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Theorem mbfi1fseq 23211
Description: A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function 𝐺 and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
mbfi1fseq.1 (𝜑𝐹 ∈ MblFn)
mbfi1fseq.2 (𝜑𝐹:ℝ⟶(0[,)+∞))
Assertion
Ref Expression
mbfi1fseq (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
Distinct variable groups:   𝑔,𝑛,𝑥,𝐹   𝜑,𝑛,𝑥
Allowed substitution hint:   𝜑(𝑔)

Proof of Theorem mbfi1fseq
Dummy variables 𝑗 𝑘 𝑚 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfi1fseq.1 . 2 (𝜑𝐹 ∈ MblFn)
2 mbfi1fseq.2 . 2 (𝜑𝐹:ℝ⟶(0[,)+∞))
3 oveq2 6535 . . . . . 6 (𝑗 = 𝑘 → (2↑𝑗) = (2↑𝑘))
43oveq2d 6543 . . . . 5 (𝑗 = 𝑘 → ((𝐹𝑧) · (2↑𝑗)) = ((𝐹𝑧) · (2↑𝑘)))
54fveq2d 6092 . . . 4 (𝑗 = 𝑘 → (⌊‘((𝐹𝑧) · (2↑𝑗))) = (⌊‘((𝐹𝑧) · (2↑𝑘))))
65, 3oveq12d 6545 . . 3 (𝑗 = 𝑘 → ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)) = ((⌊‘((𝐹𝑧) · (2↑𝑘))) / (2↑𝑘)))
7 fveq2 6088 . . . . . 6 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
87oveq1d 6542 . . . . 5 (𝑧 = 𝑦 → ((𝐹𝑧) · (2↑𝑘)) = ((𝐹𝑦) · (2↑𝑘)))
98fveq2d 6092 . . . 4 (𝑧 = 𝑦 → (⌊‘((𝐹𝑧) · (2↑𝑘))) = (⌊‘((𝐹𝑦) · (2↑𝑘))))
109oveq1d 6542 . . 3 (𝑧 = 𝑦 → ((⌊‘((𝐹𝑧) · (2↑𝑘))) / (2↑𝑘)) = ((⌊‘((𝐹𝑦) · (2↑𝑘))) / (2↑𝑘)))
116, 10cbvmpt2v 6611 . 2 (𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗))) = (𝑘 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑘))) / (2↑𝑘)))
12 eleq1 2675 . . . . . 6 (𝑦 = 𝑥 → (𝑦 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝑚[,]𝑚)))
13 oveq2 6535 . . . . . . . 8 (𝑦 = 𝑥 → (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) = (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥))
1413breq1d 4587 . . . . . . 7 (𝑦 = 𝑥 → ((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚 ↔ (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚))
1514, 13ifbieq1d 4058 . . . . . 6 (𝑦 = 𝑥 → if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦), 𝑚) = if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚))
1612, 15ifbieq1d 4058 . . . . 5 (𝑦 = 𝑥 → if(𝑦 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦), 𝑚), 0) = if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚), 0))
1716cbvmptv 4672 . . . 4 (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦), 𝑚), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚), 0))
18 negeq 10124 . . . . . . . 8 (𝑚 = 𝑘 → -𝑚 = -𝑘)
19 id 22 . . . . . . . 8 (𝑚 = 𝑘𝑚 = 𝑘)
2018, 19oveq12d 6545 . . . . . . 7 (𝑚 = 𝑘 → (-𝑚[,]𝑚) = (-𝑘[,]𝑘))
2120eleq2d 2672 . . . . . 6 (𝑚 = 𝑘 → (𝑥 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝑘[,]𝑘)))
22 oveq1 6534 . . . . . . . 8 (𝑚 = 𝑘 → (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) = (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥))
2322, 19breq12d 4590 . . . . . . 7 (𝑚 = 𝑘 → ((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚 ↔ (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘))
2423, 22, 19ifbieq12d 4062 . . . . . 6 (𝑚 = 𝑘 → if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚) = if((𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘, (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑘))
2521, 24ifbieq1d 4058 . . . . 5 (𝑚 = 𝑘 → if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚), 0) = if(𝑥 ∈ (-𝑘[,]𝑘), if((𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘, (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑘), 0))
2625mpteq2dv 4667 . . . 4 (𝑚 = 𝑘 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑘[,]𝑘), if((𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘, (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑘), 0)))
2717, 26syl5eq 2655 . . 3 (𝑚 = 𝑘 → (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦), 𝑚), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑘[,]𝑘), if((𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘, (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑘), 0)))
2827cbvmptv 4672 . 2 (𝑚 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦), 𝑚), 0))) = (𝑘 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑘[,]𝑘), if((𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘, (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑘), 0)))
291, 2, 11, 28mbfi1fseqlem6 23210 1 (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030  wex 1694  wcel 1976  wral 2895  ifcif 4035   class class class wbr 4577  cmpt 4637  dom cdm 5028  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  𝑟 cofr 6771  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   · cmul 9797  +∞cpnf 9927  cle 9931  -cneg 10118   / cdiv 10533  cn 10867  2c2 10917  [,)cico 12004  [,]cicc 12005  cfl 12408  cexp 12677  cli 14009  MblFncmbf 23106  1citg1 23107  0𝑝c0p 23159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-ofr 6773  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-ioo 12006  df-ico 12008  df-icc 12009  df-fz 12153  df-fzo 12290  df-fl 12410  df-seq 12619  df-exp 12678  df-hash 12935  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-clim 14013  df-rlim 14014  df-sum 14211  df-rest 15852  df-topgen 15873  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-top 20463  df-bases 20464  df-topon 20465  df-cmp 20942  df-ovol 22957  df-vol 22958  df-mbf 23111  df-itg1 23112  df-0p 23160
This theorem is referenced by:  mbfi1flimlem  23212  itg2add  23249
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