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Theorem limsupre3lem 41897
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 41890 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦))))
5 simp2 1131 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗))) → 𝑦 ∈ ℝ)
6 nfv 1908 . . . . . . . . . 10 𝑗(𝜑𝑦 ∈ ℝ)
7 simp3l 1195 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑦 < (𝐹𝑗))) → 𝑘𝑗)
8 simp1r 1192 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴𝑦 < (𝐹𝑗)) → 𝑦 ∈ ℝ)
98rexrd 10685 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴𝑦 < (𝐹𝑗)) → 𝑦 ∈ ℝ*)
103ffvelrnda 6849 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝐴) → (𝐹𝑗) ∈ ℝ*)
1110adantlr 711 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) → (𝐹𝑗) ∈ ℝ*)
12113adant3 1126 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴𝑦 < (𝐹𝑗)) → (𝐹𝑗) ∈ ℝ*)
13 simp3 1132 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴𝑦 < (𝐹𝑗)) → 𝑦 < (𝐹𝑗))
149, 12, 13xrltled 12538 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴𝑦 < (𝐹𝑗)) → 𝑦 ≤ (𝐹𝑗))
15143adant3l 1174 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑦 < (𝐹𝑗))) → 𝑦 ≤ (𝐹𝑗))
167, 15jca 512 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑦 < (𝐹𝑗))) → (𝑘𝑗𝑦 ≤ (𝐹𝑗)))
17163exp 1113 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → (𝑗𝐴 → ((𝑘𝑗𝑦 < (𝐹𝑗)) → (𝑘𝑗𝑦 ≤ (𝐹𝑗)))))
186, 17reximdai 3316 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) → ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗))))
1918ralimdv 3183 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗))))
20193impia 1111 . . . . . . 7 ((𝜑𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗)))
21 breq1 5066 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ≤ (𝐹𝑗) ↔ 𝑦 ≤ (𝐹𝑗)))
2221anbi2d 628 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑘𝑗𝑥 ≤ (𝐹𝑗)) ↔ (𝑘𝑗𝑦 ≤ (𝐹𝑗))))
2322rexbidv 3302 . . . . . . . . 9 (𝑥 = 𝑦 → (∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗))))
2423ralbidv 3202 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗))))
2524rspcev 3627 . . . . . . 7 ((𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 ≤ (𝐹𝑗))) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
265, 20, 25syl2anc 584 . . . . . 6 ((𝜑𝑦 ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗))) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
27263exp 1113 . . . . 5 (𝜑 → (𝑦 ∈ ℝ → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))))
2827rexlimdv 3288 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
29 peano2rem 10947 . . . . . . 7 (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ)
3029ad2antlr 723 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝑥 − 1) ∈ ℝ)
31 nfv 1908 . . . . . . . . 9 𝑗(𝜑𝑥 ∈ ℝ)
32 simp3l 1195 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑘𝑗)
33 simp1r 1192 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑥 ∈ ℝ)
3429rexrd 10685 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ*)
3533, 34syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝑥 − 1) ∈ ℝ*)
3633rexrd 10685 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑥 ∈ ℝ*)
3710adantlr 711 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → (𝐹𝑗) ∈ ℝ*)
38373adant3 1126 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝐹𝑗) ∈ ℝ*)
3933ltm1d 11566 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝑥 − 1) < 𝑥)
40 simp3r 1196 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑥 ≤ (𝐹𝑗))
4135, 36, 38, 39, 40xrltletrd 12549 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝑥 − 1) < (𝐹𝑗))
4232, 41jca 512 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗)))
43423exp 1113 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (𝑗𝐴 → ((𝑘𝑗𝑥 ≤ (𝐹𝑗)) → (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗)))))
4431, 43reximdai 3316 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))))
4544ralimdv 3183 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))))
4645imp 407 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗)))
47 breq1 5066 . . . . . . . . . 10 (𝑦 = (𝑥 − 1) → (𝑦 < (𝐹𝑗) ↔ (𝑥 − 1) < (𝐹𝑗)))
4847anbi2d 628 . . . . . . . . 9 (𝑦 = (𝑥 − 1) → ((𝑘𝑗𝑦 < (𝐹𝑗)) ↔ (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))))
4948rexbidv 3302 . . . . . . . 8 (𝑦 = (𝑥 − 1) → (∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) ↔ ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))))
5049ralbidv 3202 . . . . . . 7 (𝑦 = (𝑥 − 1) → (∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))))
5150rspcev 3627 . . . . . 6 (((𝑥 − 1) ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗 ∧ (𝑥 − 1) < (𝐹𝑗))) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)))
5230, 46, 51syl2anc 584 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)))
5352rexlimdva2 3292 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗))))
5428, 53impbid 213 . . 3 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑦 < (𝐹𝑗)) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗𝐴 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
