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

Theorem limsupre 38708
 Description: If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)
Hypotheses
Ref Expression
limsupre.1 (𝜑𝐵 ⊆ ℝ)
limsupre.2 (𝜑 → sup(𝐵, ℝ*, < ) = +∞)
limsupre.f (𝜑𝐹:𝐵⟶ℝ)
limsupre.bnd (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
Assertion
Ref Expression
limsupre (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Distinct variable groups:   𝐵,𝑗,𝑘   𝐹,𝑏,𝑗,𝑘   𝜑,𝑏,𝑗,𝑘
Allowed substitution hint:   𝐵(𝑏)

Proof of Theorem limsupre
Dummy variables 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnfxr 9975 . . . . 5 -∞ ∈ ℝ*
21a1i 11 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ ∈ ℝ*)
3 renegcl 10223 . . . . . 6 (𝑏 ∈ ℝ → -𝑏 ∈ ℝ)
43rexrd 9968 . . . . 5 (𝑏 ∈ ℝ → -𝑏 ∈ ℝ*)
54ad2antlr 759 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -𝑏 ∈ ℝ*)
6 limsupre.f . . . . . . 7 (𝜑𝐹:𝐵⟶ℝ)
7 reex 9906 . . . . . . . . 9 ℝ ∈ V
87a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
9 limsupre.1 . . . . . . . 8 (𝜑𝐵 ⊆ ℝ)
108, 9ssexd 4733 . . . . . . 7 (𝜑𝐵 ∈ V)
11 fex 6394 . . . . . . 7 ((𝐹:𝐵⟶ℝ ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
126, 10, 11syl2anc 691 . . . . . 6 (𝜑𝐹 ∈ V)
13 limsupcl 14052 . . . . . 6 (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
1412, 13syl 17 . . . . 5 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
1514ad2antrr 758 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ∈ ℝ*)
163mnfltd 11834 . . . . 5 (𝑏 ∈ ℝ → -∞ < -𝑏)
1716ad2antlr 759 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ < -𝑏)
189ad2antrr 758 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝐵 ⊆ ℝ)
19 ressxr 9962 . . . . . . . 8 ℝ ⊆ ℝ*
2019a1i 11 . . . . . . 7 (𝜑 → ℝ ⊆ ℝ*)
216, 20fssd 5970 . . . . . 6 (𝜑𝐹:𝐵⟶ℝ*)
2221ad2antrr 758 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝐹:𝐵⟶ℝ*)
23 limsupre.2 . . . . . 6 (𝜑 → sup(𝐵, ℝ*, < ) = +∞)
2423ad2antrr 758 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → sup(𝐵, ℝ*, < ) = +∞)
25 simpr 476 . . . . . . 7 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
26 nfv 1830 . . . . . . . . 9 𝑘(𝜑𝑏 ∈ ℝ)
27 nfre1 2988 . . . . . . . . 9 𝑘𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)
2826, 27nfan 1816 . . . . . . . 8 𝑘((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
29 nfv 1830 . . . . . . . . . . . 12 𝑗(𝜑𝑏 ∈ ℝ)
30 nfv 1830 . . . . . . . . . . . 12 𝑗 𝑘 ∈ ℝ
31 nfra1 2925 . . . . . . . . . . . 12 𝑗𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)
3229, 30, 31nf3an 1819 . . . . . . . . . . 11 𝑗((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
33 simp13 1086 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
34 simp2 1055 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑗𝐵)
35 simp3 1056 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑘𝑗)
36 rspa 2914 . . . . . . . . . . . . . . . 16 ((∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ∧ 𝑗𝐵) → (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
3736imp 444 . . . . . . . . . . . . . . 15 (((∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ∧ 𝑗𝐵) ∧ 𝑘𝑗) → (abs‘(𝐹𝑗)) ≤ 𝑏)
3833, 34, 35, 37syl21anc 1317 . . . . . . . . . . . . . 14 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (abs‘(𝐹𝑗)) ≤ 𝑏)
39 simp11l 1165 . . . . . . . . . . . . . . . 16 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝜑)
406ffvelrnda 6267 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝐵) → (𝐹𝑗) ∈ ℝ)
4139, 34, 40syl2anc 691 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (𝐹𝑗) ∈ ℝ)
42 simp11r 1166 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑏 ∈ ℝ)
4341, 42absled 14017 . . . . . . . . . . . . . 14 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → ((abs‘(𝐹𝑗)) ≤ 𝑏 ↔ (-𝑏 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ 𝑏)))
4438, 43mpbid 221 . . . . . . . . . . . . 13 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (-𝑏 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ 𝑏))
4544simpld 474 . . . . . . . . . . . 12 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → -𝑏 ≤ (𝐹𝑗))
46453exp 1256 . . . . . . . . . . 11 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑗𝐵 → (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
4732, 46ralrimi 2940 . . . . . . . . . 10 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
48473exp 1256 . . . . . . . . 9 ((𝜑𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))))
