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Theorem limsupre3lem 42740
 Description: Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupre3lem.1 𝑗𝐹
limsupre3lem.2 (𝜑𝐴 ⊆ ℝ)
limsupre3lem.3 (𝜑𝐹:𝐴⟶ℝ*)
Assertion
Ref Expression
limsupre3lem (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))))
Distinct variable groups:   𝐴,𝑗,𝑘,𝑥   𝑘,𝐹,𝑥   𝜑,𝑗,𝑘,𝑥
Allowed substitution hint:   𝐹(𝑗)

Proof of Theorem limsupre3lem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 limsupre3lem.1 . . 3 𝑗𝐹
2 limsupre3lem.2 . . 3 (𝜑𝐴 ⊆ ℝ)
3 limsupre3lem.3 . . 3 (𝜑𝐹:𝐴⟶ℝ*)
41, 2, 3limsupre2 42733 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦))))
5 simp2 1134 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗))) → 𝑦 ∈ ℝ)
6 nfv 1915 . . . . . . . . . 10 𝑗(𝜑𝑦 ∈ ℝ)
7 simp3l 1198 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑦 < (𝐹𝑗))) → 𝑘𝑗)
8 simp1r 1195 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴𝑦 < (𝐹𝑗)) → 𝑦 ∈ ℝ)
98rexrd 10729 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴𝑦 < (𝐹𝑗)) → 𝑦 ∈ ℝ*)
103ffvelrnda 6842 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝐴) → (𝐹𝑗) ∈ ℝ*)
1110adantlr 714 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) → (𝐹𝑗) ∈ ℝ*)
12113adant3 1129 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴𝑦 < (𝐹𝑗)) → (𝐹𝑗) ∈ ℝ*)
13 simp3 1135 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴𝑦 < (𝐹𝑗)) → 𝑦 < (𝐹𝑗))
149, 12, 13xrltled 12584 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴𝑦 < (𝐹𝑗)) → 𝑦 ≤ (𝐹𝑗))
15143adant3l 1177 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑦 < (𝐹𝑗))) → 𝑦 ≤ (𝐹𝑗))
167, 15jca 515 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑦 < (𝐹𝑗))) → (𝑘𝑗𝑦 ≤ (𝐹𝑗)))
17163exp 1116 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → (𝑗𝐴 → ((𝑘𝑗𝑦 < (𝐹𝑗)) → (𝑘𝑗𝑦 ≤ (𝐹𝑗)))))
186, 17reximdai 3235 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) → ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗))))
1918ralimdv 3109 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗))))
20193impia 1114 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗)))
21 breq1 5035 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ≤ (𝐹𝑗) ↔ 𝑦 ≤ (𝐹𝑗)))
2221anbi2d 631 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑘𝑗𝑥 ≤ (𝐹𝑗)) ↔ (𝑘𝑗𝑦 ≤ (𝐹𝑗))))
2322rexbidv 3221 . . . . . . . . 9 (𝑥 = 𝑦 → (∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗))))
2423ralbidv 3126 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗))))
2524rspcev 3541 . . . . . . 7 ((𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗))) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
265, 20, 25syl2anc 587 . . . . . 6 ((𝜑𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗))) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
27263exp 1116 . . . . 5 (𝜑 → (𝑦 ∈ ℝ → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))))
2827rexlimdv 3207 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
29 peano2rem 10991 . . . . . . 7 (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ)
3029ad2antlr 726 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝑥 − 1) ∈ ℝ)
31 nfv 1915 . . . . . . . . 9 𝑗(𝜑𝑥 ∈ ℝ)
32 simp3l 1198 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑘𝑗)
33 simp1r 1195 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑥 ∈ ℝ)
3429rexrd 10729 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ*)
3533, 34syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝑥 − 1) ∈ ℝ*)
3633rexrd 10729 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑥 ∈ ℝ*)
3710adantlr 714 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → (𝐹𝑗) ∈ ℝ*)
38373adant3 1129 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝐹𝑗) ∈ ℝ*)
3933ltm1d 11610 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝑥 − 1) < 𝑥)
40 simp3r 1199 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑥 ≤ (𝐹𝑗))
4135, 36, 38, 39, 40xrltletrd 12595 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝑥 − 1) < (𝐹𝑗))
4232, 41jca 515 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗)))
43423exp 1116 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (𝑗𝐴 → ((𝑘𝑗𝑥 ≤ (𝐹𝑗)) → (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗)))))
4431, 43reximdai 3235 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))))
4544ralimdv 3109 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))))
4645imp 410 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗)))
47 breq1 5035 . . . . . . . . . 10 (𝑦 = (𝑥 − 1) → (𝑦 < (𝐹𝑗) ↔ (𝑥 − 1) < (𝐹𝑗)))
4847anbi2d 631 . . . . . . . . 9 (𝑦 = (𝑥 − 1) → ((𝑘𝑗𝑦 < (𝐹𝑗)) ↔ (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))))
4948rexbidv 3221 . . . . . . . 8 (𝑦 = (𝑥 − 1) → (∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) ↔ ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))))
5049ralbidv 3126 . . . . . . 7 (𝑦 = (𝑥 − 1) → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))))
5150rspcev 3541 . . . . . 6 (((𝑥 − 1) ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)))
