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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfsssmf | Structured version Visualization version GIF version |
Description: If a function is measurable w.r.t. to a sigma-algebra, then it is measurable w.r.t. to a larger sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfsssmf.r | β’ (π β π β SAlg) |
smfsssmf.s | β’ (π β π β SAlg) |
smfsssmf.i | β’ (π β π β π) |
smfsssmf.f | β’ (π β πΉ β (SMblFnβπ )) |
Ref | Expression |
---|---|
smfsssmf | β’ (π β πΉ β (SMblFnβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1909 | . 2 β’ β²ππ | |
2 | smfsssmf.s | . 2 β’ (π β π β SAlg) | |
3 | smfsssmf.r | . . . 4 β’ (π β π β SAlg) | |
4 | smfsssmf.f | . . . 4 β’ (π β πΉ β (SMblFnβπ )) | |
5 | eqid 2728 | . . . 4 β’ dom πΉ = dom πΉ | |
6 | 3, 4, 5 | smfdmss 46150 | . . 3 β’ (π β dom πΉ β βͺ π ) |
7 | smfsssmf.i | . . . 4 β’ (π β π β π) | |
8 | 7 | unissd 4922 | . . 3 β’ (π β βͺ π β βͺ π) |
9 | 6, 8 | sstrd 3992 | . 2 β’ (π β dom πΉ β βͺ π) |
10 | 3, 4, 5 | smff 46149 | . 2 β’ (π β πΉ:dom πΉβΆβ) |
11 | ssrest 23100 | . . . . 5 β’ ((π β SAlg β§ π β π) β (π βΎt dom πΉ) β (π βΎt dom πΉ)) | |
12 | 2, 7, 11 | syl2anc 582 | . . . 4 β’ (π β (π βΎt dom πΉ) β (π βΎt dom πΉ)) |
13 | 12 | adantr 479 | . . 3 β’ ((π β§ π β β) β (π βΎt dom πΉ) β (π βΎt dom πΉ)) |
14 | 3 | adantr 479 | . . . 4 β’ ((π β§ π β β) β π β SAlg) |
15 | 4 | adantr 479 | . . . 4 β’ ((π β§ π β β) β πΉ β (SMblFnβπ )) |
16 | simpr 483 | . . . 4 β’ ((π β§ π β β) β π β β) | |
17 | 14, 15, 5, 16 | smfpreimalt 46148 | . . 3 β’ ((π β§ π β β) β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ)) |
18 | 13, 17 | sseldd 3983 | . 2 β’ ((π β§ π β β) β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ)) |
19 | 1, 2, 9, 10, 18 | issmfd 46152 | 1 β’ (π β πΉ β (SMblFnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β wcel 2098 {crab 3430 β wss 3949 βͺ cuni 4912 class class class wbr 5152 dom cdm 5682 βcfv 6553 (class class class)co 7426 βcr 11145 < clt 11286 βΎt crest 17409 SAlgcsalg 45725 SMblFncsmblfn 46112 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-er 8731 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-ioo 13368 df-ico 13370 df-rest 17411 df-smblfn 46113 |
This theorem is referenced by: bormflebmf 46170 |
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