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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmf | Structured version Visualization version GIF version |
Description: The predicate "πΉ is a real-valued measurable function w.r.t. to the sigma-algebra π". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of πΉ is required to be a subset of the underlying set of π. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
issmf.s | β’ (π β π β SAlg) |
issmf.d | β’ π· = dom πΉ |
Ref | Expression |
---|---|
issmf | β’ (π β (πΉ β (SMblFnβπ) β (π· β βͺ π β§ πΉ:π·βΆβ β§ βπ β β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmf.s | . . 3 β’ (π β π β SAlg) | |
2 | issmf.d | . . 3 β’ π· = dom πΉ | |
3 | 1, 2 | issmflem 46115 | . 2 β’ (π β (πΉ β (SMblFnβπ) β (π· β βͺ π β§ πΉ:π·βΆβ β§ βπ β β {π¦ β π· β£ (πΉβπ¦) < π} β (π βΎt π·)))) |
4 | breq2 5152 | . . . . . . . 8 β’ (π = π β ((πΉβπ¦) < π β (πΉβπ¦) < π)) | |
5 | 4 | rabbidv 3437 | . . . . . . 7 β’ (π = π β {π¦ β π· β£ (πΉβπ¦) < π} = {π¦ β π· β£ (πΉβπ¦) < π}) |
6 | 5 | eleq1d 2814 | . . . . . 6 β’ (π = π β ({π¦ β π· β£ (πΉβπ¦) < π} β (π βΎt π·) β {π¦ β π· β£ (πΉβπ¦) < π} β (π βΎt π·))) |
7 | fveq2 6897 | . . . . . . . . . 10 β’ (π¦ = π₯ β (πΉβπ¦) = (πΉβπ₯)) | |
8 | 7 | breq1d 5158 | . . . . . . . . 9 β’ (π¦ = π₯ β ((πΉβπ¦) < π β (πΉβπ₯) < π)) |
9 | 8 | cbvrabv 3439 | . . . . . . . 8 β’ {π¦ β π· β£ (πΉβπ¦) < π} = {π₯ β π· β£ (πΉβπ₯) < π} |
10 | 9 | eleq1i 2820 | . . . . . . 7 β’ ({π¦ β π· β£ (πΉβπ¦) < π} β (π βΎt π·) β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)) |
11 | 10 | a1i 11 | . . . . . 6 β’ (π = π β ({π¦ β π· β£ (πΉβπ¦) < π} β (π βΎt π·) β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·))) |
12 | 6, 11 | bitrd 279 | . . . . 5 β’ (π = π β ({π¦ β π· β£ (πΉβπ¦) < π} β (π βΎt π·) β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·))) |
13 | 12 | cbvralvw 3231 | . . . 4 β’ (βπ β β {π¦ β π· β£ (πΉβπ¦) < π} β (π βΎt π·) β βπ β β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)) |
14 | 13 | 3anbi3i 1157 | . . 3 β’ ((π· β βͺ π β§ πΉ:π·βΆβ β§ βπ β β {π¦ β π· β£ (πΉβπ¦) < π} β (π βΎt π·)) β (π· β βͺ π β§ πΉ:π·βΆβ β§ βπ β β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·))) |
15 | 14 | a1i 11 | . 2 β’ (π β ((π· β βͺ π β§ πΉ:π·βΆβ β§ βπ β β {π¦ β π· β£ (πΉβπ¦) < π} β (π βΎt π·)) β (π· β βͺ π β§ πΉ:π·βΆβ β§ βπ β β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)))) |
16 | 3, 15 | bitrd 279 | 1 β’ (π β (πΉ β (SMblFnβπ) β (π· β βͺ π β§ πΉ:π·βΆβ β§ βπ β β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1085 = wceq 1534 β wcel 2099 βwral 3058 {crab 3429 β wss 3947 βͺ cuni 4908 class class class wbr 5148 dom cdm 5678 βΆwf 6544 βcfv 6548 (class class class)co 7420 βcr 11137 < clt 11278 βΎt crest 17401 SAlgcsalg 45696 SMblFncsmblfn 46083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8724 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-ioo 13360 df-ico 13362 df-smblfn 46084 |
This theorem is referenced by: smfpreimalt 46119 smff 46120 smfdmss 46121 issmff 46122 issmfd 46123 issmflelem 46132 issmfgtlem 46143 issmfgelem 46157 |
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