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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 6721 | . . . 4 β’ (π β dom πΉ = π·) |
3 | issmfd.d | . . . 4 β’ (π β π· β βͺ π) | |
4 | 2, 3 | eqsstrd 4015 | . . 3 β’ (π β dom πΉ β βͺ π) |
5 | 1 | ffdmd 6741 | . . 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 3441 | . . . . . . . 8 β’ (π β {π₯ β dom πΉ β£ (πΉβπ₯) < π} = {π₯ β π· β£ (πΉβπ₯) < π}) |
9 | 2 | oveq2d 7420 | . . . . . . . 8 β’ (π β (π βΎt dom πΉ) = (π βΎt π·)) |
10 | 8, 9 | eleq12d 2821 | . . . . . . 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 3248 | . . 3 β’ (π β βπ β β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ)) |
15 | 4, 5, 14 | 3jca 1125 | . 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 2726 | . . 3 β’ dom πΉ = dom πΉ | |
18 | 16, 17 | issmf 45997 | . 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 1084 β²wnf 1777 β wcel 2098 βwral 3055 {crab 3426 β wss 3943 βͺ cuni 4902 class class class wbr 5141 dom cdm 5669 βΆwf 6532 βcfv 6536 (class class class)co 7404 βcr 11108 < clt 11249 βΎt crest 17373 SAlgcsalg 45577 SMblFncsmblfn 45964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-ioo 13331 df-ico 13333 df-smblfn 45965 |
This theorem is referenced by: sssmf 46007 mbfresmf 46008 cnfsmf 46009 incsmf 46011 smfsssmf 46012 smfres 46059 smfco 46071 |
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