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Mirrors > Home > MPE Home > Th. List > mbflimlem | Structured version Visualization version GIF version |
Description: The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) |
Ref | Expression |
---|---|
mbflim.1 | β’ π = (β€β₯βπ) |
mbflim.2 | β’ (π β π β β€) |
mbflim.4 | β’ ((π β§ π₯ β π΄) β (π β π β¦ π΅) β πΆ) |
mbflim.5 | β’ ((π β§ π β π) β (π₯ β π΄ β¦ π΅) β MblFn) |
mbflimlem.6 | β’ ((π β§ (π β π β§ π₯ β π΄)) β π΅ β β) |
Ref | Expression |
---|---|
mbflimlem | β’ (π β (π₯ β π΄ β¦ πΆ) β MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbflim.1 | . . . . 5 β’ π = (β€β₯βπ) | |
2 | mbflimlem.6 | . . . . . . 7 β’ ((π β§ (π β π β§ π₯ β π΄)) β π΅ β β) | |
3 | 2 | anass1rs 653 | . . . . . 6 β’ (((π β§ π₯ β π΄) β§ π β π) β π΅ β β) |
4 | 3 | fmpttd 7120 | . . . . 5 β’ ((π β§ π₯ β π΄) β (π β π β¦ π΅):πβΆβ) |
5 | mbflim.2 | . . . . . . 7 β’ (π β π β β€) | |
6 | 5 | adantr 479 | . . . . . 6 β’ ((π β§ π₯ β π΄) β π β β€) |
7 | mbflim.4 | . . . . . . 7 β’ ((π β§ π₯ β π΄) β (π β π β¦ π΅) β πΆ) | |
8 | climrel 15468 | . . . . . . . 8 β’ Rel β | |
9 | 8 | releldmi 5944 | . . . . . . 7 β’ ((π β π β¦ π΅) β πΆ β (π β π β¦ π΅) β dom β ) |
10 | 7, 9 | syl 17 | . . . . . 6 β’ ((π β§ π₯ β π΄) β (π β π β¦ π΅) β dom β ) |
11 | 1 | climcau 15649 | . . . . . 6 β’ ((π β β€ β§ (π β π β¦ π΅) β dom β ) β βπ¦ β β+ βπ β π βπ β (β€β₯βπ)(absβ(((π β π β¦ π΅)βπ) β ((π β π β¦ π΅)βπ))) < π¦) |
12 | 6, 10, 11 | syl2anc 582 | . . . . 5 β’ ((π β§ π₯ β π΄) β βπ¦ β β+ βπ β π βπ β (β€β₯βπ)(absβ(((π β π β¦ π΅)βπ) β ((π β π β¦ π΅)βπ))) < π¦) |
13 | 1, 4, 12 | caurcvg 15655 | . . . 4 β’ ((π β§ π₯ β π΄) β (π β π β¦ π΅) β (lim supβ(π β π β¦ π΅))) |
14 | climuni 15528 | . . . 4 β’ (((π β π β¦ π΅) β (lim supβ(π β π β¦ π΅)) β§ (π β π β¦ π΅) β πΆ) β (lim supβ(π β π β¦ π΅)) = πΆ) | |
15 | 13, 7, 14 | syl2anc 582 | . . 3 β’ ((π β§ π₯ β π΄) β (lim supβ(π β π β¦ π΅)) = πΆ) |
16 | 15 | mpteq2dva 5243 | . 2 β’ (π β (π₯ β π΄ β¦ (lim supβ(π β π β¦ π΅))) = (π₯ β π΄ β¦ πΆ)) |
17 | eqid 2725 | . . 3 β’ (π₯ β π΄ β¦ (lim supβ(π β π β¦ π΅))) = (π₯ β π΄ β¦ (lim supβ(π β π β¦ π΅))) | |
18 | eqid 2725 | . . 3 β’ (π β β β¦ sup((((π β π β¦ π΅) β (π[,)+β)) β© β*), β*, < )) = (π β β β¦ sup((((π β π β¦ π΅) β (π[,)+β)) β© β*), β*, < )) | |
19 | 4 | ffvelcdmda 7089 | . . . 4 β’ (((π β§ π₯ β π΄) β§ π β π) β ((π β π β¦ π΅)βπ) β β) |
20 | 1, 6, 13, 19 | climrecl 15559 | . . 3 β’ ((π β§ π₯ β π΄) β (lim supβ(π β π β¦ π΅)) β β) |
21 | mbflim.5 | . . 3 β’ ((π β§ π β π) β (π₯ β π΄ β¦ π΅) β MblFn) | |
22 | 1, 17, 18, 5, 20, 21, 2 | mbflimsup 25613 | . 2 β’ (π β (π₯ β π΄ β¦ (lim supβ(π β π β¦ π΅))) β MblFn) |
23 | 16, 22 | eqeltrrd 2826 | 1 β’ (π β (π₯ β π΄ β¦ πΆ) β MblFn) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3051 βwrex 3060 β© cin 3938 class class class wbr 5143 β¦ cmpt 5226 dom cdm 5672 β cima 5675 βcfv 6543 (class class class)co 7416 supcsup 9463 βcr 11137 +βcpnf 11275 β*cxr 11277 < clt 11278 β cmin 11474 β€cz 12588 β€β₯cuz 12852 β+crp 13006 [,)cico 13358 abscabs 15213 lim supclsp 15446 β cli 15460 MblFncmbf 25561 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cc 10458 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xadd 13125 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-xmet 21276 df-met 21277 df-ovol 25411 df-vol 25412 df-mbf 25566 |
This theorem is referenced by: mbflim 25615 |
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