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Theorem limsupbnd1f 40236
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 4688 . . . . . . 7 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
65imbi1d 330 . . . . . 6 (𝑘 = 𝑖 → ((𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)))
76ralbidv 3015 . . . . 5 (𝑘 = 𝑖 → (∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑗𝐵 (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)))
8 nfv 1883 . . . . . . 7 𝑙(𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)
9 nfv 1883 . . . . . . . 8 𝑗 𝑖𝑙
10 limsupbnd1f.1 . . . . . . . . . 10 𝑗𝐹
11 nfcv 2793 . . . . . . . . . 10 𝑗𝑙
1210, 11nffv 6236 . . . . . . . . 9 𝑗(𝐹𝑙)
13 nfcv 2793 . . . . . . . . 9 𝑗
14 nfcv 2793 . . . . . . . . 9 𝑗𝐴
1512, 13, 14nfbr 4732 . . . . . . . 8 𝑗(𝐹𝑙) ≤ 𝐴
169, 15nfim 1865 . . . . . . 7 𝑗(𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)
17 breq2 4689 . . . . . . . 8 (𝑗 = 𝑙 → (𝑖𝑗𝑖𝑙))
18 fveq2 6229 . . . . . . . . 9 (𝑗 = 𝑙 → (𝐹𝑗) = (𝐹𝑙))
1918breq1d 4695 . . . . . . . 8 (𝑗 = 𝑙 → ((𝐹𝑗) ≤ 𝐴 ↔ (𝐹𝑙) ≤ 𝐴))
2017, 19imbi12d 333 . . . . . . 7 (𝑗 = 𝑙 → ((𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
218, 16, 20cbvral 3197 . . . . . 6 (∀𝑗𝐵 (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
2221a1i 11 . . . . 5 (𝑘 = 𝑖 → (∀𝑗𝐵 (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
237, 22bitrd 268 . . . 4 (𝑘 = 𝑖 → (∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
2423cbvrexv 3202 . . 3 (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∃𝑖 ∈ ℝ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
254, 24sylib 208 . 2 (𝜑 → ∃𝑖 ∈ ℝ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
261, 2, 3, 25limsupbnd1 14257 1 (𝜑 → (lim sup‘𝐹) ≤ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wnfc 2780  wral 2941  wrex 2942  wss 3607   class class class wbr 4685  wf 5922  cfv 5926  cr 9973  *cxr 10111  cle 10113  lim supclsp 14245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-ico 12219  df-limsup 14246
This theorem is referenced by:  limsuppnflem  40260
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