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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 24525 | . 2 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 ∈ wcel 2112 ∖ cdif 3850 {csn 4527 ◡ccnv 5535 dom cdm 5536 ran crn 5537 “ cima 5539 ⟶wf 6354 ‘cfv 6358 Fincfn 8604 ℝcr 10693 0cc0 10694 volcvol 24314 MblFncmbf 24465 ∫1citg1 24466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-sum 15215 df-itg1 24471 |
This theorem is referenced by: i1fima 24529 i1fadd 24546 mbfmullem2 24576 itg2monolem1 24602 itg2i1fseq 24607 i1fibl 24659 itg2addnclem2 35515 ftc1anclem4 35539 ftc1anclem5 35540 ftc1anclem8 35543 |
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