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 5047 | . . . . . 6 ⊢ (𝑏 = 𝑦 → ((abs‘(𝐹‘𝑗)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑗)) ≤ 𝑦)) | |
6 | 5 | imbi2d 344 | . . . . 5 ⊢ (𝑏 = 𝑦 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
7 | 6 | ralbidv 3111 | . . . 4 ⊢ (𝑏 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
8 | breq1 5046 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗)) | |
9 | 8 | imbi1d 345 | . . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
10 | 9 | ralbidv 3111 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦))) |
11 | nfv 1922 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) | |
12 | nfv 1922 | . . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑥 | |
13 | nfcv 2900 | . . . . . . . . . 10 ⊢ Ⅎ𝑗abs | |
14 | limsupref.j | . . . . . . . . . . 11 ⊢ Ⅎ𝑗𝐹 | |
15 | nfcv 2900 | . . . . . . . . . . 11 ⊢ Ⅎ𝑗𝑥 | |
16 | 14, 15 | nffv 6716 | . . . . . . . . . 10 ⊢ Ⅎ𝑗(𝐹‘𝑥) |
17 | 13, 16 | nffv 6716 | . . . . . . . . 9 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) |
18 | nfcv 2900 | . . . . . . . . 9 ⊢ Ⅎ𝑗 ≤ | |
19 | nfcv 2900 | . . . . . . . . 9 ⊢ Ⅎ𝑗𝑦 | |
20 | 17, 18, 19 | nfbr 5090 | . . . . . . . 8 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) ≤ 𝑦 |
21 | 12, 20 | nfim 1904 | . . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦) |
22 | breq2 5047 | . . . . . . . 8 ⊢ (𝑗 = 𝑥 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑥)) | |
23 | 2fveq3 6711 | . . . . . . . . 9 ⊢ (𝑗 = 𝑥 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑥))) | |
24 | 23 | breq1d 5053 | . . . . . . . 8 ⊢ (𝑗 = 𝑥 → ((abs‘(𝐹‘𝑗)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
25 | 22, 24 | imbi12d 348 | . . . . . . 7 ⊢ (𝑗 = 𝑥 → ((𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
26 | 11, 21, 25 | cbvralw 3342 | . . . . . 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 282 | . . . 4 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))) |
29 | 7, 28 | cbvrex2vw 3365 | . . 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 221 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)) |
31 | 1, 2, 3, 30 | limsupre 42811 | 1 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 Ⅎwnfc 2880 ∀wral 3054 ∃wrex 3055 ⊆ wss 3857 class class class wbr 5043 ⟶wf 6365 ‘cfv 6369 supcsup 9045 ℝcr 10711 +∞cpnf 10847 ℝ*cxr 10849 < clt 10850 ≤ cle 10851 abscabs 14780 lim supclsp 15014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-sup 9047 df-inf 9048 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-ico 12924 df-seq 13558 df-exp 13619 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 |
This theorem is referenced by: (None) |
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