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Theorem limsupgtlem 40804
 Description: For any positive real, the superior limit of F is larger than any of its values at large enough arguments, up to that positive real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
limsupgtlem.m (𝜑𝑀 ∈ ℤ)
limsupgtlem.z 𝑍 = (ℤ𝑀)
limsupgtlem.f (𝜑𝐹:𝑍⟶ℝ)
limsupgtlem.r (𝜑 → (lim sup‘𝐹) ∈ ℝ)
limsupgtlem.x (𝜑𝑋 ∈ ℝ+)
Assertion
Ref Expression
limsupgtlem (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))
Distinct variable groups:   𝑗,𝐹,𝑘   𝑗,𝑋,𝑘   𝑗,𝑍,𝑘   𝜑,𝑗,𝑘
Allowed substitution hints:   𝑀(𝑗,𝑘)

Proof of Theorem limsupgtlem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nfv 2015 . . . 4 𝑗𝜑
2 limsupgtlem.m . . . . 5 (𝜑𝑀 ∈ ℤ)
3 limsupgtlem.z . . . . 5 𝑍 = (ℤ𝑀)
42, 3uzn0d 40447 . . . 4 (𝜑𝑍 ≠ ∅)
5 rnresss 40173 . . . . . . . 8 ran (𝐹 ↾ (ℤ𝑗)) ⊆ ran 𝐹
65a1i 11 . . . . . . 7 (𝜑 → ran (𝐹 ↾ (ℤ𝑗)) ⊆ ran 𝐹)
7 limsupgtlem.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶ℝ)
87frexr 40401 . . . . . . . 8 (𝜑𝐹:𝑍⟶ℝ*)
98frnd 6285 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ ℝ*)
106, 9sstrd 3837 . . . . . 6 (𝜑 → ran (𝐹 ↾ (ℤ𝑗)) ⊆ ℝ*)
1110supxrcld 40105 . . . . 5 (𝜑 → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ∈ ℝ*)
1211adantr 474 . . . 4 ((𝜑𝑗𝑍) → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ∈ ℝ*)
13 limsupgtlem.r . . . . . . 7 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14 nfcv 2969 . . . . . . . 8 𝑘𝐹
1514, 2, 3, 7limsupreuz 40764 . . . . . . 7 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) ∧ ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥)))
1613, 15mpbid 224 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) ∧ ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
1716simpld 490 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘))
18 rexr 10402 . . . . . . . . . 10 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
1918ad4antlr 728 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → 𝑥 ∈ ℝ*)
207ad2antrr 719 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝐹:𝑍⟶ℝ)
213uztrn2 11986 . . . . . . . . . . . . . 14 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
2221adantll 707 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
2320, 22ffvelrnd 6609 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ)
2423rexrd 10406 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
25243impa 1142 . . . . . . . . . 10 ((𝜑𝑗𝑍𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
2625ad5ant134 1488 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → (𝐹𝑘) ∈ ℝ*)
2711ad4antr 726 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ∈ ℝ*)
28 simpr 479 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → 𝑥 ≤ (𝐹𝑘))
2910ad2antrr 719 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ran (𝐹 ↾ (ℤ𝑗)) ⊆ ℝ*)
30 fvres 6452 . . . . . . . . . . . . . . 15 (𝑘 ∈ (ℤ𝑗) → ((𝐹 ↾ (ℤ𝑗))‘𝑘) = (𝐹𝑘))
3130eqcomd 2831 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ𝑗) → (𝐹𝑘) = ((𝐹 ↾ (ℤ𝑗))‘𝑘))
3231adantl 475 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) = ((𝐹 ↾ (ℤ𝑗))‘𝑘))
337ffnd 6279 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 Fn 𝑍)
3433adantr 474 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑍) → 𝐹 Fn 𝑍)
3522ssd 40069 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑍) → (ℤ𝑗) ⊆ 𝑍)
36 fnssres 6237 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝑍 ∧ (ℤ𝑗) ⊆ 𝑍) → (𝐹 ↾ (ℤ𝑗)) Fn (ℤ𝑗))
3734, 35, 36syl2anc 581 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → (𝐹 ↾ (ℤ𝑗)) Fn (ℤ𝑗))
3837adantr 474 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹 ↾ (ℤ𝑗)) Fn (ℤ𝑗))
39 simpr 479 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘 ∈ (ℤ𝑗))
