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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 6860 | . . . 4 ⊢ 𝑍 ∈ V |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑍 ∈ V) |
4 | 3 | mptexd 7178 | . 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ V) |
5 | fvexd 6861 | . 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 45155 | . . 3 ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) |
17 | fvexd 6861 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ∈ V) | |
18 | 1 | eluzelz2 43728 | . . . . 5 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
19 | 14, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
20 | eqid 2733 | . . . 4 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
21 | 1 | eleq2i 2826 | . . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
22 | 21 | biimpi 215 | . . . . . . 7 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀)) |
23 | uzss 12794 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
24 | 22, 23 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
25 | 24, 1 | sseqtrrdi 3999 | . . . . 5 ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ 𝑍) |
26 | 14, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
27 | ssid 3970 | . . . . 5 ⊢ (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑁) | |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑁)) |
29 | fvexd 6861 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) ∈ V) | |
30 | 6, 3, 17, 19, 20, 26, 28, 29 | climeqmpt 44028 | . . 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 257 | . 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) |
32 | breldmg 5869 | . 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 1372 | 1 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ dom ⇝ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 {crab 3406 Vcvv 3447 ⊆ wss 3914 ∩ ciin 4959 class class class wbr 5109 ↦ cmpt 5192 dom cdm 5637 ran crn 5638 ⟶wf 6496 ‘cfv 6500 supcsup 9384 ℝcr 11058 ℝ*cxr 11196 < clt 11197 ℤcz 12507 ℤ≥cuz 12771 lim supclsp 15361 ⇝ cli 15375 SAlgcsalg 44639 SMblFncsmblfn 45026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-ioo 13277 df-ico 13279 df-fz 13434 df-fl 13706 df-ceil 13707 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-limsup 15362 df-clim 15379 df-smblfn 45027 |
This theorem is referenced by: smflimsuplem7 45157 |
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