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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 25048.) (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 25756 | . . . 4 β’ (πΉ(βπ’βπ)πΊ β πΊ:πβΆβ) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β πΊ:πβΆβ) |
4 | 3 | feqmptd 6915 | . 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 482 | . . . 4 β’ ((π β§ π§ β π) β π β β€) |
8 | mbfulm.f | . . . . . . 7 β’ (π β πΉ:πβΆMblFn) | |
9 | 8 | ffnd 6674 | . . . . . 6 β’ (π β πΉ Fn π) |
10 | ulmf2 25759 | . . . . . 6 β’ ((πΉ Fn π β§ πΉ(βπ’βπ)πΊ) β πΉ:πβΆ(β βm π)) | |
11 | 9, 1, 10 | syl2anc 585 | . . . . 5 β’ (π β πΉ:πβΆ(β βm π)) |
12 | 11 | adantr 482 | . . . 4 β’ ((π β§ π§ β π) β πΉ:πβΆ(β βm π)) |
13 | simpr 486 | . . . 4 β’ ((π β§ π§ β π) β π§ β π) | |
14 | 5 | fvexi 6861 | . . . . . 6 β’ π β V |
15 | 14 | mptex 7178 | . . . . 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 6847 | . . . . . . . 8 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
18 | 17 | fveq1d 6849 | . . . . . . 7 β’ (π = π β ((πΉβπ)βπ§) = ((πΉβπ)βπ§)) |
19 | eqid 2737 | . . . . . . 7 β’ (π β π β¦ ((πΉβπ)βπ§)) = (π β π β¦ ((πΉβπ)βπ§)) | |
20 | fvex 6860 | . . . . . . 7 β’ ((πΉβπ)βπ§) β V | |
21 | 18, 19, 20 | fvmpt 6953 | . . . . . 6 β’ (π β π β ((π β π β¦ ((πΉβπ)βπ§))βπ) = ((πΉβπ)βπ§)) |
22 | 21 | eqcomd 2743 | . . . . 5 β’ (π β π β ((πΉβπ)βπ§) = ((π β π β¦ ((πΉβπ)βπ§))βπ)) |
23 | 22 | adantl 483 | . . . 4 β’ (((π β§ π§ β π) β§ π β π) β ((πΉβπ)βπ§) = ((π β π β¦ ((πΉβπ)βπ§))βπ)) |
24 | 1 | adantr 482 | . . . 4 β’ ((π β§ π§ β π) β πΉ(βπ’βπ)πΊ) |
25 | 5, 7, 12, 13, 16, 23, 24 | ulmclm 25762 | . . 3 β’ ((π β§ π§ β π) β (π β π β¦ ((πΉβπ)βπ§)) β (πΊβπ§)) |
26 | 11 | ffvelcdmda 7040 | . . . . . 6 β’ ((π β§ π β π) β (πΉβπ) β (β βm π)) |
27 | elmapi 8794 | . . . . . 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 6915 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) = (π§ β π β¦ ((πΉβπ)βπ§))) |
30 | 8 | ffvelcdmda 7040 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) β MblFn) |
31 | 29, 30 | eqeltrrd 2839 | . . 3 β’ ((π β§ π β π) β (π§ β π β¦ ((πΉβπ)βπ§)) β MblFn) |
32 | 28 | ffvelcdmda 7040 | . . . 4 β’ (((π β§ π β π) β§ π§ β π) β ((πΉβπ)βπ§) β β) |
33 | 32 | anasss 468 | . . 3 β’ ((π β§ (π β π β§ π§ β π)) β ((πΉβπ)βπ§) β β) |
34 | 5, 6, 25, 31, 33 | mbflim 25048 | . 2 β’ (π β (π§ β π β¦ (πΊβπ§)) β MblFn) |
35 | 4, 34 | eqeltrd 2838 | 1 β’ (π β πΊ β MblFn) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 class class class wbr 5110 β¦ cmpt 5193 Fn wfn 6496 βΆwf 6497 βcfv 6501 (class class class)co 7362 βm cmap 8772 βcc 11056 β€cz 12506 β€β₯cuz 12770 MblFncmbf 24994 βπ’culm 25751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cc 10378 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-disj 5076 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-omul 8422 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-acn 9885 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-xadd 13041 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-xmet 20805 df-met 20806 df-ovol 24844 df-vol 24845 df-mbf 24999 df-ulm 25752 |
This theorem is referenced by: iblulm 25782 |
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