4038, 39fnfvelrnd 40279 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝐹 ↾ (ℤ𝑗))‘𝑘) ∈ ran (𝐹 ↾ (ℤ𝑗)))
4132, 40eqeltrd 2906 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ran (𝐹 ↾ (ℤ𝑗)))
42 eqid 2825 . . . . . . . . . . . 12 sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )
4329, 41, 42supxrubd 40112 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
44433impa 1142 . . . . . . . . . 10 ((𝜑𝑗𝑍𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
4544ad5ant134 1488 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
4619, 26, 27, 28, 45xrletrd 12281 . . . . . . . 8 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
4746rexlimdva2 3243 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (∃𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )))
4847ralimdva 3171 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) → ∀𝑗𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )))
4948reximdva 3225 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )))
5017, 49mpd 15 . . . 4 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
51 limsupgtlem.x . . . . 5 (𝜑𝑋 ∈ ℝ+)
5251rphalfcld 12168 . . . 4 (𝜑 → (𝑋 / 2) ∈ ℝ+)
531, 4, 12, 50, 52infrpgernmpt 40489 . . 3 (𝜑 → ∃𝑗𝑍 sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2)))
54 simp3 1174 . . . . . . 7 ((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2))) → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2)))
552, 3, 8limsupvaluz 40735 . . . . . . . . . 10 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ))
5655eqcomd 2831 . . . . . . . . 9 (𝜑 → inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) = (lim sup‘𝐹))
5756oveq1d 6920 . . . . . . . 8 (𝜑 → (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) +𝑒 (𝑋 / 2)))
58573ad2ant1 1169 . . . . . . 7 ((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2))) → (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) +𝑒 (𝑋 / 2)))
5954, 58breqtrd 4899 . . . . . 6 ((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2))) → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2)))
60243adantl3 1215 . . . . . . . . . 10 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
61 simpl1 1248 . . . . . . . . . . 11 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝜑)
6261, 11syl 17 . . . . . . . . . 10 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ∈ ℝ*)
633fvexi 6447 . . . . . . . . . . . . . . 15 𝑍 ∈ V
6463a1i 11 . . . . . . . . . . . . . 14 (𝜑𝑍 ∈ V)
657, 64fexd 40111 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
6665limsupcld 40717 . . . . . . . . . . . 12 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
6751rpred 12156 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ ℝ)
6867rehalfcld 11605 . . . . . . . . . . . . 13 (𝜑 → (𝑋 / 2) ∈ ℝ)
6968rexrd 10406 . . . . . . . . . . . 12 (𝜑 → (𝑋 / 2) ∈ ℝ*)
7066, 69xaddcld 12419 . . . . . . . . . . 11 (𝜑 → ((lim sup‘𝐹) +𝑒 (𝑋 / 2)) ∈ ℝ*)
7161, 70syl 17 . . . . . . . . . 10 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → ((lim sup‘𝐹) +𝑒 (𝑋 / 2)) ∈ ℝ*)
72433adantl3 1215 . . . . . . . . . 10 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
73 simpl3 1252 . . . . . . . . . 10 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2)))
7460, 62, 71, 72, 73xrletrd 12281 . . . . . . . . 9 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2)))
7513, 68rexaddd 12353 . . . . . . . . . 10 (𝜑 → ((lim sup‘𝐹) +𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) + (𝑋 / 2)))
7661, 75syl 17 . . . . . . . . 9 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → ((lim sup‘𝐹) +𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) + (𝑋 / 2)))
7774, 76breqtrd 4899 . . . . . . . 8 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2)))
