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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 24263.) (Contributed by Mario Carneiro, 18-Mar-2015.) |
Ref | Expression |
---|---|
mbfulm.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
mbfulm.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
mbfulm.f | ⊢ (𝜑 → 𝐹:𝑍⟶MblFn) |
mbfulm.u | ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) |
Ref | Expression |
---|---|
mbfulm | ⊢ (𝜑 → 𝐺 ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfulm.u | . . . 4 ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) | |
2 | ulmcl 24963 | . . . 4 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
4 | 3 | feqmptd 6727 | . 2 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑆 ↦ (𝐺‘𝑧))) |
5 | mbfulm.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | mbfulm.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑀 ∈ ℤ) |
8 | mbfulm.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶MblFn) | |
9 | 8 | ffnd 6509 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
10 | ulmf2 24966 | . . . . . 6 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | |
11 | 9, 1, 10 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
12 | 11 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
13 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆) | |
14 | 5 | fvexi 6678 | . . . . . 6 ⊢ 𝑍 ∈ V |
15 | 14 | mptex 6980 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ∈ V |
16 | 15 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ∈ V) |
17 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) | |
18 | 17 | fveq1d 6666 | . . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑛)‘𝑧)) |
19 | eqid 2821 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) | |
20 | fvex 6677 | . . . . . . 7 ⊢ ((𝐹‘𝑛)‘𝑧) ∈ V | |
21 | 18, 19, 20 | fvmpt 6762 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧))‘𝑛) = ((𝐹‘𝑛)‘𝑧)) |
22 | 21 | eqcomd 2827 | . . . . 5 ⊢ (𝑛 ∈ 𝑍 → ((𝐹‘𝑛)‘𝑧) = ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧))‘𝑛)) |
23 | 22 | adantl 484 | . . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑧) = ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧))‘𝑛)) |
24 | 1 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝐹(⇝𝑢‘𝑆)𝐺) |
25 | 5, 7, 12, 13, 16, 23, 24 | ulmclm 24969 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) |
26 | 11 | ffvelrnda 6845 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑m 𝑆)) |
27 | elmapi 8422 | . . . . . 6 ⊢ ((𝐹‘𝑘) ∈ (ℂ ↑m 𝑆) → (𝐹‘𝑘):𝑆⟶ℂ) | |
28 | 26, 27 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘):𝑆⟶ℂ) |
29 | 28 | feqmptd 6727 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝑧 ∈ 𝑆 ↦ ((𝐹‘𝑘)‘𝑧))) |
30 | 8 | ffvelrnda 6845 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ MblFn) |
31 | 29, 30 | eqeltrrd 2914 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑧 ∈ 𝑆 ↦ ((𝐹‘𝑘)‘𝑧)) ∈ MblFn) |
32 | 28 | ffvelrnda 6845 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ) |
33 | 32 | anasss 469 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ) |
34 | 5, 6, 25, 31, 33 | mbflim 24263 | . 2 ⊢ (𝜑 → (𝑧 ∈ 𝑆 ↦ (𝐺‘𝑧)) ∈ MblFn) |
35 | 4, 34 | eqeltrd 2913 | 1 ⊢ (𝜑 → 𝐺 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 ↦ cmpt 5138 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 ℂcc 10529 ℤcz 11975 ℤ≥cuz 12237 MblFncmbf 24209 ⇝𝑢culm 24958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cc 9851 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-xmet 20532 df-met 20533 df-ovol 24059 df-vol 24060 df-mbf 24214 df-ulm 24959 |
This theorem is referenced by: iblulm 24989 |
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