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Theorem climxrre 40775
Description: If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis 𝐹 ∈ dom ⇝ is probably not enough, since in principle we could have +∞ ∈ ℂ and -∞ ∈ ℂ). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Hypotheses
Ref Expression
climxrre.m (𝜑𝑀 ∈ ℤ)
climxrre.z 𝑍 = (ℤ𝑀)
climxrre.f (𝜑𝐹:𝑍⟶ℝ*)
climxrre.a (𝜑𝐴 ∈ ℝ)
climxrre.c (𝜑𝐹𝐴)
Assertion
Ref Expression
climxrre (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
Distinct variable groups:   𝐴,𝑗   𝑗,𝐹   𝑗,𝑀   𝑗,𝑍   𝜑,𝑗

Proof of Theorem climxrre
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climxrre.m . . . . 5 (𝜑𝑀 ∈ ℤ)
21ad2antrr 717 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
3 climxrre.z . . . 4 𝑍 = (ℤ𝑀)
4 climxrre.f . . . . 5 (𝜑𝐹:𝑍⟶ℝ*)
54ad2antrr 717 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
6 climxrre.c . . . . 5 (𝜑𝐹𝐴)
76ad2antrr 717 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹𝐴)
8 simpr 479 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ∈ ℂ)
9 climxrre.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ)
109recnd 10392 . . . . . . . . 9 (𝜑𝐴 ∈ ℂ)
1110adantr 474 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ℂ) → 𝐴 ∈ ℂ)
128, 11subcld 10720 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ∈ ℂ)
13 renepnf 10411 . . . . . . . . . . 11 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
1413necomd 3054 . . . . . . . . . 10 (𝐴 ∈ ℝ → +∞ ≠ 𝐴)
159, 14syl 17 . . . . . . . . 9 (𝜑 → +∞ ≠ 𝐴)
1615adantr 474 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ≠ 𝐴)
178, 11, 16subne0d 10729 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ≠ 0)
1812, 17absrpcld 14571 . . . . . 6 ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
1918adantr 474 . . . . 5 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
20 simpr 479 . . . . . . . 8 ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ∈ ℂ)
2110adantr 474 . . . . . . . 8 ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℂ)
2220, 21subcld 10720 . . . . . . 7 ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ∈ ℂ)
239adantr 474 . . . . . . . . 9 ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℝ)
24 renemnf 10412 . . . . . . . . . 10 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
2524necomd 3054 . . . . . . . . 9 (𝐴 ∈ ℝ → -∞ ≠ 𝐴)
2623, 25syl 17 . . . . . . . 8 ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ≠ 𝐴)
2720, 21, 26subne0d 10729 . . . . . . 7 ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ≠ 0)
2822, 27absrpcld 14571 . . . . . 6 ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
2928adantlr 706 . . . . 5 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
3019, 29ifcld 4353 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ∈ ℝ+)
3119rpred 12163 . . . . . 6 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
3229rpred 12163 . . . . . 6 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
3331, 32min1d 40494 . . . . 5 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
3433adantr 474 . . . 4 ((((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
3531, 32min2d 40495 . . . . 5 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(-∞ − 𝐴)))
3635adantr 474 . . . 4 ((((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(-∞ − 𝐴)))
372, 3, 5, 7, 30, 34, 36climxrrelem 40774 . . 3 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
381ad2antrr 717 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
394ad2antrr 717 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
406ad2antrr 717 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹𝐴)
4118adantr 474 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
4218rpred 12163 . . . . . 6 ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
4342leidd 10925 . . . . 5 ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
4443ad2antrr 717 . . . 4 ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
45 pm2.21 121 . . . . . 6 (¬ -∞ ∈ ℂ → (-∞ ∈ ℂ → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴))))
4645imp 397 . . . . 5 ((¬ -∞ ∈ ℂ ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
4746adantll 705 . . . 4 ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
4838, 3, 39, 40, 41, 44, 47climxrrelem 40774 . . 3 (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
4937, 48pm2.61dan 847 . 2 ((𝜑 ∧ +∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
501ad2antrr 717 . . . 4 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
514ad2antrr 717 . . . 4 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
526ad2antrr 717 . . . 4 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹𝐴)
5328adantlr 706 . . . 4 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
54 pm2.21 121 . . . . . 6 (¬ +∞ ∈ ℂ → (+∞ ∈ ℂ → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴))))
5554imp 397 . . . . 5 ((¬ +∞ ∈ ℂ ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
5655ad4ant24 763 . . . 4 ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
5728rpred 12163 . . . . . 6 ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
5857leidd 10925 . . . . 5 ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
5958ad4ant13 757 . . . 4 ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
6050, 3, 51, 52, 53, 56, 59climxrrelem 40774 . . 3 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
61 nfv 2013 . . . . . . 7 𝑘((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ)
62 nfv 2013 . . . . . . . 8 𝑘 𝑗𝑍
63 nfra1 3150 . . . . . . . 8 𝑘𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ
