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Theorem limsupre3uzlem 39403
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 39119 . . . . 5 (ℤ𝑀) ⊆ ℝ
42, 3eqsstri 3620 . . . 4 𝑍 ⊆ ℝ
54a1i 11 . . 3 (𝜑𝑍 ⊆ ℝ)
6 limsupre3uzlem.4 . . 3 (𝜑𝐹:𝑍⟶ℝ*)
71, 5, 6limsupre3 39401 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))))
8 breq1 4626 . . . . . . . . . . 11 (𝑦 = 𝑘 → (𝑦𝑗𝑘𝑗))
98anbi1d 740 . . . . . . . . . 10 (𝑦 = 𝑘 → ((𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
109rexbidv 3047 . . . . . . . . 9 (𝑦 = 𝑘 → (∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗))))
1110cbvralv 3163 . . . . . . . 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 2937 . . . . . . . 8 𝑘𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗))
14 simpr 477 . . . . . . . . 9 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → 𝑘𝑍)
154, 14sseldi 3586 . . . . . . . . . 10 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → 𝑘 ∈ ℝ)
16 rspa 2926 . . . . . . . . . 10 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘 ∈ ℝ) → ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
1715, 16syldan 487 . . . . . . . . 9 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)))
18 nfv 1840 . . . . . . . . . . 11 𝑗 𝑘𝑍
19 nfre1 3001 . . . . . . . . . . 11 𝑗𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)
20 eqid 2621 . . . . . . . . . . . . . . 15 (ℤ𝑘) = (ℤ𝑘)
212eluzelz2 39126 . . . . . . . . . . . . . . . 16 (𝑘𝑍𝑘 ∈ ℤ)
22213ad2ant1 1080 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑘 ∈ ℤ)
232eluzelz2 39126 . . . . . . . . . . . . . . . 16 (𝑗𝑍𝑗 ∈ ℤ)
24233ad2ant2 1081 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑗 ∈ ℤ)
25 simp3 1061 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑘𝑗)
2620, 22, 24, 25eluzd 39134 . . . . . . . . . . . . . 14 ((𝑘𝑍𝑗𝑍𝑘𝑗) → 𝑗 ∈ (ℤ𝑘))
27263adant3r 1320 . . . . . . . . . . . . 13 ((𝑘𝑍𝑗𝑍 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑗 ∈ (ℤ𝑘))
28 simp3r 1088 . . . . . . . . . . . . 13 ((𝑘𝑍𝑗𝑍 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → 𝑥 ≤ (𝐹𝑗))
29 rspe 2999 . . . . . . . . . . . . 13 ((𝑗 ∈ (ℤ𝑘) ∧ 𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3027, 28, 29syl2anc 692 . . . . . . . . . . . 12 ((𝑘𝑍𝑗𝑍 ∧ (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
31303exp 1261 . . . . . . . . . . 11 (𝑘𝑍 → (𝑗𝑍 → ((𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))))
3218, 19, 31rexlimd 3021 . . . . . . . . . 10 (𝑘𝑍 → (∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
3332imp 445 . . . . . . . . 9 ((𝑘𝑍 ∧ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3414, 17, 33syl2anc 692 . . . . . . . 8 ((∀𝑘 ∈ ℝ ∃𝑗𝑍 (𝑘𝑗𝑥 ≤ (𝐹𝑗)) ∧ 𝑘𝑍) → ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
3513, 34ralrimia 38840 . . . . . . 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 4070 . . . . . . . . . . . . 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 761 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ∈ ℤ)
42 ceilcl 12599 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → (⌈‘𝑦) ∈ ℤ)
4342ad2antlr 762 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → (⌈‘𝑦) ∈ ℤ)
44 simpr 477 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → 𝑀 ≤ (⌈‘𝑦))
452, 41, 43, 44eluzd 39134 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → (⌈‘𝑦) ∈ 𝑍)
4639, 45eqeltrd 2698 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
47 iffalse 4073 . . . . . . . . . . . . . 14 𝑀 ≤ (⌈‘𝑦) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = 𝑀)
4847adantl 482 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) = 𝑀)
4940, 2uzidd2 39142 . . . . . . . . . . . . . 14 (𝜑𝑀𝑍)
5049adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → 𝑀𝑍)
5148, 50eqeltrd 2698 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
5251adantlr 750 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ ¬ 𝑀 ≤ (⌈‘𝑦)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
5346, 52pm2.61dan 831 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
5453adantlr 750 . . . . . . . . 9 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
55 simplr 791 . . . . . . . . 9 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
56 fveq2 6158 . . . . . . . . . . 11 (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (ℤ𝑘) = (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)))
