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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupref | Structured version Visualization version GIF version |
Description: If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupref.j | ⊢ Ⅎ𝑗𝐹 |
limsupref.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
limsupref.s | ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) |
limsupref.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
limsupref.b | ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) |
Ref | Expression |
---|---|
limsupref | ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupref.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | limsupref.s | . 2 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) | |
3 | limsupref.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
4 | limsupref.b | . . 3 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) | |
5 | breq2 4790 | . . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((abs‘(𝐹‘𝑗)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑗)) ≤ 𝑦)) | |
6 | 5 | imbi2d 329 | . . . . . . 7 ⊢ (𝑏 = 𝑦 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
7 | 6 | ralbidv 3135 | . . . . . 6 ⊢ (𝑏 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
8 | 7 | rexbidv 3200 | . . . . 5 ⊢ (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
9 | breq1 4789 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗)) | |
10 | 9 | imbi1d 330 | . . . . . . . . 9 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
11 | 10 | ralbidv 3135 | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
12 | nfv 1995 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) | |
13 | nfv 1995 | . . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑥 | |
14 | nfcv 2913 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑗abs | |
15 | limsupref.j | . . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐹 | |
16 | nfcv 2913 | . . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑥 | |
17 | 15, 16 | nffv 6339 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝐹‘𝑥) |
18 | 14, 17 | nffv 6339 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) |
19 | nfcv 2913 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤ | |
20 | nfcv 2913 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦 | |
21 | 18, 19, 20 | nfbr 4833 | . . . . . . . . . . 11 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) ≤ 𝑦 |
22 | 13, 21 | nfim 1977 | . . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦) |
23 | breq2 4790 | . . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑥)) | |
24 | fveq2 6332 | . . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑥 → (𝐹‘𝑗) = (𝐹‘𝑥)) | |
25 | 24 | fveq2d 6336 | . . . . . . . . . . . 12 ⊢ (𝑗 = 𝑥 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑥))) |
26 | 25 | breq1d 4796 | . . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → ((abs‘(𝐹‘𝑗)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
27 | 23, 26 | imbi12d 333 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → ((𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
28 | 12, 22, 27 | cbvral 3316 | . . . . . . . . 9 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
29 | 28 | a1i 11 | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
30 | 11, 29 | bitrd 268 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
31 | 30 | cbvrexv 3321 | . . . . . 6 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
32 | 31 | a1i 11 | . . . . 5 ⊢ (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
33 | 8, 32 | bitrd 268 | . . . 4 ⊢ (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
34 | 33 | cbvrexv 3321 | . . 3 ⊢ (∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
35 | 4, 34 | sylib 208 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
36 | 1, 2, 3, 35 | limsupre 40391 | 1 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1631 ∈ wcel 2145 Ⅎwnfc 2900 ∀wral 3061 ∃wrex 3062 ⊆ wss 3723 class class class wbr 4786 ⟶wf 6027 ‘cfv 6031 supcsup 8502 ℝcr 10137 +∞cpnf 10273 ℝ*cxr 10275 < clt 10276 ≤ cle 10277 abscabs 14182 lim supclsp 14409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-sup 8504 df-inf 8505 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11495 df-z 11580 df-uz 11889 df-rp 12036 df-ico 12386 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 |
This theorem is referenced by: (None) |
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