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

Theorem limsupubuz 42281
Description: For a real-valued function on a set of upper integers, if the superior limit is not +∞, then the function is bounded above. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupubuz.j 𝑗𝐹
limsupubuz.z 𝑍 = (ℤ𝑀)
limsupubuz.f (𝜑𝐹:𝑍⟶ℝ)
limsupubuz.n (𝜑 → (lim sup‘𝐹) ≠ +∞)
Assertion
Ref Expression
limsupubuz (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
Distinct variable groups:   𝑥,𝐹   𝑥,𝑀   𝑗,𝑍,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑗)   𝐹(𝑗)   𝑀(𝑗)

Proof of Theorem limsupubuz
Dummy variables 𝑖 𝑘 𝑙 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1916 . . . . . 6 𝑙𝜑
2 nfcv 2982 . . . . . 6 𝑙𝐹
3 limsupubuz.z . . . . . . . 8 𝑍 = (ℤ𝑀)
4 uzssre 41959 . . . . . . . 8 (ℤ𝑀) ⊆ ℝ
53, 4eqsstri 3987 . . . . . . 7 𝑍 ⊆ ℝ
65a1i 11 . . . . . 6 (𝜑𝑍 ⊆ ℝ)
7 limsupubuz.f . . . . . . 7 (𝜑𝐹:𝑍⟶ℝ)
87frexr 41945 . . . . . 6 (𝜑𝐹:𝑍⟶ℝ*)
9 limsupubuz.n . . . . . 6 (𝜑 → (lim sup‘𝐹) ≠ +∞)
101, 2, 6, 8, 9limsupub 42272 . . . . 5 (𝜑 → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦))
1110adantr 484 . . . 4 ((𝜑𝑀 ∈ ℤ) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦))
12 nfv 1916 . . . . . . . . . . 11 𝑙 𝑀 ∈ ℤ
131, 12nfan 1901 . . . . . . . . . 10 𝑙(𝜑𝑀 ∈ ℤ)
14 nfv 1916 . . . . . . . . . 10 𝑙 𝑦 ∈ ℝ
1513, 14nfan 1901 . . . . . . . . 9 𝑙((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ)
16 nfv 1916 . . . . . . . . 9 𝑙 𝑘 ∈ ℝ
1715, 16nfan 1901 . . . . . . . 8 𝑙(((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ)
18 nfra1 3213 . . . . . . . 8 𝑙𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)
1917, 18nfan 1901 . . . . . . 7 𝑙((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦))
20 nfmpt1 5150 . . . . . . . . . . 11 𝑙(𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙))
2120nfrn 5811 . . . . . . . . . 10 𝑙ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙))
22 nfcv 2982 . . . . . . . . . 10 𝑙
23 nfcv 2982 . . . . . . . . . 10 𝑙 <
2421, 22, 23nfsup 8912 . . . . . . . . 9 𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < )
25 nfcv 2982 . . . . . . . . 9 𝑙
26 nfcv 2982 . . . . . . . . 9 𝑙𝑦
2724, 25, 26nfbr 5099 . . . . . . . 8 𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < ) ≤ 𝑦
2827, 26, 24nfif 4479 . . . . . . 7 𝑙if(sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < ) ≤ 𝑦, 𝑦, sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < ))
29 breq2 5056 . . . . . . . . . . . 12 (𝑙 = 𝑖 → (𝑘𝑙𝑘𝑖))
30 fveq2 6661 . . . . . . . . . . . . 13 (𝑙 = 𝑖 → (𝐹𝑙) = (𝐹𝑖))
3130breq1d 5062 . . . . . . . . . . . 12 (𝑙 = 𝑖 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑖) ≤ 𝑦))
3229, 31imbi12d 348 . . . . . . . . . . 11 (𝑙 = 𝑖 → ((𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦)))
3332cbvralvw 3434 . . . . . . . . . 10 (∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦))
3433biimpi 219 . . . . . . . . 9 (∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) → ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦))
3534adantl 485 . . . . . . . 8 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦))
36 simp-4r 783 . . . . . . . 8 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦)) → 𝑀 ∈ ℤ)
3735, 36syldan 594 . . . . . . 7 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → 𝑀 ∈ ℤ)
387ad4antr 731 . . . . . . . 8 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦)) → 𝐹:𝑍⟶ℝ)
3935, 38syldan 594 . . . . . . 7 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → 𝐹:𝑍⟶ℝ)
