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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 6448 | . . . 4 ⊢ 𝑍 ∈ V |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ∈ V) |
4 | 3 | mptexd 6744 | . 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ V) |
5 | fvexd 6449 | . 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 41825 | . . 3 ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) |
17 | fvexd 6449 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ∈ V) | |
18 | 1 | eluzelz2 40423 | . . . . 5 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
19 | 14, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
20 | eqid 2826 | . . . 4 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
21 | 1 | eleq2i 2899 | . . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
22 | 21 | biimpi 208 | . . . . . . 7 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀)) |
23 | uzss 11990 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
24 | 22, 23 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
25 | 24, 1 | syl6sseqr 3878 | . . . . 5 ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ 𝑍) |
26 | 14, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
27 | ssid 3849 | . . . . 5 ⊢ (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑁) | |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑁)) |
29 | fvexd 6449 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) ∈ V) | |
30 | 6, 3, 17, 19, 20, 26, 28, 29 | climeqmpt 40725 | . . 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 249 | . 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) |
32 | breldmg 5563 | . 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 1496 | 1 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ dom ⇝ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 {crab 3122 Vcvv 3415 ⊆ wss 3799 ∩ ciin 4742 class class class wbr 4874 ↦ cmpt 4953 dom cdm 5343 ran crn 5344 ⟶wf 6120 ‘cfv 6124 supcsup 8616 ℝcr 10252 ℝ*cxr 10391 < clt 10392 ℤcz 11705 ℤ≥cuz 11969 lim supclsp 14579 ⇝ cli 14593 SAlgcsalg 41320 SMblFncsmblfn 41704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-sup 8618 df-inf 8619 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-z 11706 df-uz 11970 df-q 12073 df-rp 12114 df-ioo 12468 df-ico 12470 df-fz 12621 df-fl 12889 df-ceil 12890 df-seq 13097 df-exp 13156 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-limsup 14580 df-clim 14597 df-smblfn 41705 |
This theorem is referenced by: smflimsuplem7 41827 |
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