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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 25657 | . . 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 4082 | . 2 ⊢ 𝐿1 ⊆ MblFn |
| 3 | 2 | sseli 3979 | 1 ⊢ (𝐹 ∈ 𝐿1 → 𝐹 ∈ MblFn) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ⦋csb 3899 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 ici 11157 ≤ cle 11296 / cdiv 11920 3c3 12322 ...cfz 13547 ↑cexp 14102 ℜcre 15136 MblFncmbf 25649 ∫2citg2 25651 𝐿1cibl 25652 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-ss 3968 df-ibl 25657 |
| This theorem is referenced by: iblcnlem 25824 itgcnlem 25825 itgcnval 25835 itgre 25836 itgim 25837 iblneg 25838 itgneg 25839 iblss 25840 iblss2 25841 itgge0 25846 itgss3 25850 itgless 25852 iblsub 25857 itgadd 25860 itgsub 25861 itgfsum 25862 iblabs 25864 iblmulc2 25866 itgmulc2 25869 itgabs 25870 itgsplit 25871 bddmulibl 25874 itggt0 25879 itgcn 25880 ditgswap 25894 ditgsplitlem 25895 ftc1a 26078 itgsubstlem 26089 iblulm 26450 itgulm 26451 ibladdnc 37684 itgaddnclem1 37685 itgaddnclem2 37686 itgaddnc 37687 iblsubnc 37688 itgsubnc 37689 iblabsnclem 37690 iblabsnc 37691 iblmulc2nc 37692 itgmulc2nclem2 37694 itgmulc2nc 37695 itgabsnc 37696 ftc1cnnclem 37698 ftc1anclem2 37701 ftc1anclem4 37703 ftc1anclem5 37704 ftc1anclem6 37705 ftc1anclem8 37707 |
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