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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 4930 | . . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((abs‘(𝐹‘𝑗)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑗)) ≤ 𝑦)) | |
6 | 5 | imbi2d 333 | . . . . . . 7 ⊢ (𝑏 = 𝑦 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
7 | 6 | ralbidv 3142 | . . . . . 6 ⊢ (𝑏 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
8 | 7 | rexbidv 3237 | . . . . 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 4929 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗)) | |
10 | 9 | imbi1d 334 | . . . . . . . . 9 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
11 | 10 | ralbidv 3142 | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
12 | nfv 1874 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) | |
13 | nfv 1874 | . . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑥 | |
14 | nfcv 2927 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑗abs | |
15 | limsupref.j | . . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐹 | |
16 | nfcv 2927 | . . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑥 | |
17 | 15, 16 | nffv 6507 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝐹‘𝑥) |
18 | 14, 17 | nffv 6507 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) |
19 | nfcv 2927 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤ | |
20 | nfcv 2927 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑦 | |
21 | 18, 19, 20 | nfbr 4973 | . . . . . . . . . . 11 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) ≤ 𝑦 |
22 | 13, 21 | nfim 1860 | . . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦) |
23 | breq2 4930 | . . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑥)) | |
24 | 2fveq3 6502 | . . . . . . . . . . . 12 ⊢ (𝑗 = 𝑥 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑥))) | |
25 | 24 | breq1d 4936 | . . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → ((abs‘(𝐹‘𝑗)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
26 | 23, 25 | imbi12d 337 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → ((𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
27 | 12, 22, 26 | cbvral 3374 | . . . . . . . . 9 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
28 | 27 | a1i 11 | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
29 | 11, 28 | bitrd 271 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
30 | 29 | cbvrexv 3379 | . . . . . 6 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
31 | 30 | 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))) |
32 | 8, 31 | bitrd 271 | . . . 4 ⊢ (𝑏 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
33 | 32 | cbvrexv 3379 | . . 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)) |
34 | 4, 33 | sylib 210 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
35 | 1, 2, 3, 34 | limsupre 41383 | 1 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1508 ∈ wcel 2051 Ⅎwnfc 2911 ∀wral 3083 ∃wrex 3084 ⊆ wss 3824 class class class wbr 4926 ⟶wf 6182 ‘cfv 6186 supcsup 8698 ℝcr 10333 +∞cpnf 10470 ℝ*cxr 10472 < clt 10473 ≤ cle 10474 abscabs 14453 lim supclsp 14687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-sup 8700 df-inf 8701 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-n0 11707 df-z 11793 df-uz 12058 df-rp 12204 df-ico 12559 df-seq 13184 df-exp 13244 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455 df-limsup 14688 |
This theorem is referenced by: (None) |
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