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Theorem limsupre 40443
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 10350 . . . . 5 -∞ ∈ ℝ*
21a1i 11 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ ∈ ℝ*)
3 renegcl 10598 . . . . . 6 (𝑏 ∈ ℝ → -𝑏 ∈ ℝ)
43rexrd 10343 . . . . 5 (𝑏 ∈ ℝ → -𝑏 ∈ ℝ*)
54ad2antlr 718 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -𝑏 ∈ ℝ*)
6 limsupre.f . . . . . . 7 (𝜑𝐹:𝐵⟶ℝ)
7 reex 10280 . . . . . . . . 9 ℝ ∈ V
87a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
9 limsupre.1 . . . . . . . 8 (𝜑𝐵 ⊆ ℝ)
108, 9ssexd 4966 . . . . . . 7 (𝜑𝐵 ∈ V)
11 fex 6682 . . . . . . 7 ((𝐹:𝐵⟶ℝ ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
126, 10, 11syl2anc 579 . . . . . 6 (𝜑𝐹 ∈ V)
13 limsupcl 14491 . . . . . 6 (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
1412, 13syl 17 . . . . 5 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
1514ad2antrr 717 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ∈ ℝ*)
163mnfltd 12158 . . . . 5 (𝑏 ∈ ℝ → -∞ < -𝑏)
1716ad2antlr 718 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ < -𝑏)
189ad2antrr 717 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝐵 ⊆ ℝ)
19 ressxr 10337 . . . . . . . 8 ℝ ⊆ ℝ*
2019a1i 11 . . . . . . 7 (𝜑 → ℝ ⊆ ℝ*)
216, 20fssd 6237 . . . . . 6 (𝜑𝐹:𝐵⟶ℝ*)
2221ad2antrr 717 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝐹:𝐵⟶ℝ*)
23 limsupre.2 . . . . . 6 (𝜑 → sup(𝐵, ℝ*, < ) = +∞)
2423ad2antrr 717 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → sup(𝐵, ℝ*, < ) = +∞)
25 simpr 477 . . . . . . 7 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
26 nfv 2009 . . . . . . . . 9 𝑘(𝜑𝑏 ∈ ℝ)
27 nfre1 3151 . . . . . . . . 9 𝑘𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)
2826, 27nfan 1998 . . . . . . . 8 𝑘((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
29 nfv 2009 . . . . . . . . . . . 12 𝑗(𝜑𝑏 ∈ ℝ)
30 nfv 2009 . . . . . . . . . . . 12 𝑗 𝑘 ∈ ℝ
31 nfra1 3088 . . . . . . . . . . . 12 𝑗𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)
3229, 30, 31nf3an 2000 . . . . . . . . . . 11 𝑗((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
33 simp13 1262 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
34 simp2 1167 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑗𝐵)
35 simp3 1168 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑘𝑗)
36 rspa 3077 . . . . . . . . . . . . . . . 16 ((∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ∧ 𝑗𝐵) → (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
3736imp 395 . . . . . . . . . . . . . . 15 (((∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ∧ 𝑗𝐵) ∧ 𝑘𝑗) → (abs‘(𝐹𝑗)) ≤ 𝑏)
3833, 34, 35, 37syl21anc 866 . . . . . . . . . . . . . 14 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (abs‘(𝐹𝑗)) ≤ 𝑏)
39 simp11l 1383 . . . . . . . . . . . . . . . 16 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝜑)
406ffvelrnda 6549 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝐵) → (𝐹𝑗) ∈ ℝ)
4139, 34, 40syl2anc 579 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (𝐹𝑗) ∈ ℝ)
42 simp11r 1384 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑏 ∈ ℝ)
4341, 42absled 14456 . . . . . . . . . . . . . 14 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → ((abs‘(𝐹𝑗)) ≤ 𝑏 ↔ (-𝑏 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ 𝑏)))
4438, 43mpbid 223 . . . . . . . . . . . . 13 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (-𝑏 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ 𝑏))
4544simpld 488 . . . . . . . . . . . 12 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → -𝑏 ≤ (𝐹𝑗))
46453exp 1148 . . . . . . . . . . 11 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑗𝐵 → (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
4732, 46ralrimi 3104 . . . . . . . . . 10 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
48473exp 1148 . . . . . . . . 9 ((𝜑𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))))
4948adantr 472 . . . . . . . 8 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))))
