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Mirrors > Home > MPE Home > Th. List > iblmbf | Structured version Visualization version GIF version |
Description: An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.) |
Ref | Expression |
---|---|
iblmbf | ⊢ (𝐹 ∈ 𝐿1 → 𝐹 ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ibl 23197 | . . 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 | ssrab2 3650 | . . 3 ⊢ {𝑓 ∈ MblFn ∣ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ} ⊆ MblFn | |
3 | 1, 2 | eqsstri 3598 | . 2 ⊢ 𝐿1 ⊆ MblFn |
4 | 3 | sseli 3564 | 1 ⊢ (𝐹 ∈ 𝐿1 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⦋csb 3499 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 ici 9817 ≤ cle 9954 / cdiv 10563 3c3 10948 ...cfz 12197 ↑cexp 12722 ℜcre 13685 MblFncmbf 23189 ∫2citg2 23191 𝐿1cibl 23192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-in 3547 df-ss 3554 df-ibl 23197 |
This theorem is referenced by: iblcnlem 23361 itgcnlem 23362 itgcnval 23372 itgre 23373 itgim 23374 iblneg 23375 itgneg 23376 iblss 23377 iblss2 23378 itgge0 23383 itgss3 23387 itgless 23389 iblsub 23394 itgadd 23397 itgsub 23398 itgfsum 23399 iblabs 23401 iblmulc2 23403 itgmulc2 23406 itgabs 23407 itgsplit 23408 bddmulibl 23411 itggt0 23414 itgcn 23415 ditgswap 23429 ditgsplitlem 23430 ftc1a 23604 itgsubstlem 23615 iblulm 23965 itgulm 23966 ibladdnc 32637 itgaddnclem1 32638 itgaddnclem2 32639 itgaddnc 32640 iblsubnc 32641 itgsubnc 32642 iblabsnclem 32643 iblabsnc 32644 iblmulc2nc 32645 itgmulc2nclem2 32647 itgmulc2nc 32648 itgabsnc 32649 ftc1cnnclem 32653 ftc1anclem2 32656 ftc1anclem4 32658 ftc1anclem5 32659 ftc1anclem6 32660 ftc1anclem8 32662 |
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