7868ad2antrr 719 . . . . . . . . . 10 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝑋 / 2) ∈ ℝ)
7913ad2antrr 719 . . . . . . . . . 10 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (lim sup‘𝐹) ∈ ℝ)
8023, 78, 79lesubaddd 10949 . . . . . . . . 9 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) ↔ (𝐹𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2))))
81803adantl3 1215 . . . . . . . 8 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) ↔ (𝐹𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2))))
8277, 81mpbird 249 . . . . . . 7 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
8382ralrimiva 3175 . . . . . 6 ((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
8459, 83syld3an3 1534 . . . . 5 ((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2))) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
85843exp 1154 . . . 4 (𝜑 → (𝑗𝑍 → (sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2)) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))))
861, 85reximdai 3220 . . 3 (𝜑 → (∃𝑗𝑍 sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)))
8753, 86mpd 15 . 2 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
88 simpll 785 . . . . 5 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝜑)
897ffvelrnda 6608 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
9067adantr 474 . . . . . . . . 9 ((𝜑𝑘𝑍) → 𝑋 ∈ ℝ)
9189, 90resubcld 10782 . . . . . . . 8 ((𝜑𝑘𝑍) → ((𝐹𝑘) − 𝑋) ∈ ℝ)
9291adantr 474 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − 𝑋) ∈ ℝ)
9368adantr 474 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝑋 / 2) ∈ ℝ)
9489, 93resubcld 10782 . . . . . . . 8 ((𝜑𝑘𝑍) → ((𝐹𝑘) − (𝑋 / 2)) ∈ ℝ)
9594adantr 474 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − (𝑋 / 2)) ∈ ℝ)
9613ad2antrr 719 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → (lim sup‘𝐹) ∈ ℝ)
9751rphalfltd 40479 . . . . . . . . . 10 (𝜑 → (𝑋 / 2) < 𝑋)
9897adantr 474 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝑋 / 2) < 𝑋)
9993, 90, 89, 98ltsub2dd 10965 . . . . . . . 8 ((𝜑𝑘𝑍) → ((𝐹𝑘) − 𝑋) < ((𝐹𝑘) − (𝑋 / 2)))
10099adantr 474 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − 𝑋) < ((𝐹𝑘) − (𝑋 / 2)))
101 simpr 479 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
10292, 95, 96, 100, 101ltletrd 10516 . . . . . 6 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))
103102ex 403 . . . . 5 ((𝜑𝑘𝑍) → (((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹𝑘) − 𝑋) < (lim sup‘𝐹)))
10488, 22, 103syl2anc 581 . . . 4 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹𝑘) − 𝑋) < (lim sup‘𝐹)))
105104ralimdva 3171 . . 3 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹)))
106105reximdva 3225 . 2 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹)))
10787, 106mpd 15 1 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3117  ∃wrex 3118  Vcvv 3414   ⊆ wss 3798   class class class wbr 4873   ↦ cmpt 4952  ran crn 5343   ↾ cres 5344   Fn wfn 6118  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  supcsup 8615  infcinf 8616  ℝcr 10251   + caddc 10255  ℝ*cxr 10390   < clt 10391   ≤ cle 10392   − cmin 10585   / cdiv 11009  2c2 11406  ℤcz 11704  ℤ≥cuz 11968  ℝ+crp 12112   +𝑒 cxad 12230  lim supclsp 14578 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-pre-sup 10330 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-inf 8618  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-2 11414  df-n0 11619  df-z 11705  df-uz 11969  df-rp 12113  df-xadd 12233  df-ico 12469  df-fz 12620  df-fzo 12761  df-fl 12888  df-ceil 12889  df-limsup 14579 This theorem is referenced by:  limsupgt  40805
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