7 (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
3028, 29sylibr 237 . . . . . 6 (𝜑𝑇:ℕ⟶ℝ)
3130frnd 6498 . . . . 5 (𝜑 → ran 𝑇 ⊆ ℝ)
32 1nn 11640 . . . . . . . 8 1 ∈ ℕ
3330fdmd 6501 . . . . . . . 8 (𝜑 → dom 𝑇 = ℕ)
3432, 33eleqtrrid 2900 . . . . . . 7 (𝜑 → 1 ∈ dom 𝑇)
3534ne0d 4254 . . . . . 6 (𝜑 → dom 𝑇 ≠ ∅)
36 dm0rn0 5763 . . . . . . 7 (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
3736necon3bii 3042 . . . . . 6 (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
3835, 37sylib 221 . . . . 5 (𝜑 → ran 𝑇 ≠ ∅)
39 ovoliun2.t . . . . . . . . 9 (𝜑𝑇 ∈ dom ⇝ )
401, 39eqeltrrid 2898 . . . . . . . 8 (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
416, 7, 19, 26, 40isumrecl 15116 . . . . . . 7 (𝜑 → Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
42 elfznn 12935 . . . . . . . . . . . . 13 (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
4342adantl 485 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → 𝑚 ∈ ℕ)
4443, 18syl 17 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐺𝑚) = (vol*‘𝑚 / 𝑛𝐴))
45 simpr 488 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
4645, 6eleqtrdi 2903 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
47 simpl 486 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝜑)
4847, 42, 26syl2an 598 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
4948recnd 10662 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℂ)
5044, 46, 49fsumser 15083 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘𝑚 / 𝑛𝐴) = (seq1( + , 𝐺)‘𝑘))
511fveq1i 6650 . . . . . . . . . 10 (𝑇𝑘) = (seq1( + , 𝐺)‘𝑘)
5250, 51eqtr4di 2854 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘𝑚 / 𝑛𝐴) = (𝑇𝑘))
53 fzfid 13340 . . . . . . . . . . 11 (𝜑 → (1...𝑘) ∈ Fin)
54 fz1ssnn 12937 . . . . . . . . . . . 12 (1...𝑘) ⊆ ℕ
5554a1i 11 . . . . . . . . . . 11 (𝜑 → (1...𝑘) ⊆ ℕ)
563ralrimiva 3152 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
57 nfv 1915 . . . . . . . . . . . . . . 15 𝑚 𝐴 ⊆ ℝ
58 nfcv 2958 . . . . . . . . . . . . . . . 16 𝑛
5911, 58nfss 3910 . . . . . . . . . . . . . . 15 𝑛𝑚 / 𝑛𝐴 ⊆ ℝ
6013sseq1d 3949 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ 𝑚 / 𝑛𝐴 ⊆ ℝ))
6157, 59, 60cbvralw 3390 . . . . . . . . . . . . . 14 (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
6256, 61sylib 221 . . . . . . . . . . . . 13 (𝜑 → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
6362r19.21bi 3176 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
64 ovolge0 24089 . . . . . . . . . . . 12 (𝑚 / 𝑛𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝑚 / 𝑛𝐴))
6563, 64syl 17 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 0 ≤ (vol*‘𝑚 / 𝑛𝐴))
666, 7, 53, 55, 19, 26, 65, 40isumless 15196 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...𝑘)(vol*‘𝑚 / 𝑛𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴))
6766adantr 484 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘𝑚 / 𝑛𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴))
6852, 67eqbrtrrd 5057 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴))
6968ralrimiva 3152 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴))
70 brralrspcev 5093 . . . . . . 7 ((Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ 𝑥)
7141, 69, 70syl2anc 587 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ 𝑥)
7230ffnd 6492 . . . . . . . 8 (𝜑𝑇 Fn ℕ)
73 breq1 5036 . . . . . . . . 9 (𝑧 = (𝑇𝑘) → (𝑧𝑥 ↔ (𝑇𝑘) ≤ 𝑥))
7473ralrn 6835 . . . . . . . 8 (𝑇 Fn ℕ → (∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ 𝑥))
7572, 74syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ 𝑥))
7675rexbidv 3259 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ 𝑥))
7771, 76mpbird 260 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥)
78 supxrre 12712 . . . . 5 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
7931, 38, 77, 78syl3anc 1368 . . . 4 (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
806, 1, 7, 19, 26, 65, 71isumsup 15198 . . . 4 (𝜑 → Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) = sup(ran 𝑇, ℝ, < ))
8179, 80eqtr4d 2839 . . 3 (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴))
829, 12, 14cbvsumi 15050 . . 3 Σ𝑛 ∈ ℕ (vol*‘𝐴) = Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴)
8381, 82eqtr4di 2854 . 2 (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑛 ∈ ℕ (vol*‘𝐴))
845, 83breqtrd 5059 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  Vcvv 3444  csb 3831  wss 3884  c0 4246   ciun 4884   class class class wbr 5033  cmpt 5113  dom cdm 5523  ran crn 5524   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  supcsup 8892  cr 10529  0cc0 10530  1c1 10531   + caddc 10533  *cxr 10667   < clt 10668  cle 10669  cn 11629  cuz 12235  ...cfz 12889  seqcseq 13368  cli 14837  Σcsu 15038  vol*covol 24070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cc 9850  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-ioo 12734  df-ico 12736  df-fz 12890  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-rlim 14842  df-sum 15039  df-ovol 24072
This theorem is referenced by:  ovoliunnul  24115  vitalilem5  24220
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