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Theorem limsupbnd1f 45609
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 5169 . . . . . . 7 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
65imbi1d 341 . . . . . 6 (𝑘 = 𝑖 → ((𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)))
76ralbidv 3184 . . . . 5 (𝑘 = 𝑖 → (∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑗𝐵 (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)))
8 nfv 1913 . . . . . . 7 𝑙(𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)
9 nfv 1913 . . . . . . . 8 𝑗 𝑖𝑙
10 limsupbnd1f.1 . . . . . . . . . 10 𝑗𝐹
11 nfcv 2908 . . . . . . . . . 10 𝑗𝑙
1210, 11nffv 6932 . . . . . . . . 9 𝑗(𝐹𝑙)
13 nfcv 2908 . . . . . . . . 9 𝑗
14 nfcv 2908 . . . . . . . . 9 𝑗𝐴
1512, 13, 14nfbr 5213 . . . . . . . 8 𝑗(𝐹𝑙) ≤ 𝐴
169, 15nfim 1895 . . . . . . 7 𝑗(𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)
17 breq2 5170 . . . . . . . 8 (𝑗 = 𝑙 → (𝑖𝑗𝑖𝑙))
18 fveq2 6922 . . . . . . . . 9 (𝑗 = 𝑙 → (𝐹𝑗) = (𝐹𝑙))
1918breq1d 5176 . . . . . . . 8 (𝑗 = 𝑙 → ((𝐹𝑗) ≤ 𝐴 ↔ (𝐹𝑙) ≤ 𝐴))
2017, 19imbi12d 344 . . . . . . 7 (𝑗 = 𝑙 → ((𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
218, 16, 20cbvralw 3312 . . . . . 6 (∀𝑗𝐵 (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
2221a1i 11 . . . . 5 (𝑘 = 𝑖 → (∀𝑗𝐵 (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
237, 22bitrd 279 . . . 4 (𝑘 = 𝑖 → (∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
2423cbvrexvw 3244 . . 3 (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∃𝑖 ∈ ℝ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
254, 24sylib 218 . 2 (𝜑 → ∃𝑖 ∈ ℝ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
261, 2, 3, 25limsupbnd1 15530 1 (𝜑 → (lim sup‘𝐹) ≤ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2108  wnfc 2893  wral 3067  wrex 3076  wss 3976   class class class wbr 5166  wf 6571  cfv 6575  cr 11185  *cxr 11325  cle 11327  lim supclsp 15518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-cnex 11242  ax-resscn 11243  ax-1cn 11244  ax-icn 11245  ax-addcl 11246  ax-addrcl 11247  ax-mulcl 11248  ax-mulrcl 11249  ax-mulcom 11250  ax-addass 11251  ax-mulass 11252  ax-distr 11253  ax-i2m1 11254  ax-1ne0 11255  ax-1rid 11256  ax-rnegex 11257  ax-rrecex 11258  ax-cnre 11259  ax-pre-lttri 11260  ax-pre-lttrn 11261  ax-pre-ltadd 11262  ax-pre-mulgt0 11263  ax-pre-sup 11264
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-er 8765  df-en 9006  df-dom 9007  df-sdom 9008  df-sup 9513  df-inf 9514  df-pnf 11328  df-mnf 11329  df-xr 11330  df-ltxr 11331  df-le 11332  df-sub 11524  df-neg 11525  df-ico 13415  df-limsup 15519
This theorem is referenced by:  limsuppnflem  45633
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