MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovoliun2 Structured version   Visualization version   GIF version

Theorem ovoliun2 23483
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 23482.) (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 23482 . 2 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
6 nnuz 11937 . . . . . . . 8 ℕ = (ℤ‘1)
7 1zzd 11670 . . . . . . . 8 (𝜑 → 1 ∈ ℤ)
8 fvex 6418 . . . . . . . . . . 11 (vol*‘𝑚 / 𝑛𝐴) ∈ V
9 nfcv 2947 . . . . . . . . . . . . . 14 𝑚(vol*‘𝐴)
10 nfcv 2947 . . . . . . . . . . . . . . 15 𝑛vol*
11 nfcsb1v 3741 . . . . . . . . . . . . . . 15 𝑛𝑚 / 𝑛𝐴
1210, 11nffv 6415 . . . . . . . . . . . . . 14 𝑛(vol*‘𝑚 / 𝑛𝐴)
13 csbeq1a 3734 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
1413fveq2d 6409 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘𝑚 / 𝑛𝐴))
159, 12, 14cbvmpt 4939 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
162, 15eqtri 2827 . . . . . . . . . . . 12 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
1716fvmpt2 6509 . . . . . . . . . . 11 ((𝑚 ∈ ℕ ∧ (vol*‘𝑚 / 𝑛𝐴) ∈ V) → (𝐺𝑚) = (vol*‘𝑚 / 𝑛𝐴))
188, 17mpan2 674 . . . . . . . . . 10 (𝑚 ∈ ℕ → (𝐺𝑚) = (vol*‘𝑚 / 𝑛𝐴))
1918adantl 469 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) = (vol*‘𝑚 / 𝑛𝐴))
204ralrimiva 3153 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
219nfel1 2962 . . . . . . . . . . . 12 𝑚(vol*‘𝐴) ∈ ℝ
2212nfel1 2962 . . . . . . . . . . . 12 𝑛(vol*‘𝑚 / 𝑛𝐴) ∈ ℝ
2314eleq1d 2869 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ))
2421, 22, 23cbvral 3355 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
2520, 24sylib 209 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
2625r19.21bi 3119 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
2719, 26eqeltrd 2884 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → (𝐺𝑚) ∈ ℝ)
286, 7, 27serfre 13049 . . . . . . 7 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
291feq1i 6244 . . . . . . 7 (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
3028, 29sylibr 225 . . . . . 6 (𝜑𝑇:ℕ⟶ℝ)
3130frnd 6260 . . . . 5 (𝜑 → ran 𝑇 ⊆ ℝ)
32 1nn 11313 . . . . . . . 8 1 ∈ ℕ
3330fdmd 6262 . . . . . . . 8 (𝜑 → dom 𝑇 = ℕ)
3432, 33syl5eleqr 2891 . . . . . . 7 (𝜑 → 1 ∈ dom 𝑇)
3534ne0d 4120 . . . . . 6 (𝜑 → dom 𝑇 ≠ ∅)
36 dm0rn0 5540 . . . . . . 7 (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
3736necon3bii 3029 . . . . . 6 (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
3835, 37sylib 209 . . . . 5 (𝜑 → ran 𝑇 ≠ ∅)
39 ovoliun2.t . . . . . . . . 9 (𝜑𝑇 ∈ dom ⇝ )
401, 39syl5eqelr 2889 . . . . . . . 8 (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
416, 7, 19, 26, 40isumrecl 14715 . . . . . . 7 (𝜑 → Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
42 elfznn 12589 . . . . . . . . . . . . 13 (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
4342adantl 469 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → 𝑚 ∈ ℕ)
4443, 18syl 17 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐺𝑚) = (vol*‘𝑚 / 𝑛𝐴))
45 simpr 473 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
4645, 6syl6eleq 2894 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
47 simpl 470 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝜑)
4847, 42, 26syl2an 585 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
4948recnd 10350 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℂ)
5044, 46, 49fsumser 14680 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘𝑚 / 𝑛𝐴) = (seq1( + , 𝐺)‘𝑘))
511fveq1i 6406 . . . . . . . . . 10 (𝑇𝑘) = (seq1( + , 𝐺)‘𝑘)
5250, 51syl6eqr 2857 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘𝑚 / 𝑛𝐴) = (𝑇𝑘))
53 fzfid 12992 . . . . . . . . . . 11 (𝜑 → (1...𝑘) ∈ Fin)
54 elfznn 12589 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
5554ssriv 3799 . . . . . . . . . . . 12 (1...𝑘) ⊆ ℕ
5655a1i 11 . . . . . . . . . . 11 (𝜑 → (1...𝑘) ⊆ ℕ)
573ralrimiva 3153 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
58 nfv 2008 . . . . . . . . . . . . . . 15 𝑚 𝐴 ⊆ ℝ
59 nfcv 2947 . . . . . . . . . . . . . . . 16 𝑛
6011, 59nfss 3788 . . . . . . . . . . . . . . 15 𝑛𝑚 / 𝑛𝐴 ⊆ ℝ
6113sseq1d 3826 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ 𝑚 / 𝑛𝐴 ⊆ ℝ))
