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Theorem limsupre 40191
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 10134 . . . . 5 -∞ ∈ ℝ*
21a1i 11 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ ∈ ℝ*)
3 renegcl 10382 . . . . . 6 (𝑏 ∈ ℝ → -𝑏 ∈ ℝ)
43rexrd 10127 . . . . 5 (𝑏 ∈ ℝ → -𝑏 ∈ ℝ*)
54ad2antlr 763 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -𝑏 ∈ ℝ*)
6 limsupre.f . . . . . . 7 (𝜑𝐹:𝐵⟶ℝ)
7 reex 10065 . . . . . . . . 9 ℝ ∈ V
87a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
9 limsupre.1 . . . . . . . 8 (𝜑𝐵 ⊆ ℝ)
108, 9ssexd 4838 . . . . . . 7 (𝜑𝐵 ∈ V)
11 fex 6530 . . . . . . 7 ((𝐹:𝐵⟶ℝ ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
126, 10, 11syl2anc 694 . . . . . 6 (𝜑𝐹 ∈ V)
13 limsupcl 14248 . . . . . 6 (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
1412, 13syl 17 . . . . 5 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
1514ad2antrr 762 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ∈ ℝ*)
163mnfltd 11996 . . . . 5 (𝑏 ∈ ℝ → -∞ < -𝑏)
1716ad2antlr 763 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ < -𝑏)
189ad2antrr 762 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝐵 ⊆ ℝ)
19 ressxr 10121 . . . . . . . 8 ℝ ⊆ ℝ*
2019a1i 11 . . . . . . 7 (𝜑 → ℝ ⊆ ℝ*)
216, 20fssd 6095 . . . . . 6 (𝜑𝐹:𝐵⟶ℝ*)
2221ad2antrr 762 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝐹:𝐵⟶ℝ*)
23 limsupre.2 . . . . . 6 (𝜑 → sup(𝐵, ℝ*, < ) = +∞)
2423ad2antrr 762 . . . . 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 1883 . . . . . . . . 9 𝑘(𝜑𝑏 ∈ ℝ)
27 nfre1 3034 . . . . . . . . 9 𝑘𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)
2826, 27nfan 1868 . . . . . . . 8 𝑘((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
29 nfv 1883 . . . . . . . . . . . 12 𝑗(𝜑𝑏 ∈ ℝ)
30 nfv 1883 . . . . . . . . . . . 12 𝑗 𝑘 ∈ ℝ
31 nfra1 2970 . . . . . . . . . . . 12 𝑗𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)
3229, 30, 31nf3an 1871 . . . . . . . . . . 11 𝑗((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
33 simp13 1113 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
34 simp2 1082 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑗𝐵)
35 simp3 1083 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑘𝑗)
36 rspa 2959 . . . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . . 14 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (abs‘(𝐹𝑗)) ≤ 𝑏)
39 simp11l 1192 . . . . . . . . . . . . . . . 16 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝜑)
406ffvelrnda 6399 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝐵) → (𝐹𝑗) ∈ ℝ)
4139, 34, 40syl2anc 694 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (𝐹𝑗) ∈ ℝ)
42 simp11r 1193 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑏 ∈ ℝ)
4341, 42absled 14213 . . . . . . . . . . . . . 14 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → ((abs‘(𝐹𝑗)) ≤ 𝑏 ↔ (-𝑏 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ 𝑏)))
4438, 43mpbid 222 . . . . . . . . . . . . 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 1283 . . . . . . . . . . 11 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑗𝐵 → (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
4732, 46ralrimi 2986 . . . . . . . . . 10 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
48473exp 1283 . . . . . . . . 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 3041 . . . . . . 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 4689 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑖𝑗))
53 fveq2 6229 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝐹𝑖) = (𝐹𝑗))
5453breq2d 4697 . . . . . . . . . 10 (𝑖 = 𝑗 → (-𝑏 ≤ (𝐹𝑖) ↔ -𝑏 ≤ (𝐹𝑗)))
5552, 54imbi12d 333 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ (𝑗 → -𝑏 ≤ (𝐹𝑗))))
5655cbvralv 3201 . . . . . . . 8 (∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∀𝑗𝐵 (𝑗 → -𝑏 ≤ (𝐹𝑗)))
57 breq1 4688 . . . . . . . . . 10 ( = 𝑘 → (𝑗𝑘𝑗))
5857imbi1d 330 . . . . . . . . 9 ( = 𝑘 → ((𝑗 → -𝑏 ≤ (𝐹𝑗)) ↔ (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
5958ralbidv 3015 . . . . . . . 8 ( = 𝑘 → (∀𝑗𝐵 (𝑗 → -𝑏 ≤ (𝐹𝑗)) ↔ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
6056, 59syl5bb 272 . . . . . . 7 ( = 𝑘 → (∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
6160cbvrexv 3202 . . . . . 6 (∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
6251, 61sylibr 224 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)))
6318, 22, 5, 24, 62limsupbnd2 14258 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -𝑏 ≤ (lim sup‘𝐹))
642, 5, 15, 17, 63xrltletrd 12030 . . 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 3107 . 2 (𝜑 → -∞ < (lim sup‘𝐹))
67 rexr 10123 . . . . 5 (𝑏 ∈ ℝ → 𝑏 ∈ ℝ*)
6867ad2antlr 763 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝑏 ∈ ℝ*)
69 pnfxr 10130 . . . . 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 1283 . . . . . . . . . . 11 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑗𝐵 → (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
7332, 72ralrimi 2986 . . . . . . . . . 10 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
74733exp 1283 . . . . . . . . 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 3041 . . . . . . 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 4695 . . . . . . . . . 10 (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑏 ↔ (𝐹𝑗) ≤ 𝑏))
7952, 78imbi12d 333 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ (𝑗 → (𝐹𝑗) ≤ 𝑏)))
8079cbvralv 3201 . . . . . . . 8 (∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑗 → (𝐹𝑗) ≤ 𝑏))
8157imbi1d 330 . . . . . . . . 9 ( = 𝑘 → ((𝑗 → (𝐹𝑗) ≤ 𝑏) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8281ralbidv 3015 . . . . . . . 8 ( = 𝑘 → (∀𝑗𝐵 (𝑗 → (𝐹𝑗) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8380, 82syl5bb 272 . . . . . . 7 ( = 𝑘 → (∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8483cbvrexv 3202 . . . . . 6 (∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
8577, 84sylibr 224 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏))
8618, 22, 68, 85limsupbnd1 14257 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ≤ 𝑏)
87 ltpnf 11992 . . . . 5 (𝑏 ∈ ℝ → 𝑏 < +∞)
8887ad2antlr 763 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝑏 < +∞)
8915, 68, 70, 86, 88xrlelttrd 12029 . . 3 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) < +∞)
9089, 65r19.29a 3107 . 2 (𝜑 → (lim sup‘𝐹) < +∞)
91 xrrebnd 12037 . . 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 977 1 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607   class class class wbr 4685  wf 5922  cfv 5926  supcsup 8387  cr 9973  +∞cpnf 10109  -∞cmnf 10110  *cxr 10111   < clt 10112  cle 10113  -cneg 10305  abscabs 14018  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-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-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  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-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246
This theorem is referenced by:  limsupref  40235  ioodvbdlimc1lem2  40465  ioodvbdlimc2lem  40467
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