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Theorem limsupre 45661
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 11319 . . . . 5 -∞ ∈ ℝ*
21a1i 11 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ ∈ ℝ*)
3 renegcl 11573 . . . . . 6 (𝑏 ∈ ℝ → -𝑏 ∈ ℝ)
43rexrd 11312 . . . . 5 (𝑏 ∈ ℝ → -𝑏 ∈ ℝ*)
54ad2antlr 727 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -𝑏 ∈ ℝ*)
6 limsupre.f . . . . . . 7 (𝜑𝐹:𝐵⟶ℝ)
7 reex 11247 . . . . . . . . 9 ℝ ∈ V
87a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
9 limsupre.1 . . . . . . . 8 (𝜑𝐵 ⊆ ℝ)
108, 9ssexd 5323 . . . . . . 7 (𝜑𝐵 ∈ V)
116, 10fexd 7248 . . . . . 6 (𝜑𝐹 ∈ V)
12 limsupcl 15510 . . . . . 6 (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*)
1311, 12syl 17 . . . . 5 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
1413ad2antrr 726 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ∈ ℝ*)
153mnfltd 13167 . . . . 5 (𝑏 ∈ ℝ → -∞ < -𝑏)
1615ad2antlr 727 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ < -𝑏)
179ad2antrr 726 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝐵 ⊆ ℝ)
18 ressxr 11306 . . . . . . . 8 ℝ ⊆ ℝ*
1918a1i 11 . . . . . . 7 (𝜑 → ℝ ⊆ ℝ*)
206, 19fssd 6752 . . . . . 6 (𝜑𝐹:𝐵⟶ℝ*)
2120ad2antrr 726 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝐹:𝐵⟶ℝ*)
22 limsupre.2 . . . . . 6 (𝜑 → sup(𝐵, ℝ*, < ) = +∞)
2322ad2antrr 726 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → sup(𝐵, ℝ*, < ) = +∞)
24 simpr 484 . . . . . . 7 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
25 nfv 1913 . . . . . . . . 9 𝑘(𝜑𝑏 ∈ ℝ)
26 nfre1 3284 . . . . . . . . 9 𝑘𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)
2725, 26nfan 1898 . . . . . . . 8 𝑘((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
28 nfv 1913 . . . . . . . . . . . 12 𝑗(𝜑𝑏 ∈ ℝ)
29 nfv 1913 . . . . . . . . . . . 12 𝑗 𝑘 ∈ ℝ
30 nfra1 3283 . . . . . . . . . . . 12 𝑗𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)
3128, 29, 30nf3an 1900 . . . . . . . . . . 11 𝑗((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
32 simp13 1205 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
33 simp2 1137 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑗𝐵)
34 simp3 1138 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑘𝑗)
35 rspa 3247 . . . . . . . . . . . . . . . 16 ((∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ∧ 𝑗𝐵) → (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
3635imp 406 . . . . . . . . . . . . . . 15 (((∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) ∧ 𝑗𝐵) ∧ 𝑘𝑗) → (abs‘(𝐹𝑗)) ≤ 𝑏)
3732, 33, 34, 36syl21anc 837 . . . . . . . . . . . . . 14 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (abs‘(𝐹𝑗)) ≤ 𝑏)
38 simp11l 1284 . . . . . . . . . . . . . . . 16 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝜑)
396ffvelcdmda 7103 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝐵) → (𝐹𝑗) ∈ ℝ)
4038, 33, 39syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (𝐹𝑗) ∈ ℝ)
41 simp11r 1285 . . . . . . . . . . . . . . 15 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → 𝑏 ∈ ℝ)
4240, 41absled 15470 . . . . . . . . . . . . . 14 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → ((abs‘(𝐹𝑗)) ≤ 𝑏 ↔ (-𝑏 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ 𝑏)))
4337, 42mpbid 232 . . . . . . . . . . . . 13 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (-𝑏 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ 𝑏))
4443simpld 494 . . . . . . . . . . . 12 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → -𝑏 ≤ (𝐹𝑗))
45443exp 1119 . . . . . . . . . . 11 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑗𝐵 → (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
4631, 45ralrimi 3256 . . . . . . . . . 10 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
47463exp 1119 . . . . . . . . 9 ((𝜑𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))))
4847adantr 480 . . . . . . . 8 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))))
4927, 48reximdai 3260 . . . . . . 7 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
5024, 49mpd 15 . . . . . 6 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
51 breq2 5146 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑖𝑗))
52 fveq2 6905 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝐹𝑖) = (𝐹𝑗))
