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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 45042 | . 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 5114 | . . . . . . . 8 β’ (π = π β ((πΉβπ¦) < π β (πΉβπ¦) < π)) | |
5 | 4 | rabbidv 3418 | . . . . . . 7 β’ (π = π β {π¦ β π· β£ (πΉβπ¦) < π} = {π¦ β π· β£ (πΉβπ¦) < π}) |
6 | 5 | eleq1d 2823 | . . . . . 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 6847 | . . . . . . . . . 10 β’ (π¦ = π₯ β (πΉβπ¦) = (πΉβπ₯)) | |
8 | 7 | breq1d 5120 | . . . . . . . . 9 β’ (π¦ = π₯ β ((πΉβπ¦) < π β (πΉβπ₯) < π)) |
9 | 8 | cbvrabv 3420 | . . . . . . . 8 β’ {π¦ β π· β£ (πΉβπ¦) < π} = {π₯ β π· β£ (πΉβπ₯) < π} |
10 | 9 | eleq1i 2829 | . . . . . . 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 3228 | . . . 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 1160 | . . 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 1088 = wceq 1542 β wcel 2107 βwral 3065 {crab 3410 β wss 3915 βͺ cuni 4870 class class class wbr 5110 dom cdm 5638 βΆwf 6497 βcfv 6501 (class class class)co 7362 βcr 11057 < clt 11196 βΎt crest 17309 SAlgcsalg 44623 SMblFncsmblfn 45010 |
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-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 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-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-er 8655 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-ioo 13275 df-ico 13277 df-smblfn 45011 |
This theorem is referenced by: smfpreimalt 45046 smff 45047 smfdmss 45048 issmff 45049 issmfd 45050 issmflelem 45059 issmfgtlem 45070 issmfgelem 45084 |
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