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Theorem limsupbnd1f 44861
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 5151 . . . . . . 7 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
65imbi1d 341 . . . . . 6 (𝑘 = 𝑖 → ((𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)))
76ralbidv 3176 . . . . 5 (𝑘 = 𝑖 → (∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∀𝑗𝐵 (𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)))
8 nfv 1916 . . . . . . 7 𝑙(𝑖𝑗 → (𝐹𝑗) ≤ 𝐴)
9 nfv 1916 . . . . . . . 8 𝑗 𝑖𝑙
10 limsupbnd1f.1 . . . . . . . . . 10 𝑗𝐹
11 nfcv 2902 . . . . . . . . . 10 𝑗𝑙
1210, 11nffv 6901 . . . . . . . . 9 𝑗(𝐹𝑙)
13 nfcv 2902 . . . . . . . . 9 𝑗
14 nfcv 2902 . . . . . . . . 9 𝑗𝐴
1512, 13, 14nfbr 5195 . . . . . . . 8 𝑗(𝐹𝑙) ≤ 𝐴
169, 15nfim 1898 . . . . . . 7 𝑗(𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)
17 breq2 5152 . . . . . . . 8 (𝑗 = 𝑙 → (𝑖𝑗𝑖𝑙))
18 fveq2 6891 . . . . . . . . 9 (𝑗 = 𝑙 → (𝐹𝑗) = (𝐹𝑙))
1918breq1d 5158 . . . . . . . 8 (𝑗 = 𝑙 → ((𝐹𝑗) ≤ 𝐴 ↔ (𝐹𝑙) ≤ 𝐴))
2017, 19imbi12d 344 . . . . . . 7 (𝑗 = 𝑙 → ((𝑖𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴)))
218, 16, 20cbvralw 3302 . . . . . 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 3234 . . 3 (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝐴) ↔ ∃𝑖 ∈ ℝ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
254, 24sylib 217 . 2 (𝜑 → ∃𝑖 ∈ ℝ ∀𝑙𝐵 (𝑖𝑙 → (𝐹𝑙) ≤ 𝐴))
261, 2, 3, 25limsupbnd1 15433 1 (𝜑 → (lim sup‘𝐹) ≤ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  wnfc 2882  wral 3060  wrex 3069  wss 3948   class class class wbr 5148  wf 6539  cfv 6543  cr 11115  *cxr 11254  cle 11256  lim supclsp 15421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-ico 13337  df-limsup 15422
This theorem is referenced by:  limsuppnflem  44885
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