55 simplr 765 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦)) → 𝑦 ∈ ℝ)
5611adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) < 𝑦) → (𝐹𝑗) ∈ ℝ*)
57 rexr 10681 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
5857ad3antlr 727 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) < 𝑦) → 𝑦 ∈ ℝ*)
59 simpr 485 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) < 𝑦) → (𝐹𝑗) < 𝑦)
6056, 58, 59xrltled 12538 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) < 𝑦) → (𝐹𝑗) ≤ 𝑦)
6160ex 413 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) → ((𝐹𝑗) < 𝑦 → (𝐹𝑗) ≤ 𝑦))
6261imim2d 57 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝐴) → ((𝑘𝑗 → (𝐹𝑗) < 𝑦) → (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
6362ralimdva 3182 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) → ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
6463reximdv 3278 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) → ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
6564imp 407 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦)) → ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦))
66 breq2 5067 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑦))
6766imbi2d 342 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
6867ralbidv 3202 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
6968rexbidv 3302 . . . . . . 7 (𝑥 = 𝑦 → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)))
7069rspcev 3627 . . . . . 6 ((𝑦 ∈ ℝ ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
7155, 65, 70syl2anc 584 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
7271rexlimdva2 3292 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
73 peano2re 10807 . . . . . . 7 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
7473ad2antlr 723 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)) → (𝑥 + 1) ∈ ℝ)
7537adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → (𝐹𝑗) ∈ ℝ*)
76 rexr 10681 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
7776ad3antlr 727 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → 𝑥 ∈ ℝ*)
7873rexrd 10685 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ*)
7978ad3antlr 727 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → (𝑥 + 1) ∈ ℝ*)
80 simpr 485 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → (𝐹𝑗) ≤ 𝑥)
81 ltp1 11474 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
8281ad3antlr 727 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → 𝑥 < (𝑥 + 1))
8375, 77, 79, 80, 82xrlelttrd 12548 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) ∧ (𝐹𝑗) ≤ 𝑥) → (𝐹𝑗) < (𝑥 + 1))
8483ex 413 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → ((𝐹𝑗) ≤ 𝑥 → (𝐹𝑗) < (𝑥 + 1)))
8584imim2d 57 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝐴) → ((𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) → (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
8685ralimdva 3182 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) → ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
8786reximdv 3278 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) → ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
8887imp 407 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1)))
89 breq2 5067 . . . . . . . . . 10 (𝑦 = (𝑥 + 1) → ((𝐹𝑗) < 𝑦 ↔ (𝐹𝑗) < (𝑥 + 1)))
9089imbi2d 342 . . . . . . . . 9 (𝑦 = (𝑥 + 1) → ((𝑘𝑗 → (𝐹𝑗) < 𝑦) ↔ (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
9190ralbidv 3202 . . . . . . . 8 (𝑦 = (𝑥 + 1) → (∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) ↔ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
9291rexbidv 3302 . . . . . . 7 (𝑦 = (𝑥 + 1) → (∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))))
9392rspcev 3627 . . . . . 6 (((𝑥 + 1) ∈ ℝ ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < (𝑥 + 1))) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦))
9474, 88, 93syl2anc 584 . . . . 5 (((𝜑𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦))
9594rexlimdva2 3292 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦)))
9672, 95impbid 213 . . 3 (𝜑 → (∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) < 𝑦) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐴 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
9754, 96anbi12d 630 . 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 280 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wnfc 2966  wral 3143  wrex 3144  wss 3940   class class class wbr 5063  wf 6350  cfv 6354  (class class class)co 7150  cr 10530  1c1 10532   + caddc 10534  *cxr 10668   < clt 10669  cle 10670  cmin 10864  lim supclsp 14822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-inf 8901  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-ico 12739  df-limsup 14823
This theorem is referenced by:  limsupre3  41898
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