5756rexeqdv 3138 . . . . . . . . . 10 (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹𝑗)))
5857rspcva 3297 . . . . . . . . 9 ((if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) → ∃𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹𝑗))
5954, 55, 58syl2anc 692 . . . . . . . 8 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → ∃𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))𝑥 ≤ (𝐹𝑗))
60 nfv 1840 . . . . . . . . . . 11 𝑗𝜑
6118nfci 2751 . . . . . . . . . . . 12 𝑗𝑍
6261, 19nfral 2941 . . . . . . . . . . 11 𝑗𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)
6360, 62nfan 1825 . . . . . . . . . 10 𝑗(𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
64 nfv 1840 . . . . . . . . . 10 𝑗 𝑦 ∈ ℝ
6563, 64nfan 1825 . . . . . . . . 9 𝑗((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ)
66 nfre1 3001 . . . . . . . . 9 𝑗𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗))
6740ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ∈ ℤ)
68 eluzelz 11657 . . . . . . . . . . . . . . 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 11442 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀 ∈ ℝ)
714, 53sseldi 3586 . . . . . . . . . . . . . . . 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 11442 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗 ∈ ℝ)
744, 49sseldi 3586 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ ℝ)
7574adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → 𝑀 ∈ ℝ)
7642zred 11442 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ → (⌈‘𝑦) ∈ ℝ)
7776adantl 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → (⌈‘𝑦) ∈ ℝ)
78 max1 11975 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℝ ∧ (⌈‘𝑦) ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
7975, 77, 78syl2anc 692 . . . . . . . . . . . . . . . 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 11660 . . . . . . . . . . . . . . . 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 10154 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑀𝑗)
842, 67, 69, 83eluzd 39134 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗𝑍)
85843adant3 1079 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑗𝑍)
86 simplr 791 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦 ∈ ℝ)
87 simpr 477 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
88 ceilge 12601 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ → 𝑦 ≤ (⌈‘𝑦))
8988adantl 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → 𝑦 ≤ (⌈‘𝑦))
90 max2 11977 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℝ ∧ (⌈‘𝑦) ∈ ℝ) → (⌈‘𝑦) ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
9175, 77, 90syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ ℝ) → (⌈‘𝑦) ≤ if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))
9287, 77, 71, 89, 91letrd 10154 . . . . . . . . . . . . . . . 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 10154 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦𝑗)
95943adant3 1079 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑦𝑗)
96 simp3 1061 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → 𝑥 ≤ (𝐹𝑗))
9795, 96jca 554 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
98 rspe 2999 . . . . . . . . . . . 12 ((𝑗𝑍 ∧ (𝑦𝑗𝑥 ≤ (𝐹𝑗))) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
9985, 97, 98syl2anc 692 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) ∧ 𝑥 ≤ (𝐹𝑗)) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
100993exp 1261 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝑥 ≤ (𝐹𝑗) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))))
101100adantlr 750 . . . . . . . . 9 (((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) ∧ 𝑦 ∈ ℝ) → (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝑥 ≤ (𝐹𝑗) → ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))))
10265, 66, 101rexlimd 3021 . . . . . . . 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 2962 . . . . . 6 ((𝜑 ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)) → ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)))
105104ex 450 . . . . 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 3047 . . 3 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℝ ∃𝑗𝑍 (𝑦𝑗𝑥 ≤ (𝐹𝑗)) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
