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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 5150 | . . . . . 6 ⊢ (𝑏 = 𝑦 → ((abs‘(𝐹‘𝑗)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑗)) ≤ 𝑦)) | |
6 | 5 | imbi2d 341 | . . . . 5 ⊢ (𝑏 = 𝑦 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
7 | 6 | ralbidv 3178 | . . . 4 ⊢ (𝑏 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
8 | breq1 5149 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗)) | |
9 | 8 | imbi1d 342 | . . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
10 | 9 | ralbidv 3178 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
11 | nfv 1918 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) | |
12 | nfv 1918 | . . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑥 | |
13 | nfcv 2904 | . . . . . . . . . 10 ⊢ Ⅎ𝑗abs | |
14 | limsupref.j | . . . . . . . . . . 11 ⊢ Ⅎ𝑗𝐹 | |
15 | nfcv 2904 | . . . . . . . . . . 11 ⊢ Ⅎ𝑗𝑥 | |
16 | 14, 15 | nffv 6897 | . . . . . . . . . 10 ⊢ Ⅎ𝑗(𝐹‘𝑥) |
17 | 13, 16 | nffv 6897 | . . . . . . . . 9 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) |
18 | nfcv 2904 | . . . . . . . . 9 ⊢ Ⅎ𝑗 ≤ | |
19 | nfcv 2904 | . . . . . . . . 9 ⊢ Ⅎ𝑗𝑦 | |
20 | 17, 18, 19 | nfbr 5193 | . . . . . . . 8 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) ≤ 𝑦 |
21 | 12, 20 | nfim 1900 | . . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦) |
22 | breq2 5150 | . . . . . . . 8 ⊢ (𝑗 = 𝑥 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑥)) | |
23 | 2fveq3 6892 | . . . . . . . . 9 ⊢ (𝑗 = 𝑥 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑥))) | |
24 | 23 | breq1d 5156 | . . . . . . . 8 ⊢ (𝑗 = 𝑥 → ((abs‘(𝐹‘𝑗)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
25 | 22, 24 | imbi12d 345 | . . . . . . 7 ⊢ (𝑗 = 𝑥 → ((𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
26 | 11, 21, 25 | cbvralw 3304 | . . . . . 6 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
27 | 26 | a1i 11 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
28 | 10, 27 | bitrd 279 | . . . 4 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
29 | 7, 28 | cbvrex2vw 3240 | . . 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)) |
30 | 4, 29 | sylib 217 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
31 | 1, 2, 3, 30 | limsupre 44291 | 1 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2884 ∀wral 3062 ∃wrex 3071 ⊆ wss 3946 class class class wbr 5146 ⟶wf 6535 ‘cfv 6539 supcsup 9430 ℝcr 11104 +∞cpnf 11240 ℝ*cxr 11242 < clt 11243 ≤ cle 11244 abscabs 15176 lim supclsp 15409 |
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 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-inf 9433 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-n0 12468 df-z 12554 df-uz 12818 df-rp 12970 df-ico 13325 df-seq 13962 df-exp 14023 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15410 |
This theorem is referenced by: (None) |
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