6462, 63nfan 2002 . . . . . . 7 𝑘(𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
6561, 64nfan 2002 . . . . . 6 𝑘(((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ))
66 simp-4l 801 . . . . . . . 8 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝜑)
673uztrn2 11993 . . . . . . . . . 10 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
6867adantlr 706 . . . . . . . . 9 (((𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
6968adantll 705 . . . . . . . 8 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
70 simpr 479 . . . . . . . . 9 ((𝜑𝑘𝑍) → 𝑘𝑍)
714fdmd 6291 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝑍)
7271adantr 474 . . . . . . . . 9 ((𝜑𝑘𝑍) → dom 𝐹 = 𝑍)
7370, 72eleqtrrd 2909 . . . . . . . 8 ((𝜑𝑘𝑍) → 𝑘 ∈ dom 𝐹)
7466, 69, 73syl2anc 579 . . . . . . 7 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘 ∈ dom 𝐹)
754ffvelrnda 6613 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ*)
7666, 69, 75syl2anc 579 . . . . . . . 8 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
77 rspa 3139 . . . . . . . . . . 11 ((∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℂ)
7877adantll 705 . . . . . . . . . 10 (((𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℂ)
7978adantll 705 . . . . . . . . 9 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℂ)
80 simpllr 793 . . . . . . . . 9 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → ¬ -∞ ∈ ℂ)
81 nelne2 3096 . . . . . . . . 9 (((𝐹𝑘) ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹𝑘) ≠ -∞)
8279, 80, 81syl2anc 579 . . . . . . . 8 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≠ -∞)
83 simp-4r 803 . . . . . . . . 9 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → ¬ +∞ ∈ ℂ)
84 nelne2 3096 . . . . . . . . 9 (((𝐹𝑘) ∈ ℂ ∧ ¬ +∞ ∈ ℂ) → (𝐹𝑘) ≠ +∞)
8579, 83, 84syl2anc 579 . . . . . . . 8 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≠ +∞)
8676, 82, 85xrred 40376 . . . . . . 7 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ)
8774, 86jca 507 . . . . . 6 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ ℝ))
8865, 87ralrimia 40128 . . . . 5 ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) → ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ ℝ))
894ffund 6286 . . . . . . 7 (𝜑 → Fun 𝐹)
90 ffvresb 6648 . . . . . . 7 (Fun 𝐹 → ((𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ ℝ)))
9189, 90syl 17 . . . . . 6 (𝜑 → ((𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ ℝ)))
9291ad3antrrr 721 . . . . 5 ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) → ((𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ ℝ)))
9388, 92mpbird 249 . . . 4 ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
94 r19.26 3274 . . . . . . . . 9 (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1) ↔ (∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 1))
9594simplbi 493 . . . . . . . 8 (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
9695ad2antll 720 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1))) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
97 breq2 4879 . . . . . . . . . 10 (𝑥 = 1 → ((abs‘((𝐹𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹𝑘) − 𝐴)) < 1))
9897anbi2d 622 . . . . . . . . 9 (𝑥 = 1 → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1)))
9998rexralbidv 3268 . . . . . . . 8 (𝑥 = 1 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1)))
1003fvexi 6451 . . . . . . . . . . . . 13 𝑍 ∈ V
101100a1i 11 . . . . . . . . . . . 12 (𝜑𝑍 ∈ V)
1024, 101fexd 40110 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
103 eqidd 2826 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = (𝐹𝑘))
104102, 103clim 14609 . . . . . . . . . 10 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
1056, 104mpbid 224 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
106105simprd 491 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
107 1rp 12123 . . . . . . . . 9 1 ∈ ℝ+
108107a1i 11 . . . . . . . 8 (𝜑 → 1 ∈ ℝ+)
10999, 106, 108rspcdva 3532 . . . . . . 7 (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1))
11096, 109reximddv 3226 . . . . . 6 (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
1113rexuz3 14472 . . . . . . 7 (𝑀 ∈ ℤ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ))
1121, 111syl 17 . . . . . 6 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ))
113110, 112mpbird 249 . . . . 5 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
114113ad2antrr 717 . . . 4 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
11593, 114reximddv 3226 . . 3 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
11660, 115pm2.61dan 847 . 2 ((𝜑 ∧ ¬ +∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
11749, 116pm2.61dan 847 1 (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  wne 2999  wral 3117  wrex 3118  Vcvv 3414  ifcif 4308   class class class wbr 4875  dom cdm 5346  cres 5348  Fun wfun 6121  wf 6123  cfv 6127  (class class class)co 6910  cc 10257  cr 10258  1c1 10260  +∞cpnf 10395  -∞cmnf 10396  *cxr 10397   < clt 10398  cle 10399  cmin 10592  cz 11711  cuz 11975  +crp 12119  abscabs 14358  cli 14599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336  ax-pre-sup 10337
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-sup 8623  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-3 11422  df-n0 11626  df-z 11712  df-uz 11976  df-rp 12120  df-seq 13103  df-exp 13162  df-cj 14223  df-re 14224  df-im 14225  df-sqrt 14359  df-abs 14360  df-clim 14603
This theorem is referenced by:  xlimclim2  40859
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