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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmff | 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 |
---|---|
issmff.x | β’ β²π₯πΉ |
issmff.s | β’ (π β π β SAlg) |
issmff.d | β’ π· = dom πΉ |
Ref | Expression |
---|---|
issmff | 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 | issmff.s | . . 3 β’ (π β π β SAlg) | |
2 | issmff.d | . . 3 β’ π· = dom πΉ | |
3 | 1, 2 | issmf 45055 | . 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 | nfcv 2904 | . . . . . . 7 β’ β²π¦π· | |
5 | issmff.x | . . . . . . . . 9 β’ β²π₯πΉ | |
6 | 5 | nfdm 5907 | . . . . . . . 8 β’ β²π₯dom πΉ |
7 | 2, 6 | nfcxfr 2902 | . . . . . . 7 β’ β²π₯π· |
8 | nfcv 2904 | . . . . . . . . 9 β’ β²π₯π¦ | |
9 | 5, 8 | nffv 6853 | . . . . . . . 8 β’ β²π₯(πΉβπ¦) |
10 | nfcv 2904 | . . . . . . . 8 β’ β²π₯ < | |
11 | nfcv 2904 | . . . . . . . 8 β’ β²π₯π | |
12 | 9, 10, 11 | nfbr 5153 | . . . . . . 7 β’ β²π₯(πΉβπ¦) < π |
13 | nfv 1918 | . . . . . . 7 β’ β²π¦(πΉβπ₯) < π | |
14 | fveq2 6843 | . . . . . . . 8 β’ (π¦ = π₯ β (πΉβπ¦) = (πΉβπ₯)) | |
15 | 14 | breq1d 5116 | . . . . . . 7 β’ (π¦ = π₯ β ((πΉβπ¦) < π β (πΉβπ₯) < π)) |
16 | 4, 7, 12, 13, 15 | cbvrabw 3438 | . . . . . 6 β’ {π¦ β π· β£ (πΉβπ¦) < π} = {π₯ β π· β£ (πΉβπ₯) < π} |
17 | 16 | eleq1i 2825 | . . . . 5 β’ ({π¦ β π· β£ (πΉβπ¦) < π} β (π βΎt π·) β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)) |
18 | 17 | ralbii 3093 | . . . 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·)) |
19 | 18 | 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·))) |
20 | 19 | 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·)))) |
21 | 3, 20 | 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 β²wnfc 2884 βwral 3061 {crab 3406 β wss 3911 βͺ cuni 4866 class class class wbr 5106 dom cdm 5634 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcr 11055 < clt 11194 βΎt crest 17307 SAlgcsalg 44635 SMblFncsmblfn 45022 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8651 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-ioo 13274 df-ico 13276 df-smblfn 45023 |
This theorem is referenced by: smfpreimaltf 45063 issmfdf 45064 |
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