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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 2726 | . . . 4 β’ dom πΉ = dom πΉ | |
6 | 3, 4, 5 | smfdmss 46003 | . . 3 β’ (π β dom πΉ β βͺ π ) |
7 | smfsssmf.i | . . . 4 β’ (π β π β π) | |
8 | 7 | unissd 4912 | . . 3 β’ (π β βͺ π β βͺ π) |
9 | 6, 8 | sstrd 3987 | . 2 β’ (π β dom πΉ β βͺ π) |
10 | 3, 4, 5 | smff 46002 | . 2 β’ (π β πΉ:dom πΉβΆβ) |
11 | ssrest 23030 | . . . . 5 β’ ((π β SAlg β§ π β π) β (π βΎt dom πΉ) β (π βΎt dom πΉ)) | |
12 | 2, 7, 11 | syl2anc 583 | . . . 4 β’ (π β (π βΎt dom πΉ) β (π βΎt dom πΉ)) |
13 | 12 | adantr 480 | . . 3 β’ ((π β§ π β β) β (π βΎt dom πΉ) β (π βΎt dom πΉ)) |
14 | 3 | adantr 480 | . . . 4 β’ ((π β§ π β β) β π β SAlg) |
15 | 4 | adantr 480 | . . . 4 β’ ((π β§ π β β) β πΉ β (SMblFnβπ )) |
16 | simpr 484 | . . . 4 β’ ((π β§ π β β) β π β β) | |
17 | 14, 15, 5, 16 | smfpreimalt 46001 | . . 3 β’ ((π β§ π β β) β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ)) |
18 | 13, 17 | sseldd 3978 | . 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 46005 | 1 β’ (π β πΉ β (SMblFnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β wcel 2098 {crab 3426 β wss 3943 βͺ cuni 4902 class class class wbr 5141 dom cdm 5669 βcfv 6536 (class class class)co 7404 βcr 11108 < clt 11249 βΎt crest 17372 SAlgcsalg 45578 SMblFncsmblfn 45965 |
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-rep 5278 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-reu 3371 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-rest 17374 df-smblfn 45966 |
This theorem is referenced by: bormflebmf 46023 |
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