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 6202 | . . . 4 ⊢ 𝑍 ∈ V |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ∈ V) |
4 | 3 | mptexd 6487 | . 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ V) |
5 | fvexd 6203 | . 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 41030 | . . 3 ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) |
17 | fvexd 6203 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ∈ V) | |
18 | 1 | eluzelz2 39627 | . . . . 5 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
19 | 14, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
20 | eqid 2622 | . . . 4 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
21 | 1 | eleq2i 2693 | . . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
22 | 21 | biimpi 206 | . . . . . . 7 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀)) |
23 | uzss 11708 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
24 | 22, 23 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
25 | 24, 1 | syl6sseqr 3652 | . . . . 5 ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ 𝑍) |
26 | 14, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
27 | ssid 3624 | . . . . 5 ⊢ (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑁) | |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑁)) |
29 | fvexd 6203 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) ∈ V) | |
30 | 6, 3, 17, 19, 20, 26, 28, 29 | climeqmpt 39929 | . . 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 247 | . 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) |
32 | breldmg 5330 | . 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 1326 | 1 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ dom ⇝ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 {crab 2916 Vcvv 3200 ⊆ wss 3574 ∩ ciin 4521 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ⟶wf 5884 ‘cfv 5888 supcsup 8346 ℝcr 9935 ℝ*cxr 10073 < clt 10074 ℤcz 11377 ℤ≥cuz 11687 lim supclsp 14201 ⇝ cli 14215 SAlgcsalg 40528 SMblFncsmblfn 40909 |
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-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-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-iin 4523 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-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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 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-ioo 12179 df-ico 12181 df-fz 12327 df-fl 12593 df-ceil 12594 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-smblfn 40910 |
This theorem is referenced by: smflimsuplem7 41032 |
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