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Theorem limsupre 42298
 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 10689 . . . . 5 -∞ ∈ ℝ*
21a1i 11 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ ∈ ℝ*)
3 renegcl 10940 . . . . . 6 (𝑏 ∈ ℝ → -𝑏 ∈ ℝ)
43rexrd 10682 . . . . 5 (𝑏 ∈ ℝ → -𝑏 ∈ ℝ*)
54ad2antlr 726 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -𝑏 ∈ ℝ*)
6 limsupre.f . . . . . . 7 (𝜑𝐹:𝐵⟶ℝ)
7 reex 10619 . . . . . . . . 9 ℝ ∈ V
87a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
9 limsupre.1 . . . . . . . 8 (𝜑𝐵 ⊆ ℝ)
108, 9ssexd 5192 . . . . . . 7 (𝜑𝐵 ∈ V)
11 fex 6966 . . . . . . 7 ((𝐹:𝐵⟶ℝ ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
126, 10, 11syl2anc 587 . . . . . 6 (𝜑𝐹 ∈ V)
13 limsupcl 14824 . . . . . 6 (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
1412, 13syl 17 . . . . 5 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
1514ad2antrr 725 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ∈ ℝ*)
163mnfltd 12509 . . . . 5 (𝑏 ∈ ℝ → -∞ < -𝑏)
1716ad2antlr 726 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ < -𝑏)
189ad2antrr 725 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝐵 ⊆ ℝ)
19 ressxr 10676 . . . . . . . 8 ℝ ⊆ ℝ*
2019a1i 11 . . . . . . 7 (𝜑 → ℝ ⊆ ℝ*)
216, 20fssd 6502 . . . . . 6 (𝜑𝐹:𝐵⟶ℝ*)
2221ad2antrr 725 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝐹:𝐵⟶ℝ*)
23 limsupre.2 . . . . . 6 (𝜑 → sup(𝐵, ℝ*, < ) = +∞)
2423ad2antrr 725 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → sup(𝐵, ℝ*, < ) = +∞)
25 simpr 488 . . . . . . 7 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
26 nfv 1915 . . . . . . . . 9 𝑘(𝜑𝑏 ∈ ℝ)
27 nfre1 3265 . . . . . . . . 9 𝑘𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)
2826, 27nfan 1900 . . . . . . . 8 𝑘((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
29 nfv 1915 . . . . . . . . . . . 12 𝑗(𝜑𝑏 ∈ ℝ)
30 nfv 1915 . . . . . . . . . . . 12 𝑗 𝑘 ∈ ℝ
31 nfra1 3183 . . . . . . . . . . . 12 𝑗𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)
3229, 30, 31nf3an 1902 . . . . . . . . . . 11 𝑗((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
33 simp13 1202 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
34 simp2 1134 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑗𝐵)
35 simp3 1135 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑘𝑗)
36 rspa 3171 . . . . . . . . . . . . . . . 16 ((∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ∧ 𝑗𝐵) → (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
3736imp 410 . . . . . . . . . . . . . . 15 (((∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ∧ 𝑗𝐵) ∧ 𝑘𝑗) → (abs‘(𝐹𝑗)) ≤ 𝑏)
3833, 34, 35, 37syl21anc 836 . . . . . . . . . . . . . 14 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (abs‘(𝐹𝑗)) ≤ 𝑏)
39 simp11l 1281 . . . . . . . . . . . . . . . 16 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝜑)
406ffvelrnda 6828 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝐵) → (𝐹𝑗) ∈ ℝ)
4139, 34, 40syl2anc 587 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (𝐹𝑗) ∈ ℝ)
42 simp11r 1282 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑏 ∈ ℝ)
4341, 42absled 14784 . . . . . . . . . . . . . 14 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → ((abs‘(𝐹𝑗)) ≤ 𝑏 ↔ (-𝑏 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ 𝑏)))
4438, 43mpbid 235 . . . . . . . . . . . . 13 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (-𝑏 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ 𝑏))
4544simpld 498 . . . . . . . . . . . 12 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → -𝑏 ≤ (𝐹𝑗))
46453exp 1116 . . . . . . . . . . 11 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑗𝐵 → (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
4732, 46ralrimi 3180 . . . . . . . . . 10 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
48473exp 1116 . . . . . . . . 9 ((𝜑𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))))
4948adantr 484 . . . . . . . 8 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))))
