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Mirrors > Home > MPE Home > Th. List > isi1f | Structured version Visualization version GIF version |
Description: The predicate "πΉ is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom πΉ β dom β«1 to represent this concept because β«1 is the first preparation function for our final definition β« (see df-itg 25003); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
isi1f | β’ (πΉ β dom β«1 β (πΉ β MblFn β§ (πΉ:ββΆβ β§ ran πΉ β Fin β§ (volβ(β‘πΉ β (β β {0}))) β β))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq1 6654 | . . 3 β’ (π = πΉ β (π:ββΆβ β πΉ:ββΆβ)) | |
2 | rneq 5896 | . . . 4 β’ (π = πΉ β ran π = ran πΉ) | |
3 | 2 | eleq1d 2823 | . . 3 β’ (π = πΉ β (ran π β Fin β ran πΉ β Fin)) |
4 | cnveq 5834 | . . . . . 6 β’ (π = πΉ β β‘π = β‘πΉ) | |
5 | 4 | imaeq1d 6017 | . . . . 5 β’ (π = πΉ β (β‘π β (β β {0})) = (β‘πΉ β (β β {0}))) |
6 | 5 | fveq2d 6851 | . . . 4 β’ (π = πΉ β (volβ(β‘π β (β β {0}))) = (volβ(β‘πΉ β (β β {0})))) |
7 | 6 | eleq1d 2823 | . . 3 β’ (π = πΉ β ((volβ(β‘π β (β β {0}))) β β β (volβ(β‘πΉ β (β β {0}))) β β)) |
8 | 1, 3, 7 | 3anbi123d 1437 | . 2 β’ (π = πΉ β ((π:ββΆβ β§ ran π β Fin β§ (volβ(β‘π β (β β {0}))) β β) β (πΉ:ββΆβ β§ ran πΉ β Fin β§ (volβ(β‘πΉ β (β β {0}))) β β))) |
9 | sumex 15579 | . . 3 β’ Ξ£π₯ β (ran π β {0})(π₯ Β· (volβ(β‘π β {π₯}))) β V | |
10 | df-itg1 25000 | . . 3 β’ β«1 = (π β {π β MblFn β£ (π:ββΆβ β§ ran π β Fin β§ (volβ(β‘π β (β β {0}))) β β)} β¦ Ξ£π₯ β (ran π β {0})(π₯ Β· (volβ(β‘π β {π₯})))) | |
11 | 9, 10 | dmmpti 6650 | . 2 β’ dom β«1 = {π β MblFn β£ (π:ββΆβ β§ ran π β Fin β§ (volβ(β‘π β (β β {0}))) β β)} |
12 | 8, 11 | elrab2 3653 | 1 β’ (πΉ β dom β«1 β (πΉ β MblFn β§ (πΉ:ββΆβ β§ ran πΉ β Fin β§ (volβ(β‘πΉ β (β β {0}))) β β))) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 {crab 3410 β cdif 3912 {csn 4591 β‘ccnv 5637 dom cdm 5638 ran crn 5639 β cima 5641 βΆwf 6497 βcfv 6501 (class class class)co 7362 Fincfn 8890 βcr 11057 0cc0 11058 Β· cmul 11063 Ξ£csu 15577 volcvol 24843 MblFncmbf 24994 β«1citg1 24995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-sum 15578 df-itg1 25000 |
This theorem is referenced by: i1fmbf 25055 i1ff 25056 i1frn 25057 i1fima2 25059 i1fd 25061 |
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