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Theorem limsupre2lem 40274
Description: Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupre2lem.1 𝑗𝐹
limsupre2lem.2 (𝜑𝐴 ⊆ ℝ)
limsupre2lem.3 (𝜑𝐹:𝐴⟶ℝ*)
Assertion
Ref Expression
limsupre2lem (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 < (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑥))))
Distinct variable groups:   𝐴,𝑗,𝑘,𝑥   𝑘,𝐹,𝑥   𝜑,𝑗,𝑘,𝑥
Allowed substitution hint:   𝐹(𝑗)

Proof of Theorem limsupre2lem
StepHypRef Expression
1 limsupre2lem.3 . . . . 5 (𝜑𝐹:𝐴⟶ℝ*)
2 reex 10065 . . . . . . 7 ℝ ∈ V
32a1i 11 . . . . . 6 (𝜑 → ℝ ∈ V)
4 limsupre2lem.2 . . . . . 6 (𝜑𝐴 ⊆ ℝ)
53, 4ssexd 4838 . . . . 5 (𝜑𝐴 ∈ V)
61, 5fexd 39610 . . . 4 (𝜑𝐹 ∈ V)
76limsupcld 40240 . . 3 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
8 xrre4 39951 . . 3 ((lim sup‘𝐹) ∈ ℝ* → ((lim sup‘𝐹) ∈ ℝ ↔ ((lim sup‘𝐹) ≠ -∞ ∧ (lim sup‘𝐹) ≠ +∞)))
97, 8syl 17 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ ((lim sup‘𝐹) ≠ -∞ ∧ (lim sup‘𝐹) ≠ +∞)))
10 df-ne 2824 . . . . 5 ((lim sup‘𝐹) ≠ -∞ ↔ ¬ (lim sup‘𝐹) = -∞)
1110a1i 11 . . . 4 (𝜑 → ((lim sup‘𝐹) ≠ -∞ ↔ ¬ (lim sup‘𝐹) = -∞))
12 limsupre2lem.1 . . . . . 6 𝑗𝐹
1312, 4, 1limsupmnf 40271 . . . . 5 (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
1413notbid 307 . . . 4 (𝜑 → (¬ (lim sup‘𝐹) = -∞ ↔ ¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
15 annim 440 . . . . . . . . . . . 12 ((𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥) ↔ ¬ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
1615rexbii 3070 . . . . . . . . . . 11 (∃𝑗𝐴 (𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑗𝐴 ¬ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
17 rexnal 3024 . . . . . . . . . . 11 (∃𝑗𝐴 ¬ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ¬ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
1816, 17bitri 264 . . . . . . . . . 10 (∃𝑗𝐴 (𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥) ↔ ¬ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
1918ralbii 3009 . . . . . . . . 9 (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥) ↔ ∀𝑘 ∈ ℝ ¬ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
20 ralnex 3021 . . . . . . . . 9 (∀𝑘 ∈ ℝ ¬ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ¬ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
2119, 20bitri 264 . . . . . . . 8 (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥) ↔ ¬ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
2221rexbii 3070 . . . . . . 7 (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ¬ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
23 rexnal 3024 . . . . . . 7 (∃𝑥 ∈ ℝ ¬ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
2422, 23bitr2i 265 . . . . . 6 (¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥))
2524a1i 11 . . . . 5 (𝜑 → (¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥)))
26 simplr 807 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → 𝑥 ∈ ℝ)
2726rexrd 10127 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → 𝑥 ∈ ℝ*)
281adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℝ) → 𝐹:𝐴⟶ℝ*)
2928ffvelrnda 6399 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → (𝐹𝑗) ∈ ℝ*)
3027, 29xrltnled 39892 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → (𝑥 < (𝐹𝑗) ↔ ¬ (𝐹𝑗) ≤ 𝑥))
3130bicomd 213 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → (¬ (𝐹𝑗) ≤ 𝑥𝑥 < (𝐹𝑗)))
3231anbi2d 740 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → ((𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥) ↔ (𝑘𝑗𝑥 < (𝐹𝑗))))
3332rexbidva 3078 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (∃𝑗𝐴 (𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑗𝐴 (𝑘𝑗𝑥 < (𝐹𝑗))))
3433ralbidv 3015 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 < (𝐹𝑗))))
3534rexbidva 3078 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ ¬ (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 < (𝐹𝑗))))
3625, 35bitrd 268 . . . 4 (𝜑 → (¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 < (𝐹𝑗))))
3711, 14, 363bitrd 294 . . 3 (𝜑 → ((lim sup‘𝐹) ≠ -∞ ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 < (𝐹𝑗))))
38 df-ne 2824 . . . . 5 ((lim sup‘𝐹) ≠ +∞ ↔ ¬ (lim sup‘𝐹) = +∞)
3938a1i 11 . . . 4 (𝜑 → ((lim sup‘𝐹) ≠ +∞ ↔ ¬ (lim sup‘𝐹) = +∞))
4012, 4, 1limsuppnf 40261 . . . . 5 (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
4140notbid 307 . . . 4 (𝜑 → (¬ (lim sup‘𝐹) = +∞ ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
4229, 27xrltnled 39892 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → ((𝐹𝑗) < 𝑥 ↔ ¬ 𝑥 ≤ (𝐹𝑗)))
4342imbi2d 329 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → ((𝑘𝑗 → (𝐹𝑗) < 𝑥) ↔ (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗))))
4443ralbidva 3014 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗))))
4544rexbidv 3081 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗))))
4645rexbidva 3078 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑥) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗))))
47 imnan 437 . . . . . . . . . . . 12 ((𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗)) ↔ ¬ (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
4847ralbii 3009 . . . . . . . . . . 11 (∀𝑗𝐴 (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗)) ↔ ∀𝑗𝐴 ¬ (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
49 ralnex 3021 . . . . . . . . . . 11 (∀𝑗𝐴 ¬ (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ↔ ¬ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
5048, 49bitri 264 . . . . . . . . . 10 (∀𝑗𝐴 (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗)) ↔ ¬ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
5150rexbii 3070 . . . . . . . . 9 (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑘 ∈ ℝ ¬ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
52 rexnal 3024 . . . . . . . . 9 (∃𝑘 ∈ ℝ ¬ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ↔ ¬ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
5351, 52bitri 264 . . . . . . . 8 (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗)) ↔ ¬ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
5453rexbii 3070 . . . . . . 7 (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑥 ∈ ℝ ¬ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
55 rexnal 3024 . . . . . . 7 (∃𝑥 ∈ ℝ ¬ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
5654, 55bitri 264 . . . . . 6 (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
5756a1i 11 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → ¬ 𝑥 ≤ (𝐹𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
5846, 57bitr2d 269 . . . 4 (𝜑 → (¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑥)))
5939, 41, 583bitrd 294 . . 3 (𝜑 → ((lim sup‘𝐹) ≠ +∞ ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑥)))
6037, 59anbi12d 747 . 2 (𝜑 → (((lim sup‘𝐹) ≠ -∞ ∧ (lim sup‘𝐹) ≠ +∞) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 < (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑥))))
619, 60bitrd 268 1 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 < (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wnfc 2780  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607   class class class wbr 4685  wf 5922  cfv 5926  cr 9973  +∞cpnf 10109  -∞cmnf 10110  *cxr 10111   < clt 10112  cle 10113  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-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  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-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-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  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-ico 12219  df-limsup 14246
This theorem is referenced by:  limsupre2  40275
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