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Theorem limsupgtlem 40327
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 1883 . . . 4 𝑗𝜑
2 limsupgtlem.m . . . . 5 (𝜑𝑀 ∈ ℤ)
3 limsupgtlem.z . . . . 5 𝑍 = (ℤ𝑀)
42, 3uzn0d 39965 . . . 4 (𝜑𝑍 ≠ ∅)
5 rnresss 39679 . . . . . . . 8 ran (𝐹 ↾ (ℤ𝑗)) ⊆ ran 𝐹
65a1i 11 . . . . . . 7 (𝜑 → ran (𝐹 ↾ (ℤ𝑗)) ⊆ ran 𝐹)
7 limsupgtlem.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶ℝ)
87frexr 39917 . . . . . . . 8 (𝜑𝐹:𝑍⟶ℝ*)
98frnd 39740 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ ℝ*)
106, 9sstrd 3646 . . . . . 6 (𝜑 → ran (𝐹 ↾ (ℤ𝑗)) ⊆ ℝ*)
1110supxrcld 39604 . . . . 5 (𝜑 → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ∈ ℝ*)
1211adantr 480 . . . 4 ((𝜑𝑗𝑍) → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ∈ ℝ*)
13 limsupgtlem.r . . . . . . 7 (𝜑 → (lim sup‘𝐹) ∈ ℝ)
14 nfcv 2793 . . . . . . . 8 𝑘𝐹
1514, 2, 3, 7limsupreuz 40287 . . . . . . 7 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) ∧ ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥)))
1613, 15mpbid 222 . . . . . 6 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) ∧ ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
1716simpld 474 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘))
18 rexr 10123 . . . . . . . . . 10 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
1918ad4antlr 771 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → 𝑥 ∈ ℝ*)
207ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝐹:𝑍⟶ℝ)
213uztrn2 11743 . . . . . . . . . . . . . 14 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
2221adantll 750 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
2320, 22ffvelrnd 6400 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ)
2423rexrd 10127 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
25243impa 1278 . . . . . . . . . 10 ((𝜑𝑗𝑍𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
2625ad5ant134 1353 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → (𝐹𝑘) ∈ ℝ*)
2711ad4antr 769 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ∈ ℝ*)
28 simpr 476 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → 𝑥 ≤ (𝐹𝑘))
2910ad2antrr 762 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ran (𝐹 ↾ (ℤ𝑗)) ⊆ ℝ*)
30 fvres 6245 . . . . . . . . . . . . . . 15 (𝑘 ∈ (ℤ𝑗) → ((𝐹 ↾ (ℤ𝑗))‘𝑘) = (𝐹𝑘))
3130eqcomd 2657 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ𝑗) → (𝐹𝑘) = ((𝐹 ↾ (ℤ𝑗))‘𝑘))
3231adantl 481 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) = ((𝐹 ↾ (ℤ𝑗))‘𝑘))
337ffnd 6084 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 Fn 𝑍)
3433adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑍) → 𝐹 Fn 𝑍)
3522ssd 39566 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑍) → (ℤ𝑗) ⊆ 𝑍)
36 fnssres 6042 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝑍 ∧ (ℤ𝑗) ⊆ 𝑍) → (𝐹 ↾ (ℤ𝑗)) Fn (ℤ𝑗))
3734, 35, 36syl2anc 694 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → (𝐹 ↾ (ℤ𝑗)) Fn (ℤ𝑗))
3837adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹 ↾ (ℤ𝑗)) Fn (ℤ𝑗))
39 simpr 476 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘 ∈ (ℤ𝑗))
4038, 39fnfvelrnd 39793 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝐹 ↾ (ℤ𝑗))‘𝑘) ∈ ran (𝐹 ↾ (ℤ𝑗)))
4132, 40eqeltrd 2730 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ran (𝐹 ↾ (ℤ𝑗)))
42 eqid 2651 . . . . . . . . . . . 12 sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )
4329, 41, 42supxrubd 39611 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
44433impa 1278 . . . . . . . . . 10 ((𝜑𝑗𝑍𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
4544ad5ant134 1353 . . . . . . . . 9 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
4619, 26, 27, 28, 45xrletrd 12031 . . . . . . . 8 (((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑥 ≤ (𝐹𝑘)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
4746rexlimdva2 39653 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (∃𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )))
4847ralimdva 2991 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘) → ∀𝑗𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )))
4948reximdva 3046 . . . . 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 11922 . . . 4 (𝜑 → (𝑋 / 2) ∈ ℝ+)
531, 4, 12, 50, 52infrpgernmpt 40008 . . 3 (𝜑 → ∃𝑗𝑍 sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2)))
54 simp3 1083 . . . . . . 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 40258 . . . . . . . . . 10 (𝜑 → (lim sup‘𝐹) = inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ))
5655eqcomd 2657 . . . . . . . . 9 (𝜑 → inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) = (lim sup‘𝐹))
