Theorem mbfi1fseq 23488

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 6658	. . . . . 6 ⊢ (𝑗 = 𝑘 → (2↑𝑗) = (2↑𝑘))
4	3	oveq2d 6666	. . . . 5 ⊢ (𝑗 = 𝑘 → ((𝐹‘𝑧) · (2↑𝑗)) = ((𝐹‘𝑧) · (2↑𝑘)))
5	4	fveq2d 6195	. . . 4 ⊢ (𝑗 = 𝑘 → (⌊‘((𝐹‘𝑧) · (2↑𝑗))) = (⌊‘((𝐹‘𝑧) · (2↑𝑘))))
6	5, 3	oveq12d 6668	. . 3 ⊢ (𝑗 = 𝑘 → ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)) = ((⌊‘((𝐹‘𝑧) · (2↑𝑘))) / (2↑𝑘)))
7		fveq2 6191	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
8	7	oveq1d 6665	. . . . 5 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) · (2↑𝑘)) = ((𝐹‘𝑦) · (2↑𝑘)))
9	8	fveq2d 6195	. . . 4 ⊢ (𝑧 = 𝑦 → (⌊‘((𝐹‘𝑧) · (2↑𝑘))) = (⌊‘((𝐹‘𝑦) · (2↑𝑘))))
10	9	oveq1d 6665	. . 3 ⊢ (𝑧 = 𝑦 → ((⌊‘((𝐹‘𝑧) · (2↑𝑘))) / (2↑𝑘)) = ((⌊‘((𝐹‘𝑦) · (2↑𝑘))) / (2↑𝑘)))
11	6, 10	cbvmpt2v 6735	. 2 ⊢ (𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗))) = (𝑘 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑘))) / (2↑𝑘)))
12		eleq1 2689	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝑚[,]𝑚)))
13		oveq2 6658	. . . . . . . 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))
14	13	breq1d 4663	. . . . . . 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))
15	14, 13	ifbieq1d 4109	. . . . . 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))
16	12, 15	ifbieq1d 4109	. . . . 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))
17	16	cbvmptv 4750	. . . 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))
18		negeq 10273	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → -𝑚 = -𝑘)
19		id 22	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → 𝑚 = 𝑘)
20	18, 19	oveq12d 6668	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (-𝑚[,]𝑚) = (-𝑘[,]𝑘))
21	20	eleq2d 2687	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑥 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝑘[,]𝑘)))
22		oveq1 6657	. . . . . . . 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))
23	22, 19	breq12d 4666	. . . . . . 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))
24	23, 22, 19	ifbieq12d 4113	. . . . . 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))
25	21, 24	ifbieq1d 4109	. . . . 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))
26	25	mpteq2dv 4745	. . . 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)))
27	17, 26	syl5eq 2668	. . 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)))
28	27	cbvmptv 4750	. 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)))
29	1, 2, 11, 28	mbfi1fseqlem6 23487	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 384   ∧ w3a 1037  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ∘_𝑟 cofr 6896  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941  +∞cpnf 10071   ≤ cle 10075  -cneg 10267   / cdiv 10684  ℕcn 11020  2c2 11070  [,)cico 12177  [,]cicc 12178  ⌊cfl 12591  ↑cexp 12860   ⇝ cli 14215  MblFncmbf 23383  ∫₁citg1 23384  0_𝑝c0p 23436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-0p 23437

This theorem is referenced by:  mbfi1flimlem  23489  itg2add  23526