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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupgt | Structured version Visualization version GIF version |
Description: Given a sequence of real numbers, there exists an upper part of the sequence that's appxoximated from below by the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsupgt.k | ⊢ Ⅎ𝑘𝐹 |
limsupgt.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
limsupgt.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
limsupgt.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
limsupgt.r | ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
limsupgt.x | ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
Ref | Expression |
---|---|
limsupgt | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupgt.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | limsupgt.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | limsupgt.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
4 | limsupgt.r | . . 3 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) | |
5 | limsupgt.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
6 | 1, 2, 3, 4, 5 | limsupgtlem 41523 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹)) |
7 | limsupgt.k | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐹 | |
8 | nfcv 2925 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑙 | |
9 | 7, 8 | nffv 6506 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
10 | nfcv 2925 | . . . . . . . 8 ⊢ Ⅎ𝑘 − | |
11 | nfcv 2925 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑋 | |
12 | 9, 10, 11 | nfov 7004 | . . . . . . 7 ⊢ Ⅎ𝑘((𝐹‘𝑙) − 𝑋) |
13 | nfcv 2925 | . . . . . . 7 ⊢ Ⅎ𝑘 < | |
14 | nfcv 2925 | . . . . . . . 8 ⊢ Ⅎ𝑘lim sup | |
15 | 14, 7 | nffv 6506 | . . . . . . 7 ⊢ Ⅎ𝑘(lim sup‘𝐹) |
16 | 12, 13, 15 | nfbr 4972 | . . . . . 6 ⊢ Ⅎ𝑘((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) |
17 | nfv 1874 | . . . . . 6 ⊢ Ⅎ𝑙((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹) | |
18 | fveq2 6496 | . . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
19 | 18 | oveq1d 6989 | . . . . . . 7 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) − 𝑋) = ((𝐹‘𝑘) − 𝑋)) |
20 | 19 | breq1d 4935 | . . . . . 6 ⊢ (𝑙 = 𝑘 → (((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) ↔ ((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))) |
21 | 16, 17, 20 | cbvral 3372 | . . . . 5 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑖)((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) ↔ ∀𝑘 ∈ (ℤ≥‘𝑖)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) ↔ ∀𝑘 ∈ (ℤ≥‘𝑖)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))) |
23 | fveq2 6496 | . . . . 5 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
24 | 23 | raleqdv 3348 | . . . 4 ⊢ (𝑖 = 𝑗 → (∀𝑘 ∈ (ℤ≥‘𝑖)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))) |
25 | 22, 24 | bitrd 271 | . . 3 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹))) |
26 | 25 | cbvrexv 3377 | . 2 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)((𝐹‘𝑙) − 𝑋) < (lim sup‘𝐹) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) |
27 | 6, 26 | sylib 210 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1508 ∈ wcel 2051 Ⅎwnfc 2909 ∀wral 3081 ∃wrex 3082 class class class wbr 4925 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 ℝcr 10332 < clt 10472 − cmin 10668 ℤcz 11791 ℤ≥cuz 12056 ℝ+crp 12202 lim supclsp 14686 |
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 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
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 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-sup 8699 df-inf 8700 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-n0 11706 df-z 11792 df-uz 12057 df-rp 12203 df-xadd 12323 df-ico 12558 df-fz 12707 df-fzo 12848 df-fl 12975 df-ceil 12976 df-limsup 14687 |
This theorem is referenced by: liminfltlem 41550 liminflimsupclim 41553 |
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