5756oveq1d 6705 . . . . . . . 8 (𝜑 → (inf(ran (𝑗𝑍 ↦ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < )), ℝ*, < ) +𝑒 (𝑋 / 2)) = ((lim sup‘𝐹) +𝑒 (𝑋 / 2)))
58573ad2ant1 1102 . . . . . . 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 4711 . . . . . 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 1239 . . . . . . . . . 10 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
61 simpl1 1084 . . . . . . . . . . 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 6240 . . . . . . . . . . . . . . 15 𝑍 ∈ V
6463a1i 11 . . . . . . . . . . . . . 14 (𝜑𝑍 ∈ V)
657, 64fexd 39610 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
6665limsupcld 40240 . . . . . . . . . . . 12 (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
6751rpred 11910 . . . . . . . . . . . . . 14 (𝜑𝑋 ∈ ℝ)
6867rehalfcld 11317 . . . . . . . . . . . . 13 (𝜑 → (𝑋 / 2) ∈ ℝ)
6968rexrd 10127 . . . . . . . . . . . 12 (𝜑 → (𝑋 / 2) ∈ ℝ*)
7066, 69xaddcld 12169 . . . . . . . . . . 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 1239 . . . . . . . . . 10 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ))
73 simpl3 1086 . . . . . . . . . 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 12031 . . . . . . . . 9 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2)))
7513, 68rexaddd 12103 . . . . . . . . . 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 4711 . . . . . . . 8 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2)))
7868ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝑋 / 2) ∈ ℝ)
7913ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (lim sup‘𝐹) ∈ ℝ)
8023, 78, 79lesubaddd 10662 . . . . . . . . 9 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) ↔ (𝐹𝑘) ≤ ((lim sup‘𝐹) + (𝑋 / 2))))
81803adantl3 1239 . . . . . . . 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 247 . . . . . . 7 (((𝜑𝑗𝑍 ∧ sup(ran (𝐹 ↾ (ℤ𝑗)), ℝ*, < ) ≤ ((lim sup‘𝐹) +𝑒 (𝑋 / 2))) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
8382ralrimiva 2995 . . . . . 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 1283 . . . 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 3041 . . 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 805 . . . . 5 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝜑)
897ffvelrnda 6399 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
9067adantr 480 . . . . . . . . 9 ((𝜑𝑘𝑍) → 𝑋 ∈ ℝ)
9189, 90resubcld 10496 . . . . . . . 8 ((𝜑𝑘𝑍) → ((𝐹𝑘) − 𝑋) ∈ ℝ)
9291adantr 480 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − 𝑋) ∈ ℝ)
9368adantr 480 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝑋 / 2) ∈ ℝ)
9489, 93resubcld 10496 . . . . . . . 8 ((𝜑𝑘𝑍) → ((𝐹𝑘) − (𝑋 / 2)) ∈ ℝ)
9594adantr 480 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − (𝑋 / 2)) ∈ ℝ)
9613ad2antrr 762 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → (lim sup‘𝐹) ∈ ℝ)
9751rphalfltd 39997 . . . . . . . . . 10 (𝜑 → (𝑋 / 2) < 𝑋)
9897adantr 480 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝑋 / 2) < 𝑋)
9993, 90, 89, 98ltsub2dd 10678 . . . . . . . 8 ((𝜑𝑘𝑍) → ((𝐹𝑘) − 𝑋) < ((𝐹𝑘) − (𝑋 / 2)))
10099adantr 480 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − 𝑋) < ((𝐹𝑘) − (𝑋 / 2)))
101 simpr 476 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹))
10292, 95, 96, 100, 101ltletrd 10235 . . . . . 6 (((𝜑𝑘𝑍) ∧ ((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹)) → ((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))
103102ex 449 . . . . 5 ((𝜑𝑘𝑍) → (((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹𝑘) − 𝑋) < (lim sup‘𝐹)))
10488, 22, 103syl2anc 694 . . . 4 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ((𝐹𝑘) − 𝑋) < (lim sup‘𝐹)))
105104ralimdva 2991 . . 3 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − (𝑋 / 2)) ≤ (lim sup‘𝐹) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹)))
106105reximdva 3046 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607   class class class wbr 4685  cmpt 4762  ran crn 5144  cres 5145   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  supcsup 8387  infcinf 8388  cr 9973   + caddc 9977  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  2c2 11108  cz 11415  cuz 11725  +crp 11870   +𝑒 cxad 11982  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-int 4508  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-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  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-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-xadd 11985  df-ico 12219  df-fz 12365  df-fzo 12505  df-fl 12633  df-ceil 12634  df-limsup 14246
This theorem is referenced by:  limsupgt  40328
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