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| 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 25704.) (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 26425 | . . . 4 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
| 3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) | 
| 4 | 3 | feqmptd 6976 | . 2 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑆 ↦ (𝐺‘𝑧))) | 
| 5 | mbfulm.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 6 | mbfulm.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 7 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑀 ∈ ℤ) | 
| 8 | mbfulm.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶MblFn) | |
| 9 | 8 | ffnd 6736 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑍) | 
| 10 | ulmf2 26428 | . . . . . 6 ⊢ ((𝐹 Fn 𝑍 ∧ 𝐹(⇝𝑢‘𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | |
| 11 | 9, 1, 10 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | 
| 12 | 11 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | 
| 13 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆) | |
| 14 | 5 | fvexi 6919 | . . . . . 6 ⊢ 𝑍 ∈ V | 
| 15 | 14 | mptex 7244 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ∈ V | 
| 16 | 15 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ∈ V) | 
| 17 | fveq2 6905 | . . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) | |
| 18 | 17 | fveq1d 6907 | . . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑛)‘𝑧)) | 
| 19 | eqid 2736 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) | |
| 20 | fvex 6918 | . . . . . . 7 ⊢ ((𝐹‘𝑛)‘𝑧) ∈ V | |
| 21 | 18, 19, 20 | fvmpt 7015 | . . . . . 6 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧))‘𝑛) = ((𝐹‘𝑛)‘𝑧)) | 
| 22 | 21 | eqcomd 2742 | . . . . 5 ⊢ (𝑛 ∈ 𝑍 → ((𝐹‘𝑛)‘𝑧) = ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧))‘𝑛)) | 
| 23 | 22 | adantl 481 | . . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑧) = ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧))‘𝑛)) | 
| 24 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝐹(⇝𝑢‘𝑆)𝐺) | 
| 25 | 5, 7, 12, 13, 16, 23, 24 | ulmclm 26431 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧)) | 
| 26 | 11 | ffvelcdmda 7103 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑m 𝑆)) | 
| 27 | elmapi 8890 | . . . . . 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 6976 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝑧 ∈ 𝑆 ↦ ((𝐹‘𝑘)‘𝑧))) | 
| 30 | 8 | ffvelcdmda 7103 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ MblFn) | 
| 31 | 29, 30 | eqeltrrd 2841 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑧 ∈ 𝑆 ↦ ((𝐹‘𝑘)‘𝑧)) ∈ MblFn) | 
| 32 | 28 | ffvelcdmda 7103 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ) | 
| 33 | 32 | anasss 466 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ) | 
| 34 | 5, 6, 25, 31, 33 | mbflim 25704 | . 2 ⊢ (𝜑 → (𝑧 ∈ 𝑆 ↦ (𝐺‘𝑧)) ∈ MblFn) | 
| 35 | 4, 34 | eqeltrd 2840 | 1 ⊢ (𝜑 → 𝐺 ∈ MblFn) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 class class class wbr 5142 ↦ cmpt 5224 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 ℂcc 11154 ℤcz 12615 ℤ≥cuz 12879 MblFncmbf 25650 ⇝𝑢culm 26420 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cc 10476 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-xadd 13156 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-xmet 21358 df-met 21359 df-ovol 25500 df-vol 25501 df-mbf 25655 df-ulm 26421 | 
| This theorem is referenced by: iblulm 26451 | 
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