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Theorem limsupre3uzlem 40285
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 smaller or equal than the function, infinitely often; 2. there is a real number that is eventually larger or equal than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupre3uzlem.1 𝑗𝐹
limsupre3uzlem.2 (𝜑𝑀 ∈ ℤ)
limsupre3uzlem.3 𝑍 = (ℤ𝑀)
limsupre3uzlem.4 (𝜑𝐹:𝑍⟶ℝ*)
Assertion
Ref Expression
limsupre3uzlem (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
Distinct variable groups:   𝑘,𝐹,𝑥   𝑗,𝑀,𝑘   𝑗,𝑍,𝑘,𝑥   𝜑,𝑗,𝑘,𝑥
Allowed substitution hints:   𝐹(𝑗)   𝑀(𝑥)

Proof of Theorem limsupre3uzlem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 limsupre3uzlem.1 . . 3 𝑗𝐹
2 limsupre3uzlem.3 . . . . 5 𝑍 = (ℤ𝑀)
3 uzssre 39933 . . . . 5 (ℤ𝑀) ⊆ ℝ
42, 3eqsstri 3668 . . . 4 𝑍 ⊆ ℝ
54a1i 11 . . 3 (𝜑𝑍 ⊆ ℝ)
6 limsupre3uzlem.4 . . 3 (𝜑𝐹:𝑍⟶ℝ*)
71, 5, 6limsupre3 40283 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))))
8 breq1 4688 . . . . . . . . . . 11 (𝑦 = 𝑘 → (𝑦𝑗𝑘𝑗))
98anbi1d 741 . . . . . . . . . 10 (𝑦 = 𝑘 → ((𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
109rexbidv 3081 . . . . . . . . 9 (𝑦 = 𝑘 → (∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
1110cbvralv 3201 . . . . . . . 8 (∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
1211biimpi 206 . . . . . . 7 (∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
13 nfra1 2970 . . . . . . . 8 𝑘𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗))
14 simpr 476 . . . . . . . . 9 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → 𝑘𝑍)
154, 14sseldi 3634 . . . . . . . . . 10 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → 𝑘 ∈ ℝ)
16 rspa 2959 . . . . . . . . . 10 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘 ∈ ℝ) → ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
1715, 16syldan 486 . . . . . . . . 9 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
18 nfv 1883 . . . . . . . . . . 11 𝑗 𝑘𝑍
19 nfre1 3034 . . . . . . . . . . 11 𝑗𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)
20 eqid 2651 . . . . . . . . . . . . . . 15 (ℤ𝑘) = (ℤ𝑘)
212eluzelz2 39940 . . . . . . . . . . . . . . . 16 (𝑘𝑍𝑘 ∈ ℤ)
22213ad2ant1 1102 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑘 ∈ ℤ)
232eluzelz2 39940 . . . . . . . . . . . . . . . 16 (𝑗𝑍𝑗 ∈ ℤ)
24233ad2ant2 1103 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑗 ∈ ℤ)
25 simp3 1083 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑘𝑗)
2620, 22, 24, 25eluzd 39948 . . . . . . . . . . . . . 14 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑗 ∈ (ℤ𝑘))
27263adant3r 1363 . . . . . . . . . . . . 13 ((𝑘𝑍𝑗𝑍 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑗 ∈ (ℤ𝑘))
28 simp3r 1110 . . . . . . . . . . . . 13 ((𝑘𝑍𝑗𝑍 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑥 ≤ (𝐹𝑗))
29 rspe 3032 . . . . . . . . . . . . 13 ((𝑗 ∈ (ℤ𝑘) ∧ 𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3027, 28, 29syl2anc 694 . . . . . . . . . . . 12 ((𝑘𝑍𝑗𝑍 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
31303exp 1283 . . . . . . . . . . 11 (𝑘𝑍 → (𝑗𝑍 → ((𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))))
3218, 19, 31rexlimd 3055 . . . . . . . . . 10 (𝑘𝑍 → (∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
3332imp 444 . . . . . . . . 9 ((𝑘𝑍 ∧ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3414, 17, 33syl2anc 694 . . . . . . . 8 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3513, 34ralrimia 39629 . . . . . . 7 (∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3612, 35syl 17 . . . . . 6 (∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3736a1i 11 . . . . 5 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
38 iftrue 4125 . . . . . . . . . . . . 13 (𝑀 ≤ (⌈‘𝑦) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = (⌈‘𝑦))
3938adantl 481 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = (⌈‘𝑦))
40 limsupre3uzlem.2 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℤ)
4140ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ∈ ℤ)
42 ceilcl 12683 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → (⌈‘𝑦) ∈ ℤ)
4342ad2antlr 763 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → (⌈‘𝑦) ∈ ℤ)
44 simpr 476 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ≤ (⌈‘𝑦))
452, 41, 43, 44eluzd 39948 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → (⌈‘𝑦) ∈ 𝑍)
4639, 45eqeltrd 2730 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
47 iffalse 4128 . . . . . . . . . . . . . 14 𝑀 ≤ (⌈‘𝑦) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = 𝑀)
4847adantl 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = 𝑀)
4940, 2uzidd2 39956 . . . . . . . . . . . . . 14 (𝜑𝑀𝑍)
5049adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → 𝑀𝑍)
