Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limsupbnd1f Structured version   Visualization version   GIF version

Theorem limsupbnd1f 42323
 Description: If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupbnd1f.1 𝑗𝐹
limsupbnd1f.2 (𝜑𝐵 ⊆ ℝ)
limsupbnd1f.3 (𝜑𝐹:𝐵⟶ℝ*)
limsupbnd1f.4 (𝜑𝐴 ∈ ℝ*)
limsupbnd1f.5 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴))
Assertion
Ref Expression
limsupbnd1f (𝜑 → (lim sup‘𝐹) ≤ 𝐴)
Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑗,𝑘   𝑘,𝐹
Allowed substitution hints:   𝜑(𝑗,𝑘)   𝐹(𝑗)

Proof of Theorem limsupbnd1f
Dummy variables 𝑖 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limsupbnd1f.2 . 2 (𝜑𝐵 ⊆ ℝ)
2 limsupbnd1f.3 . 2 (𝜑𝐹:𝐵⟶ℝ*)
3 limsupbnd1f.4 . 2 (𝜑𝐴 ∈ ℝ*)
4 limsupbnd1f.5 . . 3 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴))
5 breq1 5033 . . . . . . 7 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
65imbi1d 345 . . . . . 6 (𝑘 = 𝑖 → ((𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)))
76ralbidv 3162 . . . . 5 (𝑘 = 𝑖 → (∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑗𝐵 (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)))
8 nfv 1915 . . . . . . 7 𝑙(𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)
9 nfv 1915 . . . . . . . 8 𝑗 𝑖𝑙
10 limsupbnd1f.1 . . . . . . . . . 10 𝑗𝐹
11 nfcv 2955 . . . . . . . . . 10 𝑗𝑙
1210, 11nffv 6655 . . . . . . . . 9 𝑗(𝐹𝑙)
13 nfcv 2955 . . . . . . . . 9 𝑗
14 nfcv 2955 . . . . . . . . 9 𝑗𝐴
1512, 13, 14nfbr 5077 . . . . . . . 8 𝑗(𝐹𝑙) ≤ 𝐴
169, 15nfim 1897 . . . . . . 7 𝑗(𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)
17 breq2 5034 . . . . . . . 8 (𝑗 = 𝑙 → (𝑖𝑗𝑖𝑙))
18 fveq2 6645 . . . . . . . . 9 (𝑗 = 𝑙 → (𝐹𝑗) = (𝐹𝑙))
1918breq1d 5040 . . . . . . . 8 (𝑗 = 𝑙 → ((𝐹𝑗) ≤ 𝐴 ↔ (𝐹𝑙) ≤ 𝐴))
2017, 19imbi12d 348 . . . . . . 7 (𝑗 = 𝑙 → ((𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
218, 16, 20cbvralw 3387 . . . . . 6 (∀𝑗𝐵 (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
2221a1i 11 . . . . 5 (𝑘 = 𝑖 → (∀𝑗𝐵 (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
237, 22bitrd 282 . . . 4 (𝑘 = 𝑖 → (∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
2423cbvrexvw 3397 . . 3 (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∃𝑖 ∈ ℝ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
254, 24sylib 221 . 2 (𝜑 → ∃𝑖 ∈ ℝ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
261, 2, 3, 25limsupbnd1 14831 1 (𝜑 → (lim sup‘𝐹) ≤ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  ℝcr 10525  ℝ*cxr 10663   ≤ cle 10665  lim supclsp 14819 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-ico 12732  df-limsup 14820 This theorem is referenced by:  limsuppnflem  42347
 Copyright terms: Public domain W3C validator