Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  climxrre Structured version   Visualization version   GIF version

Theorem climxrre 42179
 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 724 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
3 climxrre.z . . . 4 𝑍 = (ℤ𝑀)
4 climxrre.f . . . . 5 (𝜑𝐹:𝑍⟶ℝ*)
54ad2antrr 724 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
6 climxrre.c . . . . 5 (𝜑𝐹𝐴)
76ad2antrr 724 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹𝐴)
8 simpr 487 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ∈ ℂ)
9 climxrre.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ)
109recnd 10643 . . . . . . . . 9 (𝜑𝐴 ∈ ℂ)
1110adantr 483 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ℂ) → 𝐴 ∈ ℂ)
128, 11subcld 10971 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ∈ ℂ)
13 renepnf 10663 . . . . . . . . . . 11 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
1413necomd 3061 . . . . . . . . . 10 (𝐴 ∈ ℝ → +∞ ≠ 𝐴)
159, 14syl 17 . . . . . . . . 9 (𝜑 → +∞ ≠ 𝐴)
1615adantr 483 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ≠ 𝐴)
178, 11, 16subne0d 10980 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ≠ 0)
1812, 17absrpcld 14784 . . . . . 6 ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
1918adantr 483 . . . . 5 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
20 simpr 487 . . . . . . . 8 ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ∈ ℂ)
2110adantr 483 . . . . . . . 8 ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℂ)
2220, 21subcld 10971 . . . . . . 7 ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ∈ ℂ)
239adantr 483 . . . . . . . . 9 ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℝ)
24 renemnf 10664 . . . . . . . . . 10 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
2524necomd 3061 . . . . . . . . 9 (𝐴 ∈ ℝ → -∞ ≠ 𝐴)
2623, 25syl 17 . . . . . . . 8 ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ≠ 𝐴)
2720, 21, 26subne0d 10980 . . . . . . 7 ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ≠ 0)
2822, 27absrpcld 14784 . . . . . 6 ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
2928adantlr 713 . . . . 5 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
3019, 29ifcld 4484 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ∈ ℝ+)
3119rpred 12406 . . . . . 6 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
3229rpred 12406 . . . . . 6 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
3331, 32min1d 41898 . . . . 5 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
3433adantr 483 . . . 4 ((((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
3531, 32min2d 41899 . . . . 5 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(-∞ − 𝐴)))
3635adantr 483 . . . 4 ((((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(-∞ − 𝐴)))
372, 3, 5, 7, 30, 34, 36climxrrelem 42178 . . 3 (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
381ad2antrr 724 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
394ad2antrr 724 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
406ad2antrr 724 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹𝐴)
4118adantr 483 . . . 4 (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
4218rpred 12406 . . . . . 6 ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
4342leidd 11180 . . . . 5 ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
4443ad2antrr 724 . . . 4 ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
45 pm2.21 123 . . . . . 6 (¬ -∞ ∈ ℂ → (-∞ ∈ ℂ → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴))))
4645imp 409 . . . . 5 ((¬ -∞ ∈ ℂ ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
4746adantll 712 . . . 4 ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
4838, 3, 39, 40, 41, 44, 47climxrrelem 42178 . . 3 (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
4937, 48pm2.61dan 811 . 2 ((𝜑 ∧ +∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
501ad2antrr 724 . . . 4 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
514ad2antrr 724 . . . 4 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
526ad2antrr 724 . . . 4 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹𝐴)
5328adantlr 713 . . . 4 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
54 pm2.21 123 . . . . . 6 (¬ +∞ ∈ ℂ → (+∞ ∈ ℂ → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴))))
5554imp 409 . . . . 5 ((¬ +∞ ∈ ℂ ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
5655ad4ant24 752 . . . 4 ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
5728rpred 12406 . . . . . 6 ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
5857leidd 11180 . . . . 5 ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
5958ad4ant13 749 . . . 4 ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
6050, 3, 51, 52, 53, 56, 59climxrrelem 42178 . . 3 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
61 nfv 1915 . . . . . . 7 𝑘((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ)
62 nfv 1915 . . . . . . . 8 𝑘 𝑗𝑍
63 nfra1 3206 . . . . . . . 8 𝑘𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ
6462, 63nfan 1900 . . . . . . 7 𝑘(𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
6561, 64nfan 1900 . . . . . 6 𝑘(((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ))
66 simp-4l 781 . . . . . . . 8 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝜑)
673uztrn2 12237 . . . . . . . . . 10 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
6867adantlr 713 . . . . . . . . 9 (((𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
6968adantll 712 . . . . . . . 8 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
70 simpr 487 . . . . . . . . 9 ((𝜑𝑘𝑍) → 𝑘𝑍)
714fdmd 6495 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝑍)