5148, 50eqeltrd 2730 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
5251adantlr 751 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
5346, 52pm2.61dan 849 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
5453adantlr 751 . . . . . . . . 9 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
55 simplr 807 . . . . . . . . 9 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
56 fveq2 6229 . . . . . . . . . . 11 (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (ℤ𝑘) = (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)))
5756rexeqdv 3175 . . . . . . . . . 10 (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹𝑗)))
5857rspcva 3338 . . . . . . . . 9 ((if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹𝑗))
5954, 55, 58syl2anc 694 . . . . . . . 8 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → ∃𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹𝑗))
60 nfv 1883 . . . . . . . . . . 11 𝑗𝜑
6118nfci 2783 . . . . . . . . . . . 12 𝑗𝑍
6261, 19nfral 2974 . . . . . . . . . . 11 𝑗𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)
6360, 62nfan 1868 . . . . . . . . . 10 𝑗(𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
64 nfv 1883 . . . . . . . . . 10 𝑗 𝑦 ∈ ℝ
6563, 64nfan 1868 . . . . . . . . 9 𝑗((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ)
66 nfre1 3034 . . . . . . . . 9 𝑗𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗))
6740ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ∈ ℤ)
68 eluzelz 11735 . . . . . . . . . . . . . . 15 (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → 𝑗 ∈ ℤ)
6968adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ ℤ)
7067zred 11520 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ∈ ℝ)
714, 53sseldi 3634 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ ℝ)
7271adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ ℝ)
7369zred 11520 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ ℝ)
744, 49sseldi 3634 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ ℝ)
7574adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → 𝑀 ∈ ℝ)
7642zred 11520 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ → (⌈‘𝑦) ∈ ℝ)
7776adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → (⌈‘𝑦) ∈ ℝ)
78 max1 12054 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℝ ∧ (⌈‘𝑦) ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
7975, 77, 78syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
8079adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
81 eluzle 11738 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ≤ 𝑗)
8281adantl 481 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ≤ 𝑗)
8370, 72, 73, 80, 82letrd 10232 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀𝑗)
842, 67, 69, 83eluzd 39948 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗𝑍)
85843adant3 1101 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑗𝑍)
86 simplr 807 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ∈ ℝ)
87 simpr 476 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
88 ceilge 12685 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ → 𝑦 ≤ (⌈‘𝑦))
8988adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → 𝑦 ≤ (⌈‘𝑦))
90 max2 12056 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℝ ∧ (⌈‘𝑦) ∈ ℝ) → (⌈‘𝑦) ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
9175, 77, 90syl2anc 694 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → (⌈‘𝑦) ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
9287, 77, 71, 89, 91letrd 10232 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ℝ) → 𝑦 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
9392adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
9486, 72, 73, 93, 82letrd 10232 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦𝑗)
95943adant3 1101 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑦𝑗)
96 simp3 1083 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ (𝐹𝑗))
9795, 96jca 553 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
98 rspe 3032 . . . . . . . . . . . 12 ((𝑗𝑍 ∧ (𝑦𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
9985, 97, 98syl2anc 694 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
100993exp 1283 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝑥 ≤ (𝐹𝑗) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))))
101100adantlr 751 . . . . . . . . 9 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝑥 ≤ (𝐹𝑗) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))))
10265, 66, 101rexlimd 3055 . . . . . . . 8 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → (∃𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹𝑗) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗))))
10359, 102mpd 15 . . . . . . 7 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
