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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 25610.) (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 26330 | . . . 4 β’ (πΉ(βπ’βπ)πΊ β πΊ:πβΆβ) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β πΊ:πβΆβ) |
4 | 3 | feqmptd 6967 | . 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 6723 | . . . . . 6 β’ (π β πΉ Fn π) |
10 | ulmf2 26333 | . . . . . 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 6911 | . . . . . 6 β’ π β V |
15 | 14 | mptex 7235 | . . . . 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 6897 | . . . . . . . 8 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
18 | 17 | fveq1d 6899 | . . . . . . 7 β’ (π = π β ((πΉβπ)βπ§) = ((πΉβπ)βπ§)) |
19 | eqid 2728 | . . . . . . 7 β’ (π β π β¦ ((πΉβπ)βπ§)) = (π β π β¦ ((πΉβπ)βπ§)) | |
20 | fvex 6910 | . . . . . . 7 β’ ((πΉβπ)βπ§) β V | |
21 | 18, 19, 20 | fvmpt 7005 | . . . . . 6 β’ (π β π β ((π β π β¦ ((πΉβπ)βπ§))βπ) = ((πΉβπ)βπ§)) |
22 | 21 | eqcomd 2734 | . . . . 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 26336 | . . 3 β’ ((π β§ π§ β π) β (π β π β¦ ((πΉβπ)βπ§)) β (πΊβπ§)) |
26 | 11 | ffvelcdmda 7094 | . . . . . 6 β’ ((π β§ π β π) β (πΉβπ) β (β βm π)) |
27 | elmapi 8868 | . . . . . 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 6967 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) = (π§ β π β¦ ((πΉβπ)βπ§))) |
30 | 8 | ffvelcdmda 7094 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) β MblFn) |
31 | 29, 30 | eqeltrrd 2830 | . . 3 β’ ((π β§ π β π) β (π§ β π β¦ ((πΉβπ)βπ§)) β MblFn) |
32 | 28 | ffvelcdmda 7094 | . . . 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 25610 | . 2 β’ (π β (π§ β π β¦ (πΊβπ§)) β MblFn) |
35 | 4, 34 | eqeltrd 2829 | 1 β’ (π β πΊ β MblFn) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3471 class class class wbr 5148 β¦ cmpt 5231 Fn wfn 6543 βΆwf 6544 βcfv 6548 (class class class)co 7420 βm cmap 8845 βcc 11137 β€cz 12589 β€β₯cuz 12853 MblFncmbf 25556 βπ’culm 26325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cc 10459 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-dju 9925 df-card 9963 df-acn 9966 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-xadd 13126 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-xmet 21272 df-met 21273 df-ovol 25406 df-vol 25407 df-mbf 25561 df-ulm 26326 |
This theorem is referenced by: iblulm 26356 |
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