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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 652 | . . . . . 6 β’ (((π β§ π₯ β π΄) β§ π β π) β π΅ β β) |
4 | 3 | fmpttd 7110 | . . . . 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 480 | . . . . . 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 15442 | . . . . . . . 8 β’ Rel β | |
9 | 8 | releldmi 5941 | . . . . . . 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 15623 | . . . . . 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 583 | . . . . 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 15629 | . . . 4 β’ ((π β§ π₯ β π΄) β (π β π β¦ π΅) β (lim supβ(π β π β¦ π΅))) |
14 | climuni 15502 | . . . 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 583 | . . 3 β’ ((π β§ π₯ β π΄) β (lim supβ(π β π β¦ π΅)) = πΆ) |
16 | 15 | mpteq2dva 5241 | . 2 β’ (π β (π₯ β π΄ β¦ (lim supβ(π β π β¦ π΅))) = (π₯ β π΄ β¦ πΆ)) |
17 | eqid 2726 | . . 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 2726 | . . 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 7080 | . . . 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 15533 | . . 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 25550 | . 2 β’ (π β (π₯ β π΄ β¦ (lim supβ(π β π β¦ π΅))) β MblFn) |
23 | 16, 22 | eqeltrrd 2828 | 1 β’ (π β (π₯ β π΄ β¦ πΆ) β MblFn) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 βwrex 3064 β© cin 3942 class class class wbr 5141 β¦ cmpt 5224 dom cdm 5669 β cima 5672 βcfv 6537 (class class class)co 7405 supcsup 9437 βcr 11111 +βcpnf 11249 β*cxr 11251 < clt 11252 β cmin 11448 β€cz 12562 β€β₯cuz 12826 β+crp 12980 [,)cico 13332 abscabs 15187 lim supclsp 15420 β cli 15434 MblFncmbf 25498 |
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 7722 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
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-rmo 3370 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-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 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-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 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-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-xadd 13099 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-xmet 21233 df-met 21234 df-ovol 25348 df-vol 25349 df-mbf 25503 |
This theorem is referenced by: mbflim 25552 |
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