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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 24274 | . 2 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2110 ∖ cdif 3932 {csn 4566 ◡ccnv 5553 dom cdm 5554 ran crn 5555 “ cima 5557 ⟶wf 6350 ‘cfv 6354 Fincfn 8508 ℝcr 10535 0cc0 10536 volcvol 24063 MblFncmbf 24214 ∫1citg1 24215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-sum 15042 df-itg1 24220 |
This theorem is referenced by: i1fima 24278 i1fadd 24295 mbfmullem2 24324 itg2monolem1 24350 itg2i1fseq 24355 i1fibl 24407 itg2addnclem2 34943 ftc1anclem4 34969 ftc1anclem5 34970 ftc1anclem8 34973 |
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