40 simpllr 775 . . . . . . . 8 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦)) → 𝑦 ∈ ℝ)
4135, 40syldan 594 . . . . . . 7 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → 𝑦 ∈ ℝ)
42 simplr 768 . . . . . . . 8 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦)) → 𝑘 ∈ ℝ)
4335, 42syldan 594 . . . . . . 7 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → 𝑘 ∈ ℝ)
4433biimpri 231 . . . . . . . 8 (∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦) → ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦))
4535, 44syl 17 . . . . . . 7 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦))
46 eqid 2824 . . . . . . 7 if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘)) = if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))
47 eqid 2824 . . . . . . 7 sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < ) = sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < )
48 eqid 2824 . . . . . . 7 if(sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < ) ≤ 𝑦, 𝑦, sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < )) = if(sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < ) ≤ 𝑦, 𝑦, sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < ))
4919, 28, 37, 3, 39, 41, 43, 45, 46, 47, 48limsupubuzlem 42280 . . . . . 6 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
5049rexlimdva2 3279 . . . . 5 (((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥))
5150rexlimdva 3276 . . . 4 ((𝜑𝑀 ∈ ℤ) → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥))
5211, 51mpd 15 . . 3 ((𝜑𝑀 ∈ ℤ) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
533a1i 11 . . . . . 6 𝑀 ∈ ℤ → 𝑍 = (ℤ𝑀))
54 uz0 41975 . . . . . 6 𝑀 ∈ ℤ → (ℤ𝑀) = ∅)
5553, 54eqtrd 2859 . . . . 5 𝑀 ∈ ℤ → 𝑍 = ∅)
56 0red 10642 . . . . . 6 (𝑍 = ∅ → 0 ∈ ℝ)
57 rzal 4436 . . . . . 6 (𝑍 = ∅ → ∀𝑙𝑍 (𝐹𝑙) ≤ 0)
58 brralrspcev 5112 . . . . . 6 ((0 ∈ ℝ ∧ ∀𝑙𝑍 (𝐹𝑙) ≤ 0) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
5956, 57, 58syl2anc 587 . . . . 5 (𝑍 = ∅ → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
6055, 59syl 17 . . . 4 𝑀 ∈ ℤ → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
6160adantl 485 . . 3 ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
6252, 61pm2.61dan 812 . 2 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
63 limsupubuz.j . . . . . 6 𝑗𝐹
64 nfcv 2982 . . . . . 6 𝑗𝑙
6563, 64nffv 6671 . . . . 5 𝑗(𝐹𝑙)
66 nfcv 2982 . . . . 5 𝑗
67 nfcv 2982 . . . . 5 𝑗𝑥
6865, 66, 67nfbr 5099 . . . 4 𝑗(𝐹𝑙) ≤ 𝑥
69 nfv 1916 . . . 4 𝑙(𝐹𝑗) ≤ 𝑥
70 fveq2 6661 . . . . 5 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
7170breq1d 5062 . . . 4 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
7268, 69, 71cbvralw 3425 . . 3 (∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
7372rexbii 3241 . 2 (∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
7462, 73sylib 221 1 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wnfc 2962  wne 3014  wral 3133  wrex 3134  wss 3919  c0 4276  ifcif 4450   class class class wbr 5052  cmpt 5132  ran crn 5543  wf 6339  cfv 6343  (class class class)co 7149  supcsup 8901  cr 10534  0cc0 10535  +∞cpnf 10670   < clt 10673  cle 10674  cz 11978  cuz 12240  ...cfz 12894  cceil 13165  lim supclsp 14827
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-n0 11895  df-z 11979  df-uz 12241  df-ico 12741  df-fz 12895  df-fl 13166  df-ceil 13167  df-limsup 14828
This theorem is referenced by:  limsupubuzmpt  42287  limsupvaluz2  42306  supcnvlimsup  42308
  Copyright terms: Public domain W3C validator