4948adantr 480 . . . . . . . 8 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))))
5028, 49reximdai 2995 . . . . . . 7 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
5125, 50mpd 15 . . . . . 6 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
52 breq2 4587 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑖𝑗))
53 fveq2 6103 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝐹𝑖) = (𝐹𝑗))
5453breq2d 4595 . . . . . . . . . 10 (𝑖 = 𝑗 → (-𝑏 ≤ (𝐹𝑖) ↔ -𝑏 ≤ (𝐹𝑗)))
5552, 54imbi12d 333 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ (𝑗 → -𝑏 ≤ (𝐹𝑗))))
5655cbvralv 3147 . . . . . . . 8 (∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∀𝑗𝐵 (𝑗 → -𝑏 ≤ (𝐹𝑗)))
57 breq1 4586 . . . . . . . . . 10 ( = 𝑘 → (𝑗𝑘𝑗))
5857imbi1d 330 . . . . . . . . 9 ( = 𝑘 → ((𝑗 → -𝑏 ≤ (𝐹𝑗)) ↔ (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
5958ralbidv 2969 . . . . . . . 8 ( = 𝑘 → (∀𝑗𝐵 (𝑗 → -𝑏 ≤ (𝐹𝑗)) ↔ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
6056, 59syl5bb 271 . . . . . . 7 ( = 𝑘 → (∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
6160cbvrexv 3148 . . . . . 6 (∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
6251, 61sylibr 223 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)))
6318, 22, 5, 24, 62limsupbnd2 14062 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -𝑏 ≤ (lim sup‘𝐹))
642, 5, 15, 17, 63xrltletrd 11868 . . 3 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ < (lim sup‘𝐹))
65 limsupre.bnd . . 3 (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
6664, 65r19.29a 3060 . 2 (𝜑 → -∞ < (lim sup‘𝐹))
67 rexr 9964 . . . . 5 (𝑏 ∈ ℝ → 𝑏 ∈ ℝ*)
6867ad2antlr 759 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝑏 ∈ ℝ*)
69 pnfxr 9971 . . . . 5 +∞ ∈ ℝ*
7069a1i 11 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → +∞ ∈ ℝ*)
7144simprd 478 . . . . . . . . . . . 12 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (𝐹𝑗) ≤ 𝑏)
72713exp 1256 . . . . . . . . . . 11 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑗𝐵 → (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
7332, 72ralrimi 2940 . . . . . . . . . 10 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
74733exp 1256 . . . . . . . . 9 ((𝜑𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))))
7574adantr 480 . . . . . . . 8 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))))
7628, 75reximdai 2995 . . . . . . 7 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
7725, 76mpd 15 . . . . . 6 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
7853breq1d 4593 . . . . . . . . . 10 (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑏 ↔ (𝐹𝑗) ≤ 𝑏))
7952, 78imbi12d 333 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ (𝑗 → (𝐹𝑗) ≤ 𝑏)))
8079cbvralv 3147 . . . . . . . 8 (∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑗 → (𝐹𝑗) ≤ 𝑏))
8157imbi1d 330 . . . . . . . . 9 ( = 𝑘 → ((𝑗 → (𝐹𝑗) ≤ 𝑏) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8281ralbidv 2969 . . . . . . . 8 ( = 𝑘 → (∀𝑗𝐵 (𝑗 → (𝐹𝑗) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8380, 82syl5bb 271 . . . . . . 7 ( = 𝑘 → (∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8483cbvrexv 3148 . . . . . 6 (∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
8577, 84sylibr 223 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏))
8618, 22, 68, 85limsupbnd1 14061 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ≤ 𝑏)
87 ltpnf 11830 . . . . 5 (𝑏 ∈ ℝ → 𝑏 < +∞)
8887ad2antlr 759 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝑏 < +∞)
8915, 68, 70, 86, 88xrlelttrd 11867 . . 3 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) < +∞)
9089, 65r19.29a 3060 . 2 (𝜑 → (lim sup‘𝐹) < +∞)
91 xrrebnd 11873 . . 3 ((lim sup‘𝐹) ∈ ℝ* → ((lim sup‘𝐹) ∈ ℝ ↔ (-∞ < (lim sup‘𝐹) ∧ (lim sup‘𝐹) < +∞)))
9214, 91syl 17 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (-∞ < (lim sup‘𝐹) ∧ (lim sup‘𝐹) < +∞)))
9366, 90, 92mpbir2and 959 1 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583  ⟶wf 5800  ‘cfv 5804  supcsup 8229  ℝcr 9814  +∞cpnf 9950  -∞cmnf 9951  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  -cneg 10146  abscabs 13822  lim supclsp 14049 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050 This theorem is referenced by:  ioodvbdlimc1lem2  38822  ioodvbdlimc2lem  38824
 Copyright terms: Public domain W3C validator