Theorem iblmbf 24367

Description: An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)

Assertion

Ref	Expression
iblmbf	⊢ (𝐹 ∈ 𝐿1 → 𝐹 ∈ MblFn)

Proof of Theorem iblmbf

Dummy variables 𝑓 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ibl 24222	. . 3 ⊢ 𝐿1 = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ}
2	1	ssrab3 4056	. 2 ⊢ 𝐿1 ⊆ MblFn
3	2	sseli 3962	1 ⊢ (𝐹 ∈ 𝐿1 → 𝐹 ∈ MblFn)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  ∀wral 3138  ⦋csb 3882  ifcif 4466   class class class wbr 5065   ↦ cmpt 5145  dom cdm 5554  ‘cfv 6354  (class class class)co 7155  ℝcr 10535  0cc0 10536  ici 10538   ≤ cle 10675   / cdiv 11296  3c3 11692  ...cfz 12891  ↑cexp 13428  ℜcre 14455  MblFncmbf 24214  ∫₂citg2 24216  𝐿¹cibl 24217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-in 3942  df-ss 3951  df-ibl 24222

This theorem is referenced by:  iblcnlem  24388  itgcnlem  24389  itgcnval  24399  itgre  24400  itgim  24401  iblneg  24402  itgneg  24403  iblss  24404  iblss2  24405  itgge0  24410  itgss3  24414  itgless  24416  iblsub  24421  itgadd  24424  itgsub  24425  itgfsum  24426  iblabs  24428  iblmulc2  24430  itgmulc2  24433  itgabs  24434  itgsplit  24435  bddmulibl  24438  itggt0  24441  itgcn  24442  ditgswap  24456  ditgsplitlem  24457  ftc1a  24633  itgsubstlem  24644  iblulm  24994  itgulm  24995  ibladdnc  34948  itgaddnclem1  34949  itgaddnclem2  34950  itgaddnc  34951  iblsubnc  34952  itgsubnc  34953  iblabsnclem  34954  iblabsnc  34955  iblmulc2nc  34956  itgmulc2nclem2  34958  itgmulc2nc  34959  itgabsnc  34960  ftc1cnnclem  34964  ftc1anclem2  34967  ftc1anclem4  34969  ftc1anclem5  34970  ftc1anclem6  34971  ftc1anclem8  34973