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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  cr 10251   + caddc 10255  *cxr 10390   < clt 10391  cle 10392  cmin 10585   / cdiv 11009  2c2 11406  cz 11704  cuz 11968  +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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