Theorem mbfi1fseq 25742

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 7432	. . . . . 6 ⊢ (𝑗 = 𝑘 → (2↑𝑗) = (2↑𝑘))
4	3	oveq2d 7440	. . . . 5 ⊢ (𝑗 = 𝑘 → ((𝐹‘𝑧) · (2↑𝑗)) = ((𝐹‘𝑧) · (2↑𝑘)))
5	4	fveq2d 6905	. . . 4 ⊢ (𝑗 = 𝑘 → (⌊‘((𝐹‘𝑧) · (2↑𝑗))) = (⌊‘((𝐹‘𝑧) · (2↑𝑘))))
6	5, 3	oveq12d 7442	. . 3 ⊢ (𝑗 = 𝑘 → ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗)) = ((⌊‘((𝐹‘𝑧) · (2↑𝑘))) / (2↑𝑘)))
7		fveq2 6901	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
8	7	fvoveq1d 7446	. . . 4 ⊢ (𝑧 = 𝑦 → (⌊‘((𝐹‘𝑧) · (2↑𝑘))) = (⌊‘((𝐹‘𝑦) · (2↑𝑘))))
9	8	oveq1d 7439	. . 3 ⊢ (𝑧 = 𝑦 → ((⌊‘((𝐹‘𝑧) · (2↑𝑘))) / (2↑𝑘)) = ((⌊‘((𝐹‘𝑦) · (2↑𝑘))) / (2↑𝑘)))
10	6, 9	cbvmpov 7520	. 2 ⊢ (𝑗 ∈ ℕ, 𝑧 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑧) · (2↑𝑗))) / (2↑𝑗))) = (𝑘 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑘))) / (2↑𝑘)))
11		eleq1w 2809	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝑚[,]𝑚)))
12		oveq2 7432	. . . . . . . 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 5163	. . . . . . 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 4557	. . . . . 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 4557	. . . . 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 5266	. . . 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 11502	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → -𝑚 = -𝑘)
18		id 22	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → 𝑚 = 𝑘)
19	17, 18	oveq12d 7442	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (-𝑚[,]𝑚) = (-𝑘[,]𝑘))
20	19	eleq2d 2812	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑥 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝑘[,]𝑘)))
21		oveq1 7431	. . . . . . . 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 5166	. . . . . . 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 4561	. . . . . 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 4557	. . . . 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 5255	. . . 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 2778	. . 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 5266	. 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 25741	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 394   ∧ w3a 1084  ∃wex 1774   ∈ wcel 2099  ∀wral 3051  ifcif 4533   class class class wbr 5153   ↦ cmpt 5236  dom cdm 5682  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424   ∈ cmpo 7426   ∘_r cofr 7689  ℝcr 11157  0cc0 11158  1c1 11159   + caddc 11161   · cmul 11163  +∞cpnf 11295   ≤ cle 11299  -cneg 11495   / cdiv 11921  ℕcn 12264  2c2 12319  [,)cico 13380  [,]cicc 13381  ⌊cfl 13810  ↑cexp 14081   ⇝ cli 15486  MblFncmbf 25634  ∫₁citg1 25635  0_𝑝c0p 25689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-ofr 7691  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fi 9454  df-sup 9485  df-inf 9486  df-oi 9553  df-dju 9944  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-q 12985  df-rp 13029  df-xneg 13146  df-xadd 13147  df-xmul 13148  df-ioo 13382  df-ico 13384  df-icc 13385  df-fz 13539  df-fzo 13682  df-fl 13812  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-clim 15490  df-rlim 15491  df-sum 15691  df-rest 17437  df-topgen 17458  df-psmet 21335  df-xmet 21336  df-met 21337  df-bl 21338  df-mopn 21339  df-top 22887  df-topon 22904  df-bases 22940  df-cmp 23382  df-ovol 25484  df-vol 25485  df-mbf 25639  df-itg1 25640  df-0p 25690

This theorem is referenced by:  mbfi1flimlem  25743  itg2add  25780