5230, 46, 51syl2anc 587 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)))
5352rexlimdva2 3211 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗))))
5428, 53impbid 215 . . 3 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
55 simplr 768 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦)) → 𝑦 ∈ ℝ)
5611adantr 484 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) < 𝑦) → (𝐹𝑗) ∈ ℝ*)
57 rexr 10725 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
5857ad3antlr 730 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) < 𝑦) → 𝑦 ∈ ℝ*)
59 simpr 488 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) < 𝑦) → (𝐹𝑗) < 𝑦)
6056, 58, 59xrltled 12584 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) < 𝑦) → (𝐹𝑗) ≤ 𝑦)
6160ex 416 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) → ((𝐹𝑗) < 𝑦 → (𝐹𝑗) ≤ 𝑦))
6261imim2d 57 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) → ((𝑘𝑗 → (𝐹𝑗) < 𝑦) → (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
6362ralimdva 3108 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) → ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
6463reximdv 3197 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) → ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
6564imp 410 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦)) → ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦))
66 breq2 5036 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑦))
6766imbi2d 344 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
6867ralbidv 3126 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
6968rexbidv 3221 . . . . . . 7 (𝑥 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
7069rspcev 3541 . . . . . 6 ((𝑦 ∈ ℝ ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
7155, 65, 70syl2anc 587 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
7271rexlimdva2 3211 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
73 peano2re 10851 . . . . . . 7 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
7473ad2antlr 726 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)) → (𝑥 + 1) ∈ ℝ)
7537adantr 484 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → (𝐹𝑗) ∈ ℝ*)
76 rexr 10725 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
7776ad3antlr 730 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → 𝑥 ∈ ℝ*)
7873rexrd 10729 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ*)
7978ad3antlr 730 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → (𝑥 + 1) ∈ ℝ*)
80 simpr 488 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → (𝐹𝑗) ≤ 𝑥)
81 ltp1 11518 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
8281ad3antlr 730 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → 𝑥 < (𝑥 + 1))
8375, 77, 79, 80, 82xrlelttrd 12594 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → (𝐹𝑗) < (𝑥 + 1))
8483ex 416 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → ((𝐹𝑗) ≤ 𝑥 → (𝐹𝑗) < (𝑥 + 1)))
8584imim2d 57 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → ((𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) → (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
8685ralimdva 3108 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) → ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
8786reximdv 3197 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) → ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
8887imp 410 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1)))
89 breq2 5036 . . . . . . . . . 10 (𝑦 = (𝑥 + 1) → ((𝐹𝑗) < 𝑦 ↔ (𝐹𝑗) < (𝑥 + 1)))
9089imbi2d 344 . . . . . . . . 9 (𝑦 = (𝑥 + 1) → ((𝑘𝑗 → (𝐹𝑗) < 𝑦) ↔ (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
9190ralbidv 3126 . . . . . . . 8 (𝑦 = (𝑥 + 1) → (∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
9291rexbidv 3221 . . . . . . 7 (𝑦 = (𝑥 + 1) → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
9392rspcev 3541 . . . . . 6 (((𝑥 + 1) ∈ ℝ ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦))
9474, 88, 93syl2anc 587 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦))
9594rexlimdva2 3211 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦)))
9672, 95impbid 215 . . 3 (𝜑 → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
9754, 96anbi12d 633 . 2 (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦)) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))))
984, 97bitrd 282 1 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2899  ∀wral 3070  ∃wrex 3071   ⊆ wss 3858   class class class wbr 5032  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  ℝcr 10574  1c1 10576   + caddc 10578  ℝ*cxr 10712   < clt 10713   ≤ cle 10714   − cmin 10908  lim supclsp 14875 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-po 5443  df-so 5444  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-sup 8939  df-inf 8940  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-ico 12785  df-limsup 14876 This theorem is referenced by:  limsupre3  42741
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