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Theorem limsupubuz 40263
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 1883 . . . . . 6 𝑙𝜑
2 nfcv 2793 . . . . . 6 𝑙𝐹
3 limsupubuz.z . . . . . . . 8 𝑍 = (ℤ𝑀)
4 uzssre 39933 . . . . . . . 8 (ℤ𝑀) ⊆ ℝ
53, 4eqsstri 3668 . . . . . . 7 𝑍 ⊆ ℝ
65a1i 11 . . . . . 6 (𝜑𝑍 ⊆ ℝ)
7 limsupubuz.f . . . . . . 7 (𝜑𝐹:𝑍⟶ℝ)
87frexr 39917 . . . . . 6 (𝜑𝐹:𝑍⟶ℝ*)
9 limsupubuz.n . . . . . 6 (𝜑 → (lim sup‘𝐹) ≠ +∞)
101, 2, 6, 8, 9limsupub 40254 . . . . 5 (𝜑 → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦))
1110adantr 480 . . . 4 ((𝜑𝑀 ∈ ℤ) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦))
12 nfv 1883 . . . . . . . . . . . 12 𝑙 𝑀 ∈ ℤ
131, 12nfan 1868 . . . . . . . . . . 11 𝑙(𝜑𝑀 ∈ ℤ)
14 nfv 1883 . . . . . . . . . . 11 𝑙 𝑦 ∈ ℝ
1513, 14nfan 1868 . . . . . . . . . 10 𝑙((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ)
16 nfv 1883 . . . . . . . . . 10 𝑙 𝑘 ∈ ℝ
1715, 16nfan 1868 . . . . . . . . 9 𝑙(((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ)
18 nfra1 2970 . . . . . . . . 9 𝑙𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)
1917, 18nfan 1868 . . . . . . . 8 𝑙((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦))
20 nfmpt1 4780 . . . . . . . . . . . 12 𝑙(𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙))
2120nfrn 5400 . . . . . . . . . . 11 𝑙ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙))
22 nfcv 2793 . . . . . . . . . . 11 𝑙
23 nfcv 2793 . . . . . . . . . . 11 𝑙 <
2421, 22, 23nfsup 8398 . . . . . . . . . 10 𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < )
25 nfcv 2793 . . . . . . . . . 10 𝑙
26 nfcv 2793 . . . . . . . . . 10 𝑙𝑦
2724, 25, 26nfbr 4732 . . . . . . . . 9 𝑙sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < ) ≤ 𝑦
2827, 26, 24nfif 4148 . . . . . . . 8 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 4689 . . . . . . . . . . . . 13 (𝑙 = 𝑖 → (𝑘𝑙𝑘𝑖))
30 fveq2 6229 . . . . . . . . . . . . . 14 (𝑙 = 𝑖 → (𝐹𝑙) = (𝐹𝑖))
3130breq1d 4695 . . . . . . . . . . . . 13 (𝑙 = 𝑖 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑖) ≤ 𝑦))
3229, 31imbi12d 333 . . . . . . . . . . . 12 (𝑙 = 𝑖 → ((𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦)))
3332cbvralv 3201 . . . . . . . . . . 11 (∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) ↔ ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦))
3433biimpi 206 . . . . . . . . . 10 (∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) → ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦))
3534adantl 481 . . . . . . . . 9 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦))
36 simp-4r 824 . . . . . . . . 9 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦)) → 𝑀 ∈ ℤ)
3735, 36syldan 486 . . . . . . . 8 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → 𝑀 ∈ ℤ)
387ad4antr 769 . . . . . . . . 9 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦)) → 𝐹:𝑍⟶ℝ)
3935, 38syldan 486 . . . . . . . 8 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → 𝐹:𝑍⟶ℝ)
40 simpllr 815 . . . . . . . . 9 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦)) → 𝑦 ∈ ℝ)
4135, 40syldan 486 . . . . . . . 8 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → 𝑦 ∈ ℝ)
42 simplr 807 . . . . . . . . 9 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦)) → 𝑘 ∈ ℝ)
