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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 25623 | . 2 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))) | |
2 | 1 | simplbi 496 | 1 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2098 ∖ cdif 3946 {csn 4632 ◡ccnv 5681 dom cdm 5682 ran crn 5683 “ cima 5685 ⟶wf 6549 ‘cfv 6553 Fincfn 8970 ℝcr 11145 0cc0 11146 volcvol 25412 MblFncmbf 25563 ∫1citg1 25564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 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-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-sum 15673 df-itg1 25569 |
This theorem is referenced by: i1fima 25627 i1fadd 25644 mbfmullem2 25674 itg2monolem1 25700 itg2i1fseq 25705 i1fibl 25757 itg2addnclem2 37178 ftc1anclem4 37202 ftc1anclem5 37203 ftc1anclem8 37206 |
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