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Theorem limsupgtlem 43001
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 1922 . . . 4 𝑗𝜑
2 limsupgtlem.m . . . . 5 (𝜑𝑀 ∈ ℤ)
3 limsupgtlem.z . . . . 5 𝑍 = (ℤ𝑀)
42, 3uzn0d 42646 . . . 4 (𝜑𝑍 ≠ ∅)
5 rnresss 5892 . . . . . . . 8 ran (𝐹 ↾ (ℤ𝑗)) ⊆ ran 𝐹
65a1i 11 . . . . . . 7 (𝜑 → ran (𝐹 ↾ (ℤ𝑗)) ⊆ ran 𝐹)
7 limsupgtlem.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶ℝ)
87frexr 42605 . . . . . . . 8 (𝜑𝐹:𝑍⟶ℝ*)
98frnd 6558 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ ℝ*)
106, 9sstrd 3916 . . . . . 6 (𝜑 → ran (𝐹 ↾ (ℤ𝑗)) ⊆ ℝ*)
1110supxrcld 42338 . . . . 5 (𝜑 → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ∈ ℝ*)
1211adantr 484 . . . 4 ((𝜑𝑗𝑍) → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ∈ ℝ*)
13 limsupgtlem.r . . . . . . 7 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14 nfcv 2904 . . . . . . . 8 𝑘𝐹
1514, 2, 3, 7limsupreuz 42961 . . . . . . 7 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) ∧ ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥)))
1613, 15mpbid 235 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) ∧ ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
1716simpld 498 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘))
18 rexr 10884 . . . . . . . . . 10 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
1918ad4antlr 733 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → 𝑥 ∈ ℝ*)
207ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝐹:𝑍⟶ℝ)
213uztrn2 12462 . . . . . . . . . . . . . 14 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
2221adantll 714 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
2320, 22ffvelrnd 6910 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ)
2423rexrd 10888 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
25243impa 1112 . . . . . . . . . 10 ((𝜑𝑗𝑍𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
2625ad5ant134 1369 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → (𝐹𝑘) ∈ ℝ*)
2711ad4antr 732 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ∈ ℝ*)
28 simpr 488 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → 𝑥 ≤ (𝐹𝑘))
2910ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ran (𝐹 ↾ (ℤ𝑗)) ⊆ ℝ*)
30 fvres 6741 . . . . . . . . . . . . . . 15 (𝑘 ∈ (ℤ𝑗) → ((𝐹 ↾ (ℤ𝑗))‘𝑘) = (𝐹𝑘))
3130eqcomd 2743 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ𝑗) → (𝐹𝑘) = ((𝐹 ↾ (ℤ𝑗))‘𝑘))
3231adantl 485 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) = ((𝐹 ↾ (ℤ𝑗))‘𝑘))
337ffnd 6551 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 Fn 𝑍)
3433adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑍) → 𝐹 Fn 𝑍)
3522ssd 42311 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑍) → (ℤ𝑗) ⊆ 𝑍)
36 fnssres 6505 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝑍 ∧ (ℤ𝑗) ⊆ 𝑍) → (𝐹 ↾ (ℤ𝑗)) Fn (ℤ𝑗))
3734, 35, 36syl2anc 587 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → (𝐹 ↾ (ℤ𝑗)) Fn (ℤ𝑗))
3837adantr 484 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹 ↾ (ℤ𝑗)) Fn (ℤ𝑗))
39 simpr 488 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘 ∈ (ℤ𝑗))
4038, 39fnfvelrnd 42488 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝐹 ↾ (ℤ𝑗))‘𝑘) ∈ ran (𝐹 ↾ (ℤ𝑗)))
4132, 40eqeltrd 2838 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ran (𝐹 ↾ (ℤ𝑗)))
42 eqid 2737 . . . . . . . . . . . 12 sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )
4329, 41, 42supxrubd 42344 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
44433impa 1112 . . . . . . . . . 10 ((𝜑𝑗𝑍𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
4544ad5ant134 1369 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
4619, 26, 27, 28, 45xrletrd 12757 . . . . . . . 8 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
4746rexlimdva2 3211 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (∃𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )))
4847ralimdva 3100 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) → ∀𝑗𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )))
4948reximdva 3198 . . . . 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 12645 . . . 4 (𝜑 → (𝑋 / 2) ∈ ℝ+)
531, 4, 12, 50, 52infrpgernmpt 42688 . . 3 (𝜑 → ∃𝑗𝑍 sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2)))
54 simp3 1140 . . . . . . 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 42932 . . . . . . . . . 10 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ))
5655eqcomd 2743 . . . . . . . . 9 (𝜑 → inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) = (lim sup‘𝐹))
5756oveq1d 7233 . . . . . . . 8 (𝜑 → (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) +𝑒 (𝑋 / 2)))
58573ad2ant1 1135 . . . . . . 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 5084 . . . . . 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 1170 . . . . . . . . . 10 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
61 simpl1 1193 . . . . . . . . . . 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 6736 . . . . . . . . . . . . . . 15 𝑍 ∈ V
6463a1i 11 . . . . . . . . . . . . . 14 (𝜑𝑍 ∈ V)
657, 64fexd 7048 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
