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Mirrors > Home > MPE Home > Th. List > i1fmbf | Structured version Visualization version GIF version |
Description: Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
i1fmbf | ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isi1f 23247 | . 2 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))) | |
2 | 1 | simplbi 475 | 1 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 ∈ wcel 1977 ∖ cdif 3537 {csn 4125 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 ⟶wf 5800 ‘cfv 5804 Fincfn 7841 ℝcr 9814 0cc0 9815 volcvol 23039 MblFncmbf 23189 ∫1citg1 23190 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-sum 14265 df-itg1 23195 |
This theorem is referenced by: i1fima 23251 i1fadd 23268 mbfmullem2 23297 itg2monolem1 23323 itg2i1fseq 23328 i1fibl 23380 itg2addnclem2 32632 ftc1anclem4 32658 ftc1anclem5 32659 ftc1anclem8 32662 |
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