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Theorem limsupbnd1f 45542
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 5172 . . . . . . 7 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
65imbi1d 341 . . . . . 6 (𝑘 = 𝑖 → ((𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)))
76ralbidv 3180 . . . . 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 2904 . . . . . . . . . 10 𝑗𝑙
1210, 11nffv 6929 . . . . . . . . 9 𝑗(𝐹𝑙)
13 nfcv 2904 . . . . . . . . 9 𝑗
14 nfcv 2904 . . . . . . . . 9 𝑗𝐴
1512, 13, 14nfbr 5216 . . . . . . . 8 𝑗(𝐹𝑙) ≤ 𝐴
169, 15nfim 1895 . . . . . . 7 𝑗(𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)
17 breq2 5173 . . . . . . . 8 (𝑗 = 𝑙 → (𝑖𝑗𝑖𝑙))
18 fveq2 6919 . . . . . . . . 9 (𝑗 = 𝑙 → (𝐹𝑗) = (𝐹𝑙))
1918breq1d 5179 . . . . . . . 8 (𝑗 = 𝑙 → ((𝐹𝑗) ≤ 𝐴 ↔ (𝐹𝑙) ≤ 𝐴))
2017, 19imbi12d 344 . . . . . . 7 (𝑗 = 𝑙 → ((𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
218, 16, 20cbvralw 3307 . . . . . 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 3239 . . 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 15524 1 (𝜑 → (lim sup‘𝐹) ≤ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2103  wnfc 2888  wral 3063  wrex 3072  wss 3970   class class class wbr 5169  wf 6568  cfv 6572  cr 11179  *cxr 11319  cle 11321  lim supclsp 15512
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257  ax-pre-sup 11258
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-po 5611  df-so 5612  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-er 8759  df-en 9000  df-dom 9001  df-sdom 9002  df-sup 9507  df-inf 9508  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-ico 13409  df-limsup 15513
This theorem is referenced by:  limsuppnflem  45566
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