Theorem mbfi1fseq 25629

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 7398	. . . . . 6 ⊢ (𝑗 = 𝑘 → (2↑𝑗) = (2↑𝑘))
4	3	oveq2d 7406	. . . . 5 ⊢ (𝑗 = 𝑘 → ((𝐹‘𝑧) · (2↑𝑗)) = ((𝐹‘𝑧) · (2↑𝑘)))
5	4	fveq2d 6865	. . . 4 ⊢ (𝑗 = 𝑘 → (⌊‘((𝐹‘𝑧) · (2↑𝑗))) = (⌊‘((𝐹‘𝑧) · (2↑𝑘))))
6	5, 3	oveq12d 7408	. . 3 ⊢ (𝑗 = 𝑘 → ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)) = ((⌊‘((𝐹‘𝑧) · (2↑𝑘))) / (2↑𝑘)))
7		fveq2 6861	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
8	7	fvoveq1d 7412	. . . 4 ⊢ (𝑧 = 𝑦 → (⌊‘((𝐹‘𝑧) · (2↑𝑘))) = (⌊‘((𝐹‘𝑦) · (2↑𝑘))))
9	8	oveq1d 7405	. . 3 ⊢ (𝑧 = 𝑦 → ((⌊‘((𝐹‘𝑧) · (2↑𝑘))) / (2↑𝑘)) = ((⌊‘((𝐹‘𝑦) · (2↑𝑘))) / (2↑𝑘)))
10	6, 9	cbvmpov 7487	. 2 ⊢ (𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗))) = (𝑘 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑘))) / (2↑𝑘)))
11		eleq1w 2812	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝑚[,]𝑚)))
12		oveq2 7398	. . . . . . . 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 5120	. . . . . . 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 4516	. . . . . 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 4516	. . . . 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 5214	. . . 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 11420	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → -𝑚 = -𝑘)
18		id 22	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → 𝑚 = 𝑘)
19	17, 18	oveq12d 7408	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (-𝑚[,]𝑚) = (-𝑘[,]𝑘))
20	19	eleq2d 2815	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑥 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝑘[,]𝑘)))
21		oveq1 7397	. . . . . . . 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 5123	. . . . . . 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 4520	. . . . . 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 4516	. . . . 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 5204	. . . 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	eqtrid 2777	. . 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 5214	. 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 25628	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 395   ∧ w3a 1086  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ifcif 4491   class class class wbr 5110   ↦ cmpt 5191  dom cdm 5641  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   ∘_r cofr 7655  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078   · cmul 11080  +∞cpnf 11212   ≤ cle 11216  -cneg 11413   / cdiv 11842  ℕcn 12193  2c2 12248  [,)cico 13315  [,]cicc 13316  ⌊cfl 13759  ↑cexp 14033   ⇝ cli 15457  MblFncmbf 25522  ∫₁citg1 25523  0_𝑝c0p 25577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-ofr 7657  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fi 9369  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-ioo 13317  df-ico 13319  df-icc 13320  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-rlim 15462  df-sum 15660  df-rest 17392  df-topgen 17413  df-psmet 21263  df-xmet 21264  df-met 21265  df-bl 21266  df-mopn 21267  df-top 22788  df-topon 22805  df-bases 22840  df-cmp 23281  df-ovol 25372  df-vol 25373  df-mbf 25527  df-itg1 25528  df-0p 25578

This theorem is referenced by:  mbfi1flimlem  25630  itg2add  25667