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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfd | Structured version Visualization version GIF version |
Description: A sufficient condition for "πΉ being a real-valued measurable function w.r.t. to the sigma-algebra π". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
issmfd.a | β’ β²ππ |
issmfd.s | β’ (π β π β SAlg) |
issmfd.d | β’ (π β π· β βͺ π) |
issmfd.f | β’ (π β πΉ:π·βΆβ) |
issmfd.p | β’ ((π β§ π β β) β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)) |
Ref | Expression |
---|---|
issmfd | β’ (π β πΉ β (SMblFnβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmfd.f | . . . . 5 β’ (π β πΉ:π·βΆβ) | |
2 | 1 | fdmd 6733 | . . . 4 β’ (π β dom πΉ = π·) |
3 | issmfd.d | . . . 4 β’ (π β π· β βͺ π) | |
4 | 2, 3 | eqsstrd 4018 | . . 3 β’ (π β dom πΉ β βͺ π) |
5 | 1 | ffdmd 6754 | . . 3 β’ (π β πΉ:dom πΉβΆβ) |
6 | issmfd.a | . . . 4 β’ β²ππ | |
7 | issmfd.p | . . . . . 6 β’ ((π β§ π β β) β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)) | |
8 | 2 | rabeqdv 3444 | . . . . . . . 8 β’ (π β {π₯ β dom πΉ β£ (πΉβπ₯) < π} = {π₯ β π· β£ (πΉβπ₯) < π}) |
9 | 2 | oveq2d 7436 | . . . . . . . 8 β’ (π β (π βΎt dom πΉ) = (π βΎt π·)) |
10 | 8, 9 | eleq12d 2823 | . . . . . . 7 β’ (π β ({π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ) β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·))) |
11 | 10 | adantr 480 | . . . . . 6 β’ ((π β§ π β β) β ({π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ) β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·))) |
12 | 7, 11 | mpbird 257 | . . . . 5 β’ ((π β§ π β β) β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ)) |
13 | 12 | ex 412 | . . . 4 β’ (π β (π β β β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ))) |
14 | 6, 13 | ralrimi 3251 | . . 3 β’ (π β βπ β β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ)) |
15 | 4, 5, 14 | 3jca 1126 | . 2 β’ (π β (dom πΉ β βͺ π β§ πΉ:dom πΉβΆβ β§ βπ β β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ))) |
16 | issmfd.s | . . 3 β’ (π β π β SAlg) | |
17 | eqid 2728 | . . 3 β’ dom πΉ = dom πΉ | |
18 | 16, 17 | issmf 46116 | . 2 β’ (π β (πΉ β (SMblFnβπ) β (dom πΉ β βͺ π β§ πΉ:dom πΉβΆβ β§ βπ β β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ)))) |
19 | 15, 18 | mpbird 257 | 1 β’ (π β πΉ β (SMblFnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1085 β²wnf 1778 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 11138 < clt 11279 βΎt crest 17402 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 11195 ax-resscn 11196 ax-pre-lttri 11213 ax-pre-lttrn 11214 |
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 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-ioo 13361 df-ico 13363 df-smblfn 46084 |
This theorem is referenced by: sssmf 46126 mbfresmf 46127 cnfsmf 46128 incsmf 46130 smfsssmf 46131 smfres 46178 smfco 46190 |
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