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	⊢ (𝜑 → 𝑀 ∈ ℤ)
climxrre.z	⊢ 𝑍 = (ℤ≥‘𝑀)
climxrre.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
climxrre.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
climxrre.c	⊢ (𝜑 → 𝐹 ⇝ 𝐴)

Assertion

Ref	Expression
climxrre	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶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.

Step	Hyp	Ref	Expression
1		climxrre.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2	1	ad2antrr 717	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
3		climxrre.z	. . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		climxrre.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
5	4	ad2antrr 717	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
6		climxrre.c	. . . . 5 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
7	6	ad2antrr 717	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
8		simpr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ∈ ℂ)
9		climxrre.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
10	9	recnd 10392	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
11	10	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐴 ∈ ℂ)
12	8, 11	subcld 10720	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ∈ ℂ)
13		renepnf 10411	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
14	13	necomd 3054	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → +∞ ≠ 𝐴)
15	9, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → +∞ ≠ 𝐴)
16	15	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ≠ 𝐴)
17	8, 11, 16	subne0d 10729	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ≠ 0)
18	12, 17	absrpcld 14571	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
19	18	adantr 474	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
20		simpr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ∈ ℂ)
21	10	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℂ)
22	20, 21	subcld 10720	. . . . . . 7 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ∈ ℂ)
23	9	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℝ)
24		renemnf 10412	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
25	24	necomd 3054	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -∞ ≠ 𝐴)
26	23, 25	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ≠ 𝐴)
27	20, 21, 26	subne0d 10729	. . . . . . 7 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ≠ 0)
28	22, 27	absrpcld 14571	. . . . . 6 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
29	28	adantlr 706	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
30	19, 29	ifcld 4353	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ∈ ℝ+)
31	19	rpred 12163	. . . . . 6 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
32	29	rpred 12163	. . . . . 6 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
33	31, 32	min1d 40494	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
34	33	adantr 474	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
35	31, 32	min2d 40495	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(-∞ − 𝐴)))
36	35	adantr 474	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(-∞ − 𝐴)))
37	2, 3, 5, 7, 30, 34, 36	climxrrelem 40774	. . 3 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
38	1	ad2antrr 717	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
39	4	ad2antrr 717	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
40	6	ad2antrr 717	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
41	18	adantr 474	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
42	18	rpred 12163	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
43	42	leidd 10925	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
44	43	ad2antrr 717	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
45		pm2.21 121	. . . . . 6 ⊢ (¬ -∞ ∈ ℂ → (-∞ ∈ ℂ → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴))))
46	45	imp 397	. . . . 5 ⊢ ((¬ -∞ ∈ ℂ ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
47	46	adantll 705	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
48	38, 3, 39, 40, 41, 44, 47	climxrrelem 40774	. . 3 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
49	37, 48	pm2.61dan 847	. 2 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
50	1	ad2antrr 717	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
51	4	ad2antrr 717	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
52	6	ad2antrr 717	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
53	28	adantlr 706	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
54		pm2.21 121	. . . . . 6 ⊢ (¬ +∞ ∈ ℂ → (+∞ ∈ ℂ → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴))))
55	54	imp 397	. . . . 5 ⊢ ((¬ +∞ ∈ ℂ ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
56	55	ad4ant24 763	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
57	28	rpred 12163	. . . . . 6 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
58	57	leidd 10925	. . . . 5 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
59	58	ad4ant13 757	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
60	50, 3, 51, 52, 53, 56, 59	climxrrelem 40774	. . 3 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
61		nfv 2013	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ)
62		nfv 2013	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
63		nfra1 3150	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ
64	62, 63	nfan 2002	. . . . . . 7 ⊢ Ⅎ𝑘(𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
65	61, 64	nfan 2002	. . . . . 6 ⊢ Ⅎ𝑘(((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ))
66		simp-4l 801	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝜑)
67	3	uztrn2 11993	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
68	67	adantlr 706	. . . . . . . . 9 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
69	68	adantll 705	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
70		simpr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
71	4	fdmd 6291	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑍)
72	71	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom 𝐹 = 𝑍)
73	70, 72	eleqtrrd 2909	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ dom 𝐹)
74	66, 69, 73	syl2anc 579	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
75	4	ffvelrnda 6613	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ*)
76	66, 69, 75	syl2anc 579	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ*)
77		rspa 3139	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
78	77	adantll 705	. . . . . . . . . 10 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
79	78	adantll 705	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
80		simpllr 793	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ¬ -∞ ∈ ℂ)
81		nelne2 3096	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹‘𝑘) ≠ -∞)
82	79, 80, 81	syl2anc 579	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ≠ -∞)
83		simp-4r 803	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ¬ +∞ ∈ ℂ)
84		nelne2 3096	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ ¬ +∞ ∈ ℂ) → (𝐹‘𝑘) ≠ +∞)
85	79, 83, 84	syl2anc 579	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ≠ +∞)
86	76, 82, 85	xrred 40376	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
87	74, 86	jca 507	. . . . . 6 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
88	65, 87	ralrimia 40128	. . . . 5 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
89	4	ffund 6286	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
90		ffvresb 6648	. . . . . . 7 ⊢ (Fun 𝐹 → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
91	89, 90	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
92	91	ad3antrrr 721	. . . . 5 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
93	88, 92	mpbird 249	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
94		r19.26 3274	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1) ↔ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 1))
95	94	simplbi 493	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
96	95	ad2antll 720	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
97		breq2 4879	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))
98	97	anbi2d 622	. . . . . . . . 9 ⊢ (𝑥 = 1 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1)))
99	98	rexralbidv 3268	. . . . . . . 8 ⊢ (𝑥 = 1 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1)))
100	3	fvexi 6451	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
101	100	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ V)
102	4, 101	fexd 40110	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
103		eqidd 2826	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘))
104	102, 103	clim 14609	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
105	6, 104	mpbid 224	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
106	105	simprd 491	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
107		1rp 12123	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
108	107	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ+)
109	99, 106, 108	rspcdva 3532	. . . . . . 7 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))
110	96, 109	reximddv 3226	. . . . . 6 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
111	3	rexuz3 14472	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ))
112	1, 111	syl 17	. . . . . 6 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ))
113	110, 112	mpbird 249	. . . . 5 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
114	113	ad2antrr 717	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
115	93, 114	reximddv 3226	. . 3 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
116	60, 115	pm2.61dan 847	. 2 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
117	49, 116	pm2.61dan 847	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)

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  ℂcc 10257  ℝcr 10258  1c1 10260  +∞cpnf 10395  -∞cmnf 10396  ℝ^*cxr 10397   < clt 10398   ≤ cle 10399   − cmin 10592  ℤcz 11711  ℤ_≥cuz 11975  ℝ⁺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