6665limsupcld 42914 . . . . . . . . . . . 12 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
6751rpred 12633 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ ℝ)
6867rehalfcld 12082 . . . . . . . . . . . . 13 (𝜑 → (𝑋 / 2) ∈ ℝ)
6968rexrd 10888 . . . . . . . . . . . 12 (𝜑 → (𝑋 / 2) ∈ ℝ*)
7066, 69xaddcld 12896 . . . . . . . . . . 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 1170 . . . . . . . . . 10 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
73 simpl3 1195 . . . . . . . . . 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 12757 . . . . . . . . 9 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2)))
7513, 68rexaddd 12829 . . . . . . . . . 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 5084 . . . . . . . 8 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2)))
7868ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝑋 / 2) ∈ ℝ)
7913ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (lim sup‘𝐹) ∈ ℝ)
8023, 78, 79lesubaddd 11434 . . . . . . . . 9 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) ↔ (𝐹𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2))))
81803adantl3 1170 . . . . . . . 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 260 . . . . . . 7 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
8382ralrimiva 3105 . . . . . 6 ((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
8459, 83syld3an3 1411 . . . . 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 1121 . . . 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 3235 . . 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 767 . . . . 5 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝜑)
897ffvelrnda 6909 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
9067adantr 484 . . . . . . . . 9 ((𝜑𝑘𝑍) → 𝑋 ∈ ℝ)
9189, 90resubcld 11265 . . . . . . . 8 ((𝜑𝑘𝑍) → ((𝐹𝑘) − 𝑋) ∈ ℝ)
9291adantr 484 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − 𝑋) ∈ ℝ)
9368adantr 484 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝑋 / 2) ∈ ℝ)
9489, 93resubcld 11265 . . . . . . . 8 ((𝜑𝑘𝑍) → ((𝐹𝑘) − (𝑋 / 2)) ∈ ℝ)
9594adantr 484 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − (𝑋 / 2)) ∈ ℝ)
9613ad2antrr 726 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → (lim sup‘𝐹) ∈ ℝ)
9751rphalfltd 42678 . . . . . . . . . 10 (𝜑 → (𝑋 / 2) < 𝑋)
9897adantr 484 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝑋 / 2) < 𝑋)
9993, 90, 89, 98ltsub2dd 11450 . . . . . . . 8 ((𝜑𝑘𝑍) → ((𝐹𝑘) − 𝑋) < ((𝐹𝑘) − (𝑋 / 2)))
10099adantr 484 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − 𝑋) < ((𝐹𝑘) − (𝑋 / 2)))
101 simpr 488 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
10292, 95, 96, 100, 101ltletrd 10997 . . . . . 6 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))
103102ex 416 . . . . 5 ((𝜑𝑘𝑍) → (((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹𝑘) − 𝑋) < (lim sup‘𝐹)))
10488, 22, 103syl2anc 587 . . . 4 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹𝑘) − 𝑋) < (lim sup‘𝐹)))
105104ralimdva 3100 . . 3 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹)))
106105reximdva 3198 . 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  Vcvv 3413  wss 3871   class class class wbr 5058  cmpt 5140  ran crn 5557  cres 5558   Fn wfn 6380  wf 6381  cfv 6385  (class class class)co 7218  supcsup 9061  infcinf 9062  cr 10733   + caddc 10737  *cxr 10871   < clt 10872  cle 10873  cmin 11067   / cdiv 11494  2c2 11890  cz 12181  cuz 12443  +crp 12591   +𝑒 cxad 12707  lim supclsp 15036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5184  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327  ax-un 7528  ax-cnex 10790  ax-resscn 10791  ax-1cn 10792  ax-icn 10793  ax-addcl 10794  ax-addrcl 10795  ax-mulcl 10796  ax-mulrcl 10797  ax-mulcom 10798  ax-addass 10799  ax-mulass 10800  ax-distr 10801  ax-i2m1 10802  ax-1ne0 10803  ax-1rid 10804  ax-rnegex 10805  ax-rrecex 10806  ax-cnre 10807  ax-pre-lttri 10808  ax-pre-lttrn 10809  ax-pre-ltadd 10810  ax-pre-mulgt0 10811  ax-pre-sup 10812
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-pss 3890  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-tp 4551  df-op 4553  df-uni 4825  df-iun 4911  df-br 5059  df-opab 5121  df-mpt 5141  df-tr 5167  df-id 5460  df-eprel 5465  df-po 5473  df-so 5474  df-fr 5514  df-we 5516  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-pred 6165  df-ord 6221  df-on 6222  df-lim 6223  df-suc 6224  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393  df-riota 7175  df-ov 7221  df-oprab 7222  df-mpo 7223  df-om 7650  df-1st 7766  df-2nd 7767  df-wrecs 8052  df-recs 8113  df-rdg 8151  df-1o 8207  df-er 8396  df-en 8632  df-dom 8633  df-sdom 8634  df-fin 8635  df-sup 9063  df-inf 9064  df-pnf 10874  df-mnf 10875  df-xr 10876  df-ltxr 10877  df-le 10878  df-sub 11069  df-neg 11070  df-div 11495  df-nn 11836  df-2 11898  df-n0 12096  df-z 12182  df-uz 12444  df-rp 12592  df-xadd 12710  df-ico 12946  df-fz 13101  df-fzo 13244  df-fl 13372  df-ceil 13373  df-limsup 15037
This theorem is referenced by:  limsupgt  43002
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