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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfdmpt | 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 |
---|---|
issmfdmpt.x | β’ β²π₯π |
issmfdmpt.a | β’ β²ππ |
issmfdmpt.s | β’ (π β π β SAlg) |
issmfdmpt.i | β’ (π β π΄ β βͺ π) |
issmfdmpt.b | β’ ((π β§ π₯ β π΄) β π΅ β β) |
issmfdmpt.p | β’ ((π β§ π β β) β {π₯ β π΄ β£ π΅ < π} β (π βΎt π΄)) |
Ref | Expression |
---|---|
issmfdmpt | β’ (π β (π₯ β π΄ β¦ π΅) β (SMblFnβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfmpt1 5249 | . 2 β’ β²π₯(π₯ β π΄ β¦ π΅) | |
2 | issmfdmpt.a | . 2 β’ β²ππ | |
3 | issmfdmpt.s | . 2 β’ (π β π β SAlg) | |
4 | issmfdmpt.i | . 2 β’ (π β π΄ β βͺ π) | |
5 | issmfdmpt.x | . . 3 β’ β²π₯π | |
6 | issmfdmpt.b | . . 3 β’ ((π β§ π₯ β π΄) β π΅ β β) | |
7 | eqid 2726 | . . 3 β’ (π₯ β π΄ β¦ π΅) = (π₯ β π΄ β¦ π΅) | |
8 | 5, 6, 7 | fmptdf 7112 | . 2 β’ (π β (π₯ β π΄ β¦ π΅):π΄βΆβ) |
9 | eqidd 2727 | . . . . . . . . 9 β’ (π β (π₯ β π΄ β¦ π΅) = (π₯ β π΄ β¦ π΅)) | |
10 | 9, 6 | fvmpt2d 7005 | . . . . . . . 8 β’ ((π β§ π₯ β π΄) β ((π₯ β π΄ β¦ π΅)βπ₯) = π΅) |
11 | 10 | breq1d 5151 | . . . . . . 7 β’ ((π β§ π₯ β π΄) β (((π₯ β π΄ β¦ π΅)βπ₯) < π β π΅ < π)) |
12 | 11 | ex 412 | . . . . . 6 β’ (π β (π₯ β π΄ β (((π₯ β π΄ β¦ π΅)βπ₯) < π β π΅ < π))) |
13 | 5, 12 | ralrimi 3248 | . . . . 5 β’ (π β βπ₯ β π΄ (((π₯ β π΄ β¦ π΅)βπ₯) < π β π΅ < π)) |
14 | rabbi 3456 | . . . . 5 β’ (βπ₯ β π΄ (((π₯ β π΄ β¦ π΅)βπ₯) < π β π΅ < π) β {π₯ β π΄ β£ ((π₯ β π΄ β¦ π΅)βπ₯) < π} = {π₯ β π΄ β£ π΅ < π}) | |
15 | 13, 14 | sylib 217 | . . . 4 β’ (π β {π₯ β π΄ β£ ((π₯ β π΄ β¦ π΅)βπ₯) < π} = {π₯ β π΄ β£ π΅ < π}) |
16 | 15 | adantr 480 | . . 3 β’ ((π β§ π β β) β {π₯ β π΄ β£ ((π₯ β π΄ β¦ π΅)βπ₯) < π} = {π₯ β π΄ β£ π΅ < π}) |
17 | issmfdmpt.p | . . 3 β’ ((π β§ π β β) β {π₯ β π΄ β£ π΅ < π} β (π βΎt π΄)) | |
18 | 16, 17 | eqeltrd 2827 | . 2 β’ ((π β§ π β β) β {π₯ β π΄ β£ ((π₯ β π΄ β¦ π΅)βπ₯) < π} β (π βΎt π΄)) |
19 | 1, 2, 3, 4, 8, 18 | issmfdf 46025 | 1 β’ (π β (π₯ β π΄ β¦ π΅) β (SMblFnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β²wnf 1777 β wcel 2098 βwral 3055 {crab 3426 β wss 3943 βͺ cuni 4902 class class class wbr 5141 β¦ cmpt 5224 βcfv 6537 (class class class)co 7405 βcr 11111 < clt 11252 βΎt crest 17375 SAlgcsalg 45596 SMblFncsmblfn 45983 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 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 13334 df-ico 13336 df-smblfn 45984 |
This theorem is referenced by: smfadd 46053 smfrec 46077 smfmul 46083 |
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