5352breq2d 5154 . . . . . . . . . 10 (𝑖 = 𝑗 → (-𝑏 ≤ (𝐹𝑖) ↔ -𝑏 ≤ (𝐹𝑗)))
5451, 53imbi12d 344 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ (𝑗 → -𝑏 ≤ (𝐹𝑗))))
5554cbvralvw 3236 . . . . . . . 8 (∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∀𝑗𝐵 (𝑗 → -𝑏 ≤ (𝐹𝑗)))
56 breq1 5145 . . . . . . . . . 10 ( = 𝑘 → (𝑗𝑘𝑗))
5756imbi1d 341 . . . . . . . . 9 ( = 𝑘 → ((𝑗 → -𝑏 ≤ (𝐹𝑗)) ↔ (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
5857ralbidv 3177 . . . . . . . 8 ( = 𝑘 → (∀𝑗𝐵 (𝑗 → -𝑏 ≤ (𝐹𝑗)) ↔ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
5955, 58bitrid 283 . . . . . . 7 ( = 𝑘 → (∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗))))
6059cbvrexvw 3237 . . . . . 6 (∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → -𝑏 ≤ (𝐹𝑗)))
6150, 60sylibr 234 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → -𝑏 ≤ (𝐹𝑖)))
6217, 21, 5, 23, 61limsupbnd2 15520 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -𝑏 ≤ (lim sup‘𝐹))
632, 5, 14, 16, 62xrltletrd 13204 . . 3 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → -∞ < (lim sup‘𝐹))
64 limsupre.bnd . . 3 (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))
6563, 64r19.29a 3161 . 2 (𝜑 → -∞ < (lim sup‘𝐹))
66 rexr 11308 . . . . 5 (𝑏 ∈ ℝ → 𝑏 ∈ ℝ*)
6766ad2antlr 727 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝑏 ∈ ℝ*)
68 pnfxr 11316 . . . . 5 +∞ ∈ ℝ*
6968a1i 11 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → +∞ ∈ ℝ*)
7043simprd 495 . . . . . . . . . . . 12 ((((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) ∧ 𝑗𝐵𝑘𝑗) → (𝐹𝑗) ≤ 𝑏)
71703exp 1119 . . . . . . . . . . 11 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑗𝐵 → (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
7231, 71ralrimi 3256 . . . . . . . . . 10 (((𝜑𝑏 ∈ ℝ) ∧ 𝑘 ∈ ℝ ∧ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
73723exp 1119 . . . . . . . . 9 ((𝜑𝑏 ∈ ℝ) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))))
7473adantr 480 . . . . . . . 8 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (𝑘 ∈ ℝ → (∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))))
7527, 74reximdai 3260 . . . . . . 7 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
7624, 75mpd 15 . . . . . 6 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
7752breq1d 5152 . . . . . . . . . 10 (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑏 ↔ (𝐹𝑗) ≤ 𝑏))
7851, 77imbi12d 344 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ (𝑗 → (𝐹𝑗) ≤ 𝑏)))
7978cbvralvw 3236 . . . . . . . 8 (∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑗 → (𝐹𝑗) ≤ 𝑏))
8056imbi1d 341 . . . . . . . . 9 ( = 𝑘 → ((𝑗 → (𝐹𝑗) ≤ 𝑏) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8180ralbidv 3177 . . . . . . . 8 ( = 𝑘 → (∀𝑗𝐵 (𝑗 → (𝐹𝑗) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8279, 81bitrid 283 . . . . . . 7 ( = 𝑘 → (∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏)))
8382cbvrexvw 3237 . . . . . 6 (∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑏))
8476, 83sylibr 234 . . . . 5 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → ∃ ∈ ℝ ∀𝑖𝐵 (𝑖 → (𝐹𝑖) ≤ 𝑏))
8517, 21, 67, 84limsupbnd1 15519 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) ≤ 𝑏)
86 ltpnf 13163 . . . . 5 (𝑏 ∈ ℝ → 𝑏 < +∞)
8786ad2antlr 727 . . . 4 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → 𝑏 < +∞)
8814, 67, 69, 85, 87xrlelttrd 13203 . . 3 (((𝜑𝑏 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏)) → (lim sup‘𝐹) < +∞)
8988, 64r19.29a 3161 . 2 (𝜑 → (lim sup‘𝐹) < +∞)
90 xrrebnd 13211 . . 3 ((lim sup‘𝐹) ∈ ℝ* → ((lim sup‘𝐹) ∈ ℝ ↔ (-∞ < (lim sup‘𝐹) ∧ (lim sup‘𝐹) < +∞)))
9113, 90syl 17 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (-∞ < (lim sup‘𝐹) ∧ (lim sup‘𝐹) < +∞)))
9265, 89, 91mpbir2and 713 1 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  wss 3950   class class class wbr 5142  wf 6556  cfv 6560  supcsup 9481  cr 11155  +∞cpnf 11293  -∞cmnf 11294  *cxr 11295   < clt 11296  cle 11297  -cneg 11494  abscabs 15274  lim supclsp 15507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-ico 13394  df-seq 14044  df-exp 14104  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-limsup 15508
This theorem is referenced by:  limsupref  45705  ioodvbdlimc1lem2  45952  ioodvbdlimc2lem  45954
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