5028, 49reximdai 3158 . . . . . . 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 4813 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑖𝑗))
53 fveq2 6375 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝐹𝑖) = (𝐹𝑗))
5453breq2d 4821 . . . . . . . . . 10 (𝑖 = 𝑗 → (-𝑏 ≤ (𝐹𝑖) ↔ -𝑏 ≤ (𝐹𝑗)))
5552, 54imbi12d 335 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ (𝑗 → -𝑏 ≤ (𝐹𝑗))))
5655cbvralv 3319 . . . . . . . 8 (∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∀𝑗𝐵 (𝑗 → -𝑏 ≤ (𝐹𝑗)))
57 breq1 4812 . . . . . . . . . 10 ( = 𝑘 → (𝑗𝑘𝑗))
5857imbi1d 332 . . . . . . . . 9 ( = 𝑘 → ((𝑗 → -𝑏 ≤ (𝐹𝑗)) ↔ (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
5958ralbidv 3133 . . . . . . . 8 ( = 𝑘 → (∀𝑗𝐵 (𝑗 → -𝑏 ≤ (𝐹𝑗)) ↔ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
6056, 59syl5bb 274 . . . . . . 7 ( = 𝑘 → (∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
6160cbvrexv 3320 . . . . . 6 (∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
6251, 61sylibr 225 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)))
6318, 22, 5, 24, 62limsupbnd2 14501 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -𝑏 ≤ (lim sup‘𝐹))
642, 5, 15, 17, 63xrltletrd 12194 . . 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 3225 . 2 (𝜑 → -∞ < (lim sup‘𝐹))
67 rexr 10339 . . . . 5 (𝑏 ∈ ℝ → 𝑏 ∈ ℝ*)
6867ad2antlr 718 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝑏 ∈ ℝ*)
69 pnfxr 10346 . . . . 5 +∞ ∈ ℝ*
7069a1i 11 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → +∞ ∈ ℝ*)
7144simprd 489 . . . . . . . . . . . 12 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (𝐹𝑗) ≤ 𝑏)
72713exp 1148 . . . . . . . . . . 11 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑗𝐵 → (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
7332, 72ralrimi 3104 . . . . . . . . . 10 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
74733exp 1148 . . . . . . . . 9 ((𝜑𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))))
7574adantr 472 . . . . . . . 8 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))))
7628, 75reximdai 3158 . . . . . . 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 4819 . . . . . . . . . 10 (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑏 ↔ (𝐹𝑗) ≤ 𝑏))
7952, 78imbi12d 335 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ (𝑗 → (𝐹𝑗) ≤ 𝑏)))
8079cbvralv 3319 . . . . . . . 8 (∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑗 → (𝐹𝑗) ≤ 𝑏))
8157imbi1d 332 . . . . . . . . 9 ( = 𝑘 → ((𝑗 → (𝐹𝑗) ≤ 𝑏) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8281ralbidv 3133 . . . . . . . 8 ( = 𝑘 → (∀𝑗𝐵 (𝑗 → (𝐹𝑗) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8380, 82syl5bb 274 . . . . . . 7 ( = 𝑘 → (∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8483cbvrexv 3320 . . . . . 6 (∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
8577, 84sylibr 225 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏))
8618, 22, 68, 85limsupbnd1 14500 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ≤ 𝑏)
87 ltpnf 12154 . . . . 5 (𝑏 ∈ ℝ → 𝑏 < +∞)
8887ad2antlr 718 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝑏 < +∞)
8915, 68, 70, 86, 88xrlelttrd 12193 . . 3 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) < +∞)
9089, 65r19.29a 3225 . 2 (𝜑 → (lim sup‘𝐹) < +∞)
91 xrrebnd 12201 . . 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 704 1 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  wss 3732   class class class wbr 4809  wf 6064  cfv 6068  supcsup 8553  cr 10188  +∞cpnf 10325  -∞cmnf 10326  *cxr 10327   < clt 10328  cle 10329  -cneg 10521  abscabs 14261  lim supclsp 14488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-sup 8555  df-inf 8556  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-ico 12383  df-seq 13009  df-exp 13068  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-limsup 14489
This theorem is referenced by:  limsupref  40487  ioodvbdlimc1lem2  40717  ioodvbdlimc2lem  40719
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