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Theorem limsupre3uzlem 41881
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, infinitely often; 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
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 41534 . . . . 5 (ℤ𝑀) ⊆ ℝ
42, 3eqsstri 4005 . . . 4 𝑍 ⊆ ℝ
54a1i 11 . . 3 (𝜑𝑍 ⊆ ℝ)
6 limsupre3uzlem.4 . . 3 (𝜑𝐹:𝑍⟶ℝ*)
71, 5, 6limsupre3 41879 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))))
8 breq1 5066 . . . . . . . . . . 11 (𝑦 = 𝑘 → (𝑦𝑗𝑘𝑗))
98anbi1d 629 . . . . . . . . . 10 (𝑦 = 𝑘 → ((𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
109rexbidv 3302 . . . . . . . . 9 (𝑦 = 𝑘 → (∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
1110cbvralv 3458 . . . . . . . 8 (∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
1211biimpi 217 . . . . . . 7 (∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) → ∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
13 nfra1 3224 . . . . . . . 8 𝑘𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗))
14 simpr 485 . . . . . . . . 9 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → 𝑘𝑍)
154, 14sseldi 3969 . . . . . . . . . 10 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → 𝑘 ∈ ℝ)
16 rspa 3211 . . . . . . . . . 10 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘 ∈ ℝ) → ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
1715, 16syldan 591 . . . . . . . . 9 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
18 nfv 1908 . . . . . . . . . . 11 𝑗 𝑘𝑍
19 nfre1 3311 . . . . . . . . . . 11 𝑗𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)
20 eqid 2826 . . . . . . . . . . . . . . 15 (ℤ𝑘) = (ℤ𝑘)
212eluzelz2 41541 . . . . . . . . . . . . . . . 16 (𝑘𝑍𝑘 ∈ ℤ)
22213ad2ant1 1127 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑘 ∈ ℤ)
232eluzelz2 41541 . . . . . . . . . . . . . . . 16 (𝑗𝑍𝑗 ∈ ℤ)
24233ad2ant2 1128 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑗 ∈ ℤ)
25 simp3 1132 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑘𝑗)
2620, 22, 24, 25eluzd 41547 . . . . . . . . . . . . . 14 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑗 ∈ (ℤ𝑘))
27263adant3r 1175 . . . . . . . . . . . . 13 ((𝑘𝑍𝑗𝑍 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑗 ∈ (ℤ𝑘))
28 simp3r 1196 . . . . . . . . . . . . 13 ((𝑘𝑍𝑗𝑍 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑥 ≤ (𝐹𝑗))
29 rspe 3309 . . . . . . . . . . . . 13 ((𝑗 ∈ (ℤ𝑘) ∧ 𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3027, 28, 29syl2anc 584 . . . . . . . . . . . 12 ((𝑘𝑍𝑗𝑍 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
31303exp 1113 . . . . . . . . . . 11 (𝑘𝑍 → (𝑗𝑍 → ((𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))))
3218, 19, 31rexlimd 3322 . . . . . . . . . 10 (𝑘𝑍 → (∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
3332imp 407 . . . . . . . . 9 ((𝑘𝑍 ∧ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3414, 17, 33syl2anc 584 . . . . . . . 8 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3513, 34ralrimia 41263 . . . . . . 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 4476 . . . . . . . . . . . . 13 (𝑀 ≤ (⌈‘𝑦) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = (⌈‘𝑦))
3938adantl 482 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = (⌈‘𝑦))
40 limsupre3uzlem.2 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℤ)
4140ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ∈ ℤ)
42 ceilcl 13202 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → (⌈‘𝑦) ∈ ℤ)
4342ad2antlr 723 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → (⌈‘𝑦) ∈ ℤ)
44 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ≤ (⌈‘𝑦))
452, 41, 43, 44eluzd 41547 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → (⌈‘𝑦) ∈ 𝑍)
4639, 45eqeltrd 2918 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
47 iffalse 4479 . . . . . . . . . . . . . 14 𝑀 ≤ (⌈‘𝑦) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = 𝑀)
4847adantl 482 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = 𝑀)
4940, 2uzidd2 41555 . . . . . . . . . . . . . 14 (𝜑𝑀𝑍)
5049adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → 𝑀𝑍)
5148, 50eqeltrd 2918 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
5251adantlr 711 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
5346, 52pm2.61dan 809 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
5453adantlr 711 . . . . . . . . 9 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
55 simplr 765 . . . . . . . . 9 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
56 fveq2 6667 . . . . . . . . . . 11 (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (ℤ𝑘) = (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)))
5756rexeqdv 3422 . . . . . . . . . 10 (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹𝑗)))
5857rspcva 3625 . . . . . . . . 9 ((if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹𝑗))
5954, 55, 58syl2anc 584 . . . . . . . 8 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → ∃𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹𝑗))
60 nfv 1908 . . . . . . . . . . 11 𝑗𝜑
6118nfci 2969 . . . . . . . . . . . 12 𝑗𝑍
6261, 19nfral 3231 . . . . . . . . . . 11 𝑗𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)
6360, 62nfan 1893 . . . . . . . . . 10 𝑗(𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
64 nfv 1908 . . . . . . . . . 10 𝑗 𝑦 ∈ ℝ
6563, 64nfan 1893 . . . . . . . . 9 𝑗((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ)
66 nfre1 3311 . . . . . . . . 9 𝑗𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗))
6740ad2antrr 722 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ∈ ℤ)
68 eluzelz 12242 . . . . . . . . . . . . . . 15 (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → 𝑗 ∈ ℤ)
6968adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ ℤ)
7067zred 12076 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ∈ ℝ)
714, 53sseldi 3969 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ ℝ)
7271adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ ℝ)
7369zred 12076 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ ℝ)
744, 49sseldi 3969 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ ℝ)
7574adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → 𝑀 ∈ ℝ)
7642zred 12076 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ → (⌈‘𝑦) ∈ ℝ)
7776adantl 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → (⌈‘𝑦) ∈ ℝ)
78 max1 12568 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℝ ∧ (⌈‘𝑦) ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
7975, 77, 78syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
