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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 23435.) (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 24135 | . . . 4 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
4 | 3 | feqmptd 6249 | . 2 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑆 ↦ (𝐺‘𝑧))) |
5 | mbfulm.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | mbfulm.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑀 ∈ ℤ) |
8 | mbfulm.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶MblFn) | |
9 | ffn 6045 | . . . . . . 7 ⊢ (𝐹:𝑍⟶MblFn → 𝐹 Fn 𝑍) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
11 | ulmf2 24138 | . . . . . 6 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆)) | |
12 | 10, 1, 11 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆)) |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆)) |
14 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆) | |
15 | fvex 6201 | . . . . . . 7 ⊢ (ℤ≥‘𝑀) ∈ V | |
16 | 5, 15 | eqeltri 2697 | . . . . . 6 ⊢ 𝑍 ∈ V |
17 | 16 | mptex 6486 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ∈ V |
18 | 17 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ∈ V) |
19 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) | |
20 | 19 | fveq1d 6193 | . . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑛)‘𝑧)) |
21 | eqid 2622 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) | |
22 | fvex 6201 | . . . . . . 7 ⊢ ((𝐹‘𝑛)‘𝑧) ∈ V | |
23 | 20, 21, 22 | fvmpt 6282 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧))‘𝑛) = ((𝐹‘𝑛)‘𝑧)) |
24 | 23 | eqcomd 2628 | . . . . 5 ⊢ (𝑛 ∈ 𝑍 → ((𝐹‘𝑛)‘𝑧) = ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧))‘𝑛)) |
25 | 24 | adantl 482 | . . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑧) = ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧))‘𝑛)) |
26 | 1 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝐹(⇝𝑢‘𝑆)𝐺) |
27 | 5, 7, 13, 14, 18, 25, 26 | ulmclm 24141 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) |
28 | 12 | ffvelrnda 6359 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑𝑚 𝑆)) |
29 | elmapi 7879 | . . . . . 6 ⊢ ((𝐹‘𝑘) ∈ (ℂ ↑𝑚 𝑆) → (𝐹‘𝑘):𝑆⟶ℂ) | |
30 | 28, 29 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘):𝑆⟶ℂ) |
31 | 30 | feqmptd 6249 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝑧 ∈ 𝑆 ↦ ((𝐹‘𝑘)‘𝑧))) |
32 | 8 | ffvelrnda 6359 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ MblFn) |
33 | 31, 32 | eqeltrrd 2702 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑧 ∈ 𝑆 ↦ ((𝐹‘𝑘)‘𝑧)) ∈ MblFn) |
34 | 30 | ffvelrnda 6359 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ) |
35 | 34 | anasss 679 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ) |
36 | 5, 6, 27, 33, 35 | mbflim 23435 | . 2 ⊢ (𝜑 → (𝑧 ∈ 𝑆 ↦ (𝐺‘𝑧)) ∈ MblFn) |
37 | 4, 36 | eqeltrd 2701 | 1 ⊢ (𝜑 → 𝐺 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ↦ cmpt 4729 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℂcc 9934 ℤcz 11377 ℤ≥cuz 11687 MblFncmbf 23383 ⇝𝑢culm 24130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-xmet 19739 df-met 19740 df-ovol 23233 df-vol 23234 df-mbf 23388 df-ulm 24131 |
This theorem is referenced by: iblulm 24161 |
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