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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  Ⅎwnfc 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  ℝcr 9973  0cc0 9974  +∞cpnf 10109   < clt 10112   ≤ cle 10113  ℤcz 11415  ℤ≥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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