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Mirrors > Home > MPE Home > Th. List > mbfulm | Structured version Visualization version GIF version |
Description: A uniform limit of measurable functions is measurable. (This is just a corollary of the fact that a pointwise limit of measurable functions is measurable, see mbflim 25548.) (Contributed by Mario Carneiro, 18-Mar-2015.) |
Ref | Expression |
---|---|
mbfulm.z | β’ π = (β€β₯βπ) |
mbfulm.m | β’ (π β π β β€) |
mbfulm.f | β’ (π β πΉ:πβΆMblFn) |
mbfulm.u | β’ (π β πΉ(βπ’βπ)πΊ) |
Ref | Expression |
---|---|
mbfulm | β’ (π β πΊ β MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfulm.u | . . . 4 β’ (π β πΉ(βπ’βπ)πΊ) | |
2 | ulmcl 26268 | . . . 4 β’ (πΉ(βπ’βπ)πΊ β πΊ:πβΆβ) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β πΊ:πβΆβ) |
4 | 3 | feqmptd 6953 | . 2 β’ (π β πΊ = (π§ β π β¦ (πΊβπ§))) |
5 | mbfulm.z | . . 3 β’ π = (β€β₯βπ) | |
6 | mbfulm.m | . . 3 β’ (π β π β β€) | |
7 | 6 | adantr 480 | . . . 4 β’ ((π β§ π§ β π) β π β β€) |
8 | mbfulm.f | . . . . . . 7 β’ (π β πΉ:πβΆMblFn) | |
9 | 8 | ffnd 6711 | . . . . . 6 β’ (π β πΉ Fn π) |
10 | ulmf2 26271 | . . . . . 6 β’ ((πΉ Fn π β§ πΉ(βπ’βπ)πΊ) β πΉ:πβΆ(β βm π)) | |
11 | 9, 1, 10 | syl2anc 583 | . . . . 5 β’ (π β πΉ:πβΆ(β βm π)) |
12 | 11 | adantr 480 | . . . 4 β’ ((π β§ π§ β π) β πΉ:πβΆ(β βm π)) |
13 | simpr 484 | . . . 4 β’ ((π β§ π§ β π) β π§ β π) | |
14 | 5 | fvexi 6898 | . . . . . 6 β’ π β V |
15 | 14 | mptex 7219 | . . . . 5 β’ (π β π β¦ ((πΉβπ)βπ§)) β V |
16 | 15 | a1i 11 | . . . 4 β’ ((π β§ π§ β π) β (π β π β¦ ((πΉβπ)βπ§)) β V) |
17 | fveq2 6884 | . . . . . . . 8 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
18 | 17 | fveq1d 6886 | . . . . . . 7 β’ (π = π β ((πΉβπ)βπ§) = ((πΉβπ)βπ§)) |
19 | eqid 2726 | . . . . . . 7 β’ (π β π β¦ ((πΉβπ)βπ§)) = (π β π β¦ ((πΉβπ)βπ§)) | |
20 | fvex 6897 | . . . . . . 7 β’ ((πΉβπ)βπ§) β V | |
21 | 18, 19, 20 | fvmpt 6991 | . . . . . 6 β’ (π β π β ((π β π β¦ ((πΉβπ)βπ§))βπ) = ((πΉβπ)βπ§)) |
22 | 21 | eqcomd 2732 | . . . . 5 β’ (π β π β ((πΉβπ)βπ§) = ((π β π β¦ ((πΉβπ)βπ§))βπ)) |
23 | 22 | adantl 481 | . . . 4 β’ (((π β§ π§ β π) β§ π β π) β ((πΉβπ)βπ§) = ((π β π β¦ ((πΉβπ)βπ§))βπ)) |
24 | 1 | adantr 480 | . . . 4 β’ ((π β§ π§ β π) β πΉ(βπ’βπ)πΊ) |
25 | 5, 7, 12, 13, 16, 23, 24 | ulmclm 26274 | . . 3 β’ ((π β§ π§ β π) β (π β π β¦ ((πΉβπ)βπ§)) β (πΊβπ§)) |
26 | 11 | ffvelcdmda 7079 | . . . . . 6 β’ ((π β§ π β π) β (πΉβπ) β (β βm π)) |
27 | elmapi 8842 | . . . . . 6 β’ ((πΉβπ) β (β βm π) β (πΉβπ):πβΆβ) | |
28 | 26, 27 | syl 17 | . . . . 5 β’ ((π β§ π β π) β (πΉβπ):πβΆβ) |
29 | 28 | feqmptd 6953 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) = (π§ β π β¦ ((πΉβπ)βπ§))) |
30 | 8 | ffvelcdmda 7079 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) β MblFn) |
31 | 29, 30 | eqeltrrd 2828 | . . 3 β’ ((π β§ π β π) β (π§ β π β¦ ((πΉβπ)βπ§)) β MblFn) |
32 | 28 | ffvelcdmda 7079 | . . . 4 β’ (((π β§ π β π) β§ π§ β π) β ((πΉβπ)βπ§) β β) |
33 | 32 | anasss 466 | . . 3 β’ ((π β§ (π β π β§ π§ β π)) β ((πΉβπ)βπ§) β β) |
34 | 5, 6, 25, 31, 33 | mbflim 25548 | . 2 β’ (π β (π§ β π β¦ (πΊβπ§)) β MblFn) |
35 | 4, 34 | eqeltrd 2827 | 1 β’ (π β πΊ β MblFn) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 class class class wbr 5141 β¦ cmpt 5224 Fn wfn 6531 βΆwf 6532 βcfv 6536 (class class class)co 7404 βm cmap 8819 βcc 11107 β€cz 12559 β€β₯cuz 12823 MblFncmbf 25494 βπ’culm 26263 |
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-inf2 9635 ax-cc 10429 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 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-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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 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-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-acn 9936 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xadd 13096 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14031 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-xmet 21229 df-met 21230 df-ovol 25344 df-vol 25345 df-mbf 25499 df-ulm 26264 |
This theorem is referenced by: iblulm 26294 |
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