6258, 60, 61cbvral 3355 . . . . . . . . . . . . . 14 (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
6357, 62sylib 209 . . . . . . . . . . . . 13 (𝜑 → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
6463r19.21bi 3119 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
65 ovolge0 23458 . . . . . . . . . . . 12 (𝑚 / 𝑛𝐴 ⊆ ℝ → 0 ≤ (vol*‘𝑚 / 𝑛𝐴))
6664, 65syl 17 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 0 ≤ (vol*‘𝑚 / 𝑛𝐴))
676, 7, 53, 56, 19, 26, 66, 40isumless 14795 . . . . . . . . . 10 (𝜑 → Σ𝑚 ∈ (1...𝑘)(vol*‘𝑚 / 𝑛𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴))
6867adantr 468 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘𝑚 / 𝑛𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴))
6952, 68eqbrtrrd 4864 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑇𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴))
7069ralrimiva 3153 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴))
71 brralrspcev 4900 . . . . . . 7 ((Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ 𝑥)
7241, 70, 71syl2anc 575 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ 𝑥)
7330ffnd 6254 . . . . . . . 8 (𝜑𝑇 Fn ℕ)
74 breq1 4843 . . . . . . . . 9 (𝑧 = (𝑇𝑘) → (𝑧𝑥 ↔ (𝑇𝑘) ≤ 𝑥))
7574ralrn 6581 . . . . . . . 8 (𝑇 Fn ℕ → (∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ 𝑥))
7673, 75syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ 𝑥))
7776rexbidv 3239 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇𝑘) ≤ 𝑥))
7872, 77mpbird 248 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥)
79 supxrre 12371 . . . . 5 ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧𝑥) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
8031, 38, 78, 79syl3anc 1483 . . . 4 (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
816, 1, 7, 19, 26, 66, 72isumsup 14797 . . . 4 (𝜑 → Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) = sup(ran 𝑇, ℝ, < ))
8280, 81eqtr4d 2842 . . 3 (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴))
839, 12, 14cbvsumi 14646 . . 3 Σ𝑛 ∈ ℕ (vol*‘𝐴) = Σ𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴)
8482, 83syl6eqr 2857 . 2 (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑛 ∈ ℕ (vol*‘𝐴))
855, 84breqtrd 4866 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2158  wne 2977  wral 3095  wrex 3096  Vcvv 3390  csb 3725  wss 3766  c0 4113   ciun 4708   class class class wbr 4840  cmpt 4919  dom cdm 5308  ran crn 5309   Fn wfn 6093  wf 6094  cfv 6098  (class class class)co 6871  supcsup 8582  cr 10217  0cc0 10218  1c1 10219   + caddc 10221  *cxr 10355   < clt 10356  cle 10357  cn 11302  cuz 11900  ...cfz 12545  seqcseq 13020  cli 14434  Σcsu 14635  vol*covol 23439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176  ax-inf2 8782  ax-cc 9539  ax-cnex 10274  ax-resscn 10275  ax-1cn 10276  ax-icn 10277  ax-addcl 10278  ax-addrcl 10279  ax-mulcl 10280  ax-mulrcl 10281  ax-mulcom 10282  ax-addass 10283  ax-mulass 10284  ax-distr 10285  ax-i2m1 10286  ax-1ne0 10287  ax-1rid 10288  ax-rnegex 10289  ax-rrecex 10290  ax-cnre 10291  ax-pre-lttri 10292  ax-pre-lttrn 10293  ax-pre-ltadd 10294  ax-pre-mulgt0 10295  ax-pre-sup 10296
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-reu 3102  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-se 5268  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-isom 6107  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-1st 7395  df-2nd 7396  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-1o 7793  df-oadd 7797  df-er 7976  df-map 8091  df-pm 8092  df-en 8190  df-dom 8191  df-sdom 8192  df-fin 8193  df-sup 8584  df-inf 8585  df-oi 8651  df-card 9045  df-pnf 10358  df-mnf 10359  df-xr 10360  df-ltxr 10361  df-le 10362  df-sub 10550  df-neg 10551  df-div 10967  df-nn 11303  df-2 11360  df-3 11361  df-n0 11556  df-z 11640  df-uz 11901  df-q 12004  df-rp 12043  df-ioo 12393  df-ico 12395  df-fz 12546  df-fzo 12686  df-fl 12813  df-seq 13021  df-exp 13080  df-hash 13334  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-rlim 14439  df-sum 14636  df-ovol 23441
This theorem is referenced by:  ovoliunnul  23484  vitalilem5  23589
  Copyright terms: Public domain W3C validator