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  Ⅎwnfc 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  ℝcr 10534  0cc0 10535  +∞cpnf 10670   < clt 10673   ≤ cle 10674  ℤcz 11978  ℤ≥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