104103ralrimiva 2995 . . . . . 6 ((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) → ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
105104ex 449 . . . . 5 (𝜑 → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) → ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗))))
10637, 105impbid 202 . . . 4 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
107106rexbidv 3081 . . 3 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
10853adantr 480 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
10960, 64nfan 1868 . . . . . . . . . 10 𝑗(𝜑𝑦 ∈ ℝ)
110 nfra1 2970 . . . . . . . . . 10 𝑗𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)
111109, 110nfan 1868 . . . . . . . . 9 𝑗((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
11294adantlr 751 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦𝑗)
113 simplr 807 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
11484adantlr 751 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗𝑍)
115 rspa 2959 . . . . . . . . . . . 12 ((∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍) → (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
116113, 114, 115syl2anc 694 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
117112, 116mpd 15 . . . . . . . . . 10 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → (𝐹𝑗) ≤ 𝑥)
118117ex 449 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝐹𝑗) ≤ 𝑥))
119111, 118ralrimi 2986 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∀𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹𝑗) ≤ 𝑥)
12056raleqdv 3174 . . . . . . . . 9 (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹𝑗) ≤ 𝑥))
121120rspcev 3340 . . . . . . . 8 ((if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹𝑗) ≤ 𝑥) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
122108, 119, 121syl2anc 694 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
123122ex 449 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
124123rexlimdva 3060 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
1254sseli 3632 . . . . . . . . 9 (𝑘𝑍𝑘 ∈ ℝ)
126125ad2antlr 763 . . . . . . . 8 (((𝜑𝑘𝑍) ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → 𝑘 ∈ ℝ)
127 nfra1 2970 . . . . . . . . . . 11 𝑗𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥
12818, 127nfan 1868 . . . . . . . . . 10 𝑗(𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
129 simp1r 1106 . . . . . . . . . . . 12 (((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍𝑘𝑗) → ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
130263adant1r 1359 . . . . . . . . . . . 12 (((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍𝑘𝑗) → 𝑗 ∈ (ℤ𝑘))
131 rspa 2959 . . . . . . . . . . . 12 ((∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ≤ 𝑥)
132129, 130, 131syl2anc 694 . . . . . . . . . . 11 (((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍𝑘𝑗) → (𝐹𝑗) ≤ 𝑥)
1331323exp 1283 . . . . . . . . . 10 ((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → (𝑗𝑍 → (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
134128, 133ralrimi 2986 . . . . . . . . 9 ((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
135134adantll 750 . . . . . . . 8 (((𝜑𝑘𝑍) ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
1368imbi1d 330 . . . . . . . . . 10 (𝑦 = 𝑘 → ((𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
137136ralbidv 3015 . . . . . . . . 9 (𝑦 = 𝑘 → (∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
138137rspcev 3340 . . . . . . . 8 ((𝑘 ∈ ℝ ∧ ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
139126, 135, 138syl2anc 694 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
140139ex 449 . . . . . 6 ((𝜑𝑘𝑍) → (∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)))
141140rexlimdva 3060 . . . . 5 (𝜑 → (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)))
142124, 141impbid 202 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
143142rexbidv 3081 . . 3 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
144107, 143anbi12d 747 . 2 (𝜑 → ((∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
1457, 144bitrd 268 1 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wnfc 2780  wral 2941  wrex 2942  wss 3607  ifcif 4119   class class class wbr 4685  wf 5922  cfv 5926  cr 9973  *cxr 10111  cle 10113  cz 11415  cuz 11725  cceil 12632  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-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-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  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-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-ico 12219  df-fl 12633  df-ceil 12634  df-limsup 14246
This theorem is referenced by:  limsupre3uz  40286  limsupreuz  40287
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