Theorem mbfi1fseq 24015

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	⊢ (𝜑 → 𝐹 ∈ MblFn)
mbfi1fseq.2	⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))

Assertion

Ref	Expression
mbfi1fseq	⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘𝑟 ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))

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.

Step	Hyp	Ref	Expression
1		mbfi1fseq.1	. 2 ⊢ (𝜑 → 𝐹 ∈ MblFn)
2		mbfi1fseq.2	. 2 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
3		oveq2 6978	. . . . . 6 ⊢ (𝑗 = 𝑘 → (2↑𝑗) = (2↑𝑘))
4	3	oveq2d 6986	. . . . 5 ⊢ (𝑗 = 𝑘 → ((𝐹‘𝑧) · (2↑𝑗)) = ((𝐹‘𝑧) · (2↑𝑘)))
5	4	fveq2d 6497	. . . 4 ⊢ (𝑗 = 𝑘 → (⌊‘((𝐹‘𝑧) · (2↑𝑗))) = (⌊‘((𝐹‘𝑧) · (2↑𝑘))))
6	5, 3	oveq12d 6988	. . 3 ⊢ (𝑗 = 𝑘 → ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)) = ((⌊‘((𝐹‘𝑧) · (2↑𝑘))) / (2↑𝑘)))
7		fveq2 6493	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
8	7	fvoveq1d 6992	. . . 4 ⊢ (𝑧 = 𝑦 → (⌊‘((𝐹‘𝑧) · (2↑𝑘))) = (⌊‘((𝐹‘𝑦) · (2↑𝑘))))
9	8	oveq1d 6985	. . 3 ⊢ (𝑧 = 𝑦 → ((⌊‘((𝐹‘𝑧) · (2↑𝑘))) / (2↑𝑘)) = ((⌊‘((𝐹‘𝑦) · (2↑𝑘))) / (2↑𝑘)))
10	6, 9	cbvmpov 7059	. 2 ⊢ (𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗))) = (𝑘 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑘))) / (2↑𝑘)))
11		eleq1w 2842	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝑚[,]𝑚)))
12		oveq2 6978	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) = (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥))
13	12	breq1d 4933	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚 ↔ (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚))
14	13, 12	ifbieq1d 4367	. . . . . 6 ⊢ (𝑦 = 𝑥 → if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦), 𝑚) = if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚))
15	11, 14	ifbieq1d 4367	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦), 𝑚), 0) = if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚), 0))
16	15	cbvmptv 5022	. . . 4 ⊢ (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦), 𝑚), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚), 0))
17		negeq 10670	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → -𝑚 = -𝑘)
18		id 22	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → 𝑚 = 𝑘)
19	17, 18	oveq12d 6988	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (-𝑚[,]𝑚) = (-𝑘[,]𝑘))
20	19	eleq2d 2845	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑥 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝑘[,]𝑘)))
21		oveq1 6977	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) = (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥))
22	21, 18	breq12d 4936	. . . . . . 7 ⊢ (𝑚 = 𝑘 → ((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚 ↔ (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘))
23	22, 21, 18	ifbieq12d 4371	. . . . . 6 ⊢ (𝑚 = 𝑘 → if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚) = if((𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘, (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑘))
24	20, 23	ifbieq1d 4367	. . . . 5 ⊢ (𝑚 = 𝑘 → if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚), 0) = if(𝑥 ∈ (-𝑘[,]𝑘), if((𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘, (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑘), 0))
25	24	mpteq2dv 5017	. . . 4 ⊢ (𝑚 = 𝑘 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑚), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑘[,]𝑘), if((𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘, (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑘), 0)))
26	16, 25	syl5eq 2820	. . 3 ⊢ (𝑚 = 𝑘 → (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦), 𝑚), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑘[,]𝑘), if((𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘, (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑘), 0)))
27	26	cbvmptv 5022	. 2 ⊢ (𝑚 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (-𝑚[,]𝑚), if((𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦) ≤ 𝑚, (𝑚(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑦), 𝑚), 0))) = (𝑘 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑘[,]𝑘), if((𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥) ≤ 𝑘, (𝑘(𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)))𝑥), 𝑘), 0)))
28	1, 2, 10, 27	mbfi1fseqlem6 24014	1 ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘𝑟 ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘𝑟 ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1068  ∃wex 1742   ∈ wcel 2048  ∀wral 3082  ifcif 4344   class class class wbr 4923   ↦ cmpt 5002  dom cdm 5400  ⟶wf 6178  ‘cfv 6182  (class class class)co 6970   ∈ cmpo 6972   ∘_𝑟 cofr 7220  ℝcr 10326  0cc0 10327  1c1 10328   + caddc 10330   · cmul 10332  +∞cpnf 10463   ≤ cle 10467  -cneg 10663   / cdiv 11090  ℕcn 11431  2c2 11488  [,)cico 12549  [,]cicc 12550  ⌊cfl 12968  ↑cexp 13237   ⇝ cli 14692  MblFncmbf 23908  ∫₁citg1 23909  0_𝑝c0p 23963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-inf2 8890  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404  ax-pre-sup 10405

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-ofr 7222  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-2o 7898  df-oadd 7901  df-er 8081  df-map 8200  df-pm 8201  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-fi 8662  df-sup 8693  df-inf 8694  df-oi 8761  df-dju 9116  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-div 11091  df-nn 11432  df-2 11496  df-3 11497  df-n0 11701  df-z 11787  df-uz 12052  df-q 12156  df-rp 12198  df-xneg 12317  df-xadd 12318  df-xmul 12319  df-ioo 12551  df-ico 12553  df-icc 12554  df-fz 12702  df-fzo 12843  df-fl 12970  df-seq 13178  df-exp 13238  df-hash 13499  df-cj 14309  df-re 14310  df-im 14311  df-sqrt 14445  df-abs 14446  df-clim 14696  df-rlim 14697  df-sum 14894  df-rest 16542  df-topgen 16563  df-psmet 20229  df-xmet 20230  df-met 20231  df-bl 20232  df-mopn 20233  df-top 21196  df-topon 21213  df-bases 21248  df-cmp 21689  df-ovol 23758  df-vol 23759  df-mbf 23913  df-itg1 23914  df-0p 23964

This theorem is referenced by:  mbfi1flimlem  24016  itg2add  24053