10853adantr 481 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍)
10960, 64nfan 1825 . . . . . . . . . 10 𝑗(𝜑𝑦 ∈ ℝ)
110 nfra1 2937 . . . . . . . . . 10 𝑗𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)
111109, 110nfan 1825 . . . . . . . . 9 𝑗((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
11294adantlr 750 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑦𝑗)
113 simplr 791 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
11484adantlr 750 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))) → 𝑗𝑍)
115 rspa 2926 . . . . . . . . . . . 12 ((∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍) → (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
116113, 114, 115syl2anc 692 . . . . . . . . . . 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 450 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → (𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀)) → (𝐹𝑗) ≤ 𝑥))
119111, 118ralrimi 2953 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∀𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹𝑗) ≤ 𝑥)
12056raleqdv 3137 . . . . . . . . 9 (𝑘 = if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) → (∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹𝑗) ≤ 𝑥))
121120rspcev 3299 . . . . . . . 8 ((if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀) ∈ 𝑍 ∧ ∀𝑗 ∈ (ℤ‘if(𝑀 ≤ (⌈‘𝑦), (⌈‘𝑦), 𝑀))(𝐹𝑗) ≤ 𝑥) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
122108, 119, 121syl2anc 692 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
123122ex 450 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
124123rexlimdva 3026 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) → ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
1254sseli 3584 . . . . . . . . 9 (𝑘𝑍𝑘 ∈ ℝ)
126125ad2antlr 762 . . . . . . . 8 (((𝜑𝑘𝑍) ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → 𝑘 ∈ ℝ)
127 nfra1 2937 . . . . . . . . . . 11 𝑗𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥
12818, 127nfan 1825 . . . . . . . . . 10 𝑗(𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
129 simp1r 1084 . . . . . . . . . . . 12 (((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍𝑘𝑗) → ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
130263adant1r 1316 . . . . . . . . . . . 12 (((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍𝑘𝑗) → 𝑗 ∈ (ℤ𝑘))
131 rspa 2926 . . . . . . . . . . . 12 ((∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ≤ 𝑥)
132129, 130, 131syl2anc 692 . . . . . . . . . . 11 (((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ∧ 𝑗𝑍𝑘𝑗) → (𝐹𝑗) ≤ 𝑥)
1331323exp 1261 . . . . . . . . . 10 ((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → (𝑗𝑍 → (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
134128, 133ralrimi 2953 . . . . . . . . 9 ((𝑘𝑍 ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
135134adantll 749 . . . . . . . 8 (((𝜑𝑘𝑍) ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥))
1368imbi1d 331 . . . . . . . . . 10 (𝑦 = 𝑘 → ((𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
137136ralbidv 2982 . . . . . . . . 9 (𝑦 = 𝑘 → (∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)))
138137rspcev 3299 . . . . . . . 8 ((𝑘 ∈ ℝ ∧ ∀𝑗𝑍 (𝑘𝑗 → (𝐹𝑗) ≤ 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
139126, 135, 138syl2anc 692 . . . . . . 7 (((𝜑𝑘𝑍) ∧ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥))
140139ex 450 . . . . . 6 ((𝜑𝑘𝑍) → (∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥)))
141140rexlimdva 3026 . . . . 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 3047 . . 3 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ∀𝑗𝑍 (𝑦𝑗 → (𝐹𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
144107, 143anbi12d 746 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wnfc 2748  wral 2908  wrex 2909  wss 3560  ifcif 4064   class class class wbr 4623  wf 5853  cfv 5857  cr 9895  *cxr 10033  cle 10035  cz 11337  cuz 11647  cceil 12548  lim supclsp 14151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-ico 12139  df-fl 12549  df-ceil 12550  df-limsup 14152
This theorem is referenced by:  limsupre3uz  39404  limsupreuz  39405
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