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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfdf | Structured version Visualization version GIF version |
Description: A sufficient condition for "πΉ being a measurable function w.r.t. to the sigma-algebra π". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
issmfdf.x | β’ β²π₯πΉ |
issmfdf.a | β’ β²ππ |
issmfdf.s | β’ (π β π β SAlg) |
issmfdf.d | β’ (π β π· β βͺ π) |
issmfdf.f | β’ (π β πΉ:π·βΆβ) |
issmfdf.p | β’ ((π β§ π β β) β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)) |
Ref | Expression |
---|---|
issmfdf | β’ (π β πΉ β (SMblFnβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmfdf.f | . . . . 5 β’ (π β πΉ:π·βΆβ) | |
2 | 1 | fdmd 6727 | . . . 4 β’ (π β dom πΉ = π·) |
3 | issmfdf.d | . . . 4 β’ (π β π· β βͺ π) | |
4 | 2, 3 | eqsstrd 4019 | . . 3 β’ (π β dom πΉ β βͺ π) |
5 | 1 | ffdmd 6747 | . . 3 β’ (π β πΉ:dom πΉβΆβ) |
6 | issmfdf.a | . . . 4 β’ β²ππ | |
7 | issmfdf.p | . . . . . 6 β’ ((π β§ π β β) β {π₯ β π· β£ (πΉβπ₯) < π} β (π βΎt π·)) | |
8 | issmfdf.x | . . . . . . . . . . 11 β’ β²π₯πΉ | |
9 | 8 | nfdm 5949 | . . . . . . . . . 10 β’ β²π₯dom πΉ |
10 | nfcv 2901 | . . . . . . . . . 10 β’ β²π₯π· | |
11 | 9, 10 | rabeqf 3464 | . . . . . . . . 9 β’ (dom πΉ = π· β {π₯ β dom πΉ β£ (πΉβπ₯) < π} = {π₯ β π· β£ (πΉβπ₯) < π}) |
12 | 2, 11 | syl 17 | . . . . . . . 8 β’ (π β {π₯ β dom πΉ β£ (πΉβπ₯) < π} = {π₯ β π· β£ (πΉβπ₯) < π}) |
13 | 2 | oveq2d 7427 | . . . . . . . 8 β’ (π β (π βΎt dom πΉ) = (π βΎt π·)) |
14 | 12, 13 | eleq12d 2825 | . . . . . . 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·))) |
15 | 14 | adantr 479 | . . . . . 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·))) |
16 | 7, 15 | mpbird 256 | . . . . 5 β’ ((π β§ π β β) β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ)) |
17 | 16 | ex 411 | . . . 4 β’ (π β (π β β β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ))) |
18 | 6, 17 | ralrimi 3252 | . . 3 β’ (π β βπ β β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ)) |
19 | 4, 5, 18 | 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))) |
20 | issmfdf.s | . . 3 β’ (π β π β SAlg) | |
21 | eqid 2730 | . . 3 β’ dom πΉ = dom πΉ | |
22 | 8, 20, 21 | issmff 45748 | . 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)))) |
23 | 19, 22 | mpbird 256 | 1 β’ (π β πΉ β (SMblFnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1085 = wceq 1539 β²wnf 1783 β wcel 2104 β²wnfc 2881 βwral 3059 {crab 3430 β wss 3947 βͺ cuni 4907 class class class wbr 5147 dom cdm 5675 βΆwf 6538 βcfv 6542 (class class class)co 7411 βcr 11111 < clt 11252 βΎt crest 17370 SAlgcsalg 45322 SMblFncsmblfn 45709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-ioo 13332 df-ico 13334 df-smblfn 45710 |
This theorem is referenced by: issmfdmpt 45762 smfconst 45763 |
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