8079adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
81 eluzle 12245 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ≤ 𝑗)
8281adantl 482 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ≤ 𝑗)
8370, 72, 73, 80, 82letrd 10786 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀𝑗)
842, 67, 69, 83eluzd 41547 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗𝑍)
85843adant3 1126 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑗𝑍)
86 simplr 765 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ∈ ℝ)
87 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
88 ceilge 13204 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ → 𝑦 ≤ (⌈‘𝑦))
8988adantl 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → 𝑦 ≤ (⌈‘𝑦))
90 max2 12570 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℝ ∧ (⌈‘𝑦) ∈ ℝ) → (⌈‘𝑦) ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
9175, 77, 90syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → (⌈‘𝑦) ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
9287, 77, 71, 89, 91letrd 10786 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ℝ) → 𝑦 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
9392adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
9486, 72, 73, 93, 82letrd 10786 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦𝑗)
95943adant3 1126 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑦𝑗)
96 simp3 1132 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ (𝐹𝑗))
9795, 96jca 512 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
98 rspe 3309 . . . . . . . . . . . 12 ((𝑗𝑍 ∧ (𝑦𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
9985, 97, 98syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
100993exp 1113 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝑥 ≤ (𝐹𝑗) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))))
101100adantlr 711 . . . . . . . . 9 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝑥 ≤ (𝐹𝑗) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))))
10265, 66, 101rexlimd 3322 . . . . . . . 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 3187 . . . . . 6 ((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) → ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
105104ex 413 . . . . 5 (𝜑 → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) → ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗))))
10637, 105impbid 213 . . . 4 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
107106rexbidv 3302 . . 3 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
10853adantr 481 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
10960, 64nfan 1893 . . . . . . . . 9 𝑗(𝜑𝑦 ∈ ℝ)
110 nfra1 3224 . . . . . . . . 9 𝑗𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)
111109, 110nfan 1893 . . . . . . . 8 𝑗((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
11294adantlr 711 . . . . . . . . . 10 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦𝑗)
113 simplr 765 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
11484adantlr 711 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗𝑍)
115 rspa 3211 . . . . . . . . . . 11 ((∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍) → (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
116113, 114, 115syl2anc 584 . . . . . . . . . 10 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
117112, 116mpd 15 . . . . . . . . 9 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → (𝐹𝑗) ≤ 𝑥)
118117ex 413 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝐹𝑗) ≤ 𝑥))
119111, 118ralrimi 3221 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∀𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹𝑗) ≤ 𝑥)
12056raleqdv 3421 . . . . . . . 8 (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹𝑗) ≤ 𝑥))
121120rspcev 3627 . . . . . . 7 ((if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹𝑗) ≤ 𝑥) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
122108, 119, 121syl2anc 584 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
123122rexlimdva2 3292 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
1244sseli 3967 . . . . . . . 8 (𝑘𝑍𝑘 ∈ ℝ)
125124ad2antlr 723 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → 𝑘 ∈ ℝ)
126 nfra1 3224 . . . . . . . . . 10 𝑗𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥
12718, 126nfan 1893 . . . . . . . . 9 𝑗(𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
128 simp1r 1192 . . . . . . . . . . 11 (((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍𝑘𝑗) → ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
129263adant1r 1171 . . . . . . . . . . 11 (((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍𝑘𝑗) → 𝑗 ∈ (ℤ𝑘))
130 rspa 3211 . . . . . . . . . . 11 ((∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ≤ 𝑥)
131128, 129, 130syl2anc 584 . . . . . . . . . 10 (((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍𝑘𝑗) → (𝐹𝑗) ≤ 𝑥)
1321313exp 1113 . . . . . . . . 9 ((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → (𝑗𝑍 → (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
133127, 132ralrimi 3221 . . . . . . . 8 ((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
134133adantll 710 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
1358rspceaimv 3632 . . . . . . 7 ((𝑘 ∈ ℝ ∧ ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
136125, 134, 135syl2anc 584 . . . . . 6 (((𝜑𝑘𝑍) ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
137136rexlimdva2 3292 . . . . 5 (𝜑 → (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)))
138123, 137impbid 213 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
139138rexbidv 3302 . . 3 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
140107, 139anbi12d 630 . 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)))
1417, 140bitrd 280 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wnfc 2966  wral 3143  wrex 3144  wss 3940  ifcif 4470   class class class wbr 5063  wf 6348  cfv 6352  cr 10525  *cxr 10663  cle 10665  cz 11970  cuz 12232  cceil 13151  lim supclsp 14817
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 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
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-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  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-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-sup 8895  df-inf 8896  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-n0 11887  df-z 11971  df-uz 12233  df-ico 12734  df-fl 13152  df-ceil 13153  df-limsup 14818
This theorem is referenced by:  limsupre3uz  41882  limsupreuz  41883
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