4335, 42syldan 486 . . . . . . . 8 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → 𝑘 ∈ ℝ)
4433biimpri 218 . . . . . . . . 9 (∀𝑖𝑍 (𝑘𝑖 → (𝐹𝑖) ≤ 𝑦) → ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦))
4535, 44syl 17 . . . . . . . 8 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦))
46 eqid 2651 . . . . . . . 8 if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘)) = if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))
47 eqid 2651 . . . . . . . 8 sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < ) = sup(ran (𝑙 ∈ (𝑀...if((⌈‘𝑘) ≤ 𝑀, 𝑀, (⌈‘𝑘))) ↦ (𝐹𝑙)), ℝ, < )
48 eqid 2651 . . . . . . . 8 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 40262 . . . . . . 7 (((((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
5049exp31 629 . . . . . 6 (((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)))
5150rexlimdv 3059 . . . . 5 (((𝜑𝑀 ∈ ℤ) ∧ 𝑦 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥))
5251rexlimdva 3060 . . . 4 ((𝜑𝑀 ∈ ℤ) → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑙𝑍 (𝑘𝑙 → (𝐹𝑙) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥))
5311, 52mpd 15 . . 3 ((𝜑𝑀 ∈ ℤ) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
543a1i 11 . . . . . 6 𝑀 ∈ ℤ → 𝑍 = (ℤ𝑀))
55 uz0 39952 . . . . . 6 𝑀 ∈ ℤ → (ℤ𝑀) = ∅)
5654, 55eqtrd 2685 . . . . 5 𝑀 ∈ ℤ → 𝑍 = ∅)
57 0red 10079 . . . . . 6 (𝑍 = ∅ → 0 ∈ ℝ)
58 rzal 4106 . . . . . 6 (𝑍 = ∅ → ∀𝑙𝑍 (𝐹𝑙) ≤ 0)
59 breq2 4689 . . . . . . . 8 (𝑥 = 0 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑙) ≤ 0))
6059ralbidv 3015 . . . . . . 7 (𝑥 = 0 → (∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥 ↔ ∀𝑙𝑍 (𝐹𝑙) ≤ 0))
6160rspcev 3340 . . . . . 6 ((0 ∈ ℝ ∧ ∀𝑙𝑍 (𝐹𝑙) ≤ 0) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
6257, 58, 61syl2anc 694 . . . . 5 (𝑍 = ∅ → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
6356, 62syl 17 . . . 4 𝑀 ∈ ℤ → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
6463adantl 481 . . 3 ((𝜑 ∧ ¬ 𝑀 ∈ ℤ) → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
6553, 64pm2.61dan 849 . 2 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥)
66 limsupubuz.j . . . . . 6 𝑗𝐹
67 nfcv 2793 . . . . . 6 𝑗𝑙
6866, 67nffv 6236 . . . . 5 𝑗(𝐹𝑙)
69 nfcv 2793 . . . . 5 𝑗
70 nfcv 2793 . . . . 5 𝑗𝑥
7168, 69, 70nfbr 4732 . . . 4 𝑗(𝐹𝑙) ≤ 𝑥
72 nfv 1883 . . . 4 𝑙(𝐹𝑗) ≤ 𝑥
73 fveq2 6229 . . . . 5 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
7473breq1d 4695 . . . 4 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
7571, 72, 74cbvral 3197 . . 3 (∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
7675rexbii 3070 . 2 (∃𝑥 ∈ ℝ ∀𝑙𝑍 (𝐹𝑙) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
7765, 76sylib 208 1 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wnfc 2780  wne 2823  wral 2941  wrex 2942  wss 3607  c0 3948  ifcif 4119   class class class wbr 4685  cmpt 4762  ran crn 5144  wf 5922  cfv 5926  (class class class)co 6690  supcsup 8387  cr 9973  0cc0 9974  +∞cpnf 10109   < clt 10112  cle 10113  cz 11415  cuz 11725  ...cfz 12364  cceil 12632  lim supclsp 14245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-ico 12219  df-fz 12365  df-fl 12633  df-ceil 12634  df-limsup 14246
This theorem is referenced by:  limsupubuzmpt  40269  limsupvaluz2  40288  supcnvlimsup  40290
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