7271adantr 483 . . . . . . . . 9 ((𝜑𝑘𝑍) → dom 𝐹 = 𝑍)
7370, 72eleqtrrd 2914 . . . . . . . 8 ((𝜑𝑘𝑍) → 𝑘 ∈ dom 𝐹)
7466, 69, 73syl2anc 586 . . . . . . 7 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘 ∈ dom 𝐹)
754ffvelrnda 6823 . . . . . . . . 9 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ*)
7666, 69, 75syl2anc 586 . . . . . . . 8 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ*)
77 rspa 3193 . . . . . . . . . . 11 ((∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℂ)
7877adantll 712 . . . . . . . . . 10 (((𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℂ)
7978adantll 712 . . . . . . . . 9 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℂ)
80 simpllr 774 . . . . . . . . 9 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → ¬ -∞ ∈ ℂ)
81 nelne2 3103 . . . . . . . . 9 (((𝐹𝑘) ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹𝑘) ≠ -∞)
8279, 80, 81syl2anc 586 . . . . . . . 8 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≠ -∞)
83 simp-4r 782 . . . . . . . . 9 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → ¬ +∞ ∈ ℂ)
84 nelne2 3103 . . . . . . . . 9 (((𝐹𝑘) ∈ ℂ ∧ ¬ +∞ ∈ ℂ) → (𝐹𝑘) ≠ +∞)
8579, 83, 84syl2anc 586 . . . . . . . 8 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ≠ +∞)
8676, 82, 85xrred 41783 . . . . . . 7 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝐹𝑘) ∈ ℝ)
8774, 86jca 514 . . . . . 6 (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ𝑗)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ ℝ))
8865, 87ralrimia 41548 . . . . 5 ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) → ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ ℝ))
894ffund 6490 . . . . . . 7 (𝜑 → Fun 𝐹)
90 ffvresb 6860 . . . . . . 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 728 . . . . 5 ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) → ((𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ ℝ)))
9388, 92mpbird 259 . . . 4 ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗𝑍 ∧ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)) → (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
94 r19.26 3157 . . . . . . . . 9 (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1) ↔ (∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − 𝐴)) < 1))
9594simplbi 500 . . . . . . . 8 (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
9695ad2antll 727 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1))) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
97 breq2 5042 . . . . . . . . . 10 (𝑥 = 1 → ((abs‘((𝐹𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹𝑘) − 𝐴)) < 1))
9897anbi2d 630 . . . . . . . . 9 (𝑥 = 1 → (((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1)))
9998rexralbidv 3286 . . . . . . . 8 (𝑥 = 1 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1)))
1003fvexi 6656 . . . . . . . . . . . . 13 𝑍 ∈ V
101100a1i 11 . . . . . . . . . . . 12 (𝜑𝑍 ∈ V)
1024, 101fexd 6962 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
103 eqidd 2821 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = (𝐹𝑘))
104102, 103clim 14827 . . . . . . . . . 10 (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))))
1056, 104mpbid 234 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥)))
106105simprd 498 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 𝑥))
107 1rp 12368 . . . . . . . . 9 1 ∈ ℝ+
108107a1i 11 . . . . . . . 8 (𝜑 → 1 ∈ ℝ+)
10999, 106, 108rspcdva 3601 . . . . . . 7 (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − 𝐴)) < 1))
11096, 109reximddv 3260 . . . . . 6 (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
1113rexuz3 14684 . . . . . . 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 259 . . . . 5 (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
114113ad2antrr 724 . . . 4 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ ℂ)
11593, 114reximddv 3260 . . 3 (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
11660, 115pm2.61dan 811 . 2 ((𝜑 ∧ ¬ +∞ ∈ ℂ) → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
11749, 116pm2.61dan 811 1 (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶ℝ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3006  ∀wral 3125  ∃wrex 3126  Vcvv 3470  ifcif 4439   class class class wbr 5038  dom cdm 5527   ↾ cres 5529  Fun wfun 6321  ⟶wf 6323  ‘cfv 6327  (class class class)co 7129  ℂcc 10509  ℝcr 10510  1c1 10512  +∞cpnf 10646  -∞cmnf 10647  ℝ*cxr 10648   < clt 10649   ≤ cle 10650   − cmin 10844  ℤcz 11956  ℤ≥cuz 12218  ℝ+crp 12364  abscabs 14569   ⇝ cli 14817 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435  ax-cnex 10567  ax-resscn 10568  ax-1cn 10569  ax-icn 10570  ax-addcl 10571  ax-addrcl 10572  ax-mulcl 10573  ax-mulrcl 10574  ax-mulcom 10575  ax-addass 10576  ax-mulass 10577  ax-distr 10578  ax-i2m1 10579  ax-1ne0 10580  ax-1rid 10581  ax-rnegex 10582  ax-rrecex 10583  ax-cnre 10584  ax-pre-lttri 10585  ax-pre-lttrn 10586  ax-pre-ltadd 10587  ax-pre-mulgt0 10588  ax-pre-sup 10589 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-tr 5145  df-id 5432  df-eprel 5437  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7087  df-ov 7132  df-oprab 7133  df-mpo 7134  df-om 7555  df-2nd 7664  df-wrecs 7921  df-recs 7982  df-rdg 8020  df-er 8263  df-en 8484  df-dom 8485  df-sdom 8486  df-sup 8880  df-pnf 10651  df-mnf 10652  df-xr 10653  df-ltxr 10654  df-le 10655  df-sub 10846  df-neg 10847  df-div 11272  df-nn 11613  df-2 11675  df-3 11676  df-n0 11873  df-z 11957  df-uz 12219  df-rp 12365  df-seq 13350  df-exp 13411  df-cj 14434  df-re 14435  df-im 14436  df-sqrt 14570  df-abs 14571  df-clim 14821 This theorem is referenced by:  xlimclim2  42269
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