Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smflimsuplem6 | Structured version Visualization version GIF version |
Description: The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
smflimsuplem6.a | ⊢ Ⅎ𝑛𝜑 |
smflimsuplem6.b | ⊢ Ⅎ𝑚𝜑 |
smflimsuplem6.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
smflimsuplem6.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
smflimsuplem6.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smflimsuplem6.f | ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
smflimsuplem6.e | ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) |
smflimsuplem6.h | ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) |
smflimsuplem6.r | ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) |
smflimsuplem6.n | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
smflimsuplem6.x | ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑁)dom (𝐹‘𝑚)) |
Ref | Expression |
---|---|
smflimsuplem6 | ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ dom ⇝ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smflimsuplem6.z | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | 1 | fvexi 6677 | . . . 4 ⊢ 𝑍 ∈ V |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ∈ V) |
4 | 3 | mptexd 6978 | . 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ V) |
5 | fvexd 6678 | . 2 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V) | |
6 | smflimsuplem6.a | . . . 4 ⊢ Ⅎ𝑛𝜑 | |
7 | smflimsuplem6.b | . . . 4 ⊢ Ⅎ𝑚𝜑 | |
8 | smflimsuplem6.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
9 | smflimsuplem6.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
10 | smflimsuplem6.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) | |
11 | smflimsuplem6.e | . . . 4 ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) | |
12 | smflimsuplem6.h | . . . 4 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) | |
13 | smflimsuplem6.r | . . . 4 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) | |
14 | smflimsuplem6.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
15 | smflimsuplem6.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑁)dom (𝐹‘𝑚)) | |
16 | 6, 7, 8, 1, 9, 10, 11, 12, 13, 14, 15 | smflimsuplem5 42975 | . . 3 ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) |
17 | fvexd 6678 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ∈ V) | |
18 | 1 | eluzelz2 41552 | . . . . 5 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
19 | 14, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
20 | eqid 2818 | . . . 4 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
21 | 1 | eleq2i 2901 | . . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
22 | 21 | biimpi 217 | . . . . . . 7 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀)) |
23 | uzss 12253 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
24 | 22, 23 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
25 | 24, 1 | sseqtrrdi 4015 | . . . . 5 ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ 𝑍) |
26 | 14, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
27 | ssid 3986 | . . . . 5 ⊢ (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑁) | |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑁)) |
29 | fvexd 6678 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) ∈ V) | |
30 | 6, 3, 17, 19, 20, 26, 28, 29 | climeqmpt 41854 | . . 3 ⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) ↔ (𝑛 ∈ (ℤ≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))))) |
31 | 16, 30 | mpbird 258 | . 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) |
32 | breldmg 5771 | . 2 ⊢ (((𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ V ∧ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V ∧ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ dom ⇝ ) | |
33 | 4, 5, 31, 32 | syl3anc 1363 | 1 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ dom ⇝ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 {crab 3139 Vcvv 3492 ⊆ wss 3933 ∩ ciin 4911 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ran crn 5549 ⟶wf 6344 ‘cfv 6348 supcsup 8892 ℝcr 10524 ℝ*cxr 10662 < clt 10663 ℤcz 11969 ℤ≥cuz 12231 lim supclsp 14815 ⇝ cli 14829 SAlgcsalg 42470 SMblFncsmblfn 42854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-ioo 12730 df-ico 12732 df-fz 12881 df-fl 13150 df-ceil 13151 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-smblfn 42855 |
This theorem is referenced by: smflimsuplem7 42977 |
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