5028, 49reximdai 3270 . . . . . . 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 5034 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑖𝑗))
53 fveq2 6645 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝐹𝑖) = (𝐹𝑗))
5453breq2d 5042 . . . . . . . . . 10 (𝑖 = 𝑗 → (-𝑏 ≤ (𝐹𝑖) ↔ -𝑏 ≤ (𝐹𝑗)))
5552, 54imbi12d 348 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ (𝑗 → -𝑏 ≤ (𝐹𝑗))))
5655cbvralvw 3396 . . . . . . . 8 (∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∀𝑗𝐵 (𝑗 → -𝑏 ≤ (𝐹𝑗)))
57 breq1 5033 . . . . . . . . . 10 ( = 𝑘 → (𝑗𝑘𝑗))
5857imbi1d 345 . . . . . . . . 9 ( = 𝑘 → ((𝑗 → -𝑏 ≤ (𝐹𝑗)) ↔ (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
5958ralbidv 3162 . . . . . . . 8 ( = 𝑘 → (∀𝑗𝐵 (𝑗 → -𝑏 ≤ (𝐹𝑗)) ↔ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
6056, 59syl5bb 286 . . . . . . 7 ( = 𝑘 → (∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
6160cbvrexvw 3397 . . . . . 6 (∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
6251, 61sylibr 237 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)))
6318, 22, 5, 24, 62limsupbnd2 14834 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -𝑏 ≤ (lim sup‘𝐹))
642, 5, 15, 17, 63xrltletrd 12544 . . 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 3248 . 2 (𝜑 → -∞ < (lim sup‘𝐹))
67 rexr 10678 . . . . 5 (𝑏 ∈ ℝ → 𝑏 ∈ ℝ*)
6867ad2antlr 726 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝑏 ∈ ℝ*)
69 pnfxr 10686 . . . . 5 +∞ ∈ ℝ*
7069a1i 11 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → +∞ ∈ ℝ*)
7144simprd 499 . . . . . . . . . . . 12 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (𝐹𝑗) ≤ 𝑏)
72713exp 1116 . . . . . . . . . . 11 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑗𝐵 → (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
7332, 72ralrimi 3180 . . . . . . . . . 10 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
74733exp 1116 . . . . . . . . 9 ((𝜑𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))))
7574adantr 484 . . . . . . . 8 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))))
7628, 75reximdai 3270 . . . . . . 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 5040 . . . . . . . . . 10 (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑏 ↔ (𝐹𝑗) ≤ 𝑏))
7952, 78imbi12d 348 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ (𝑗 → (𝐹𝑗) ≤ 𝑏)))
8079cbvralvw 3396 . . . . . . . 8 (∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑗 → (𝐹𝑗) ≤ 𝑏))
8157imbi1d 345 . . . . . . . . 9 ( = 𝑘 → ((𝑗 → (𝐹𝑗) ≤ 𝑏) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8281ralbidv 3162 . . . . . . . 8 ( = 𝑘 → (∀𝑗𝐵 (𝑗 → (𝐹𝑗) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8380, 82syl5bb 286 . . . . . . 7 ( = 𝑘 → (∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8483cbvrexvw 3397 . . . . . 6 (∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
8577, 84sylibr 237 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏))
8618, 22, 68, 85limsupbnd1 14833 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ≤ 𝑏)
87 ltpnf 12505 . . . . 5 (𝑏 ∈ ℝ → 𝑏 < +∞)
8887ad2antlr 726 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝑏 < +∞)
8915, 68, 70, 86, 88xrlelttrd 12543 . . 3 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) < +∞)
9089, 65r19.29a 3248 . 2 (𝜑 → (lim sup‘𝐹) < +∞)
91 xrrebnd 12551 . . 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 712 1 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  supcsup 8890  ℝcr 10527  +∞cpnf 10663  -∞cmnf 10664  ℝ*cxr 10665   < clt 10666   ≤ cle 10667  -cneg 10862  abscabs 14587  lim supclsp 14821 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605  ax-pre-sup 10606 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-sup 8892  df-inf 8893  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-2 11690  df-3 11691  df-n0 11888  df-z 11972  df-uz 12234  df-rp 12380  df-ico 12734  df-seq 13367  df-exp 13428  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-limsup 14822 This theorem is referenced by:  limsupref  42342  ioodvbdlimc1lem2  42589  ioodvbdlimc2lem  42591
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