Theorem climxrre 42392

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 725	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
3		climxrre.z	. . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		climxrre.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
5	4	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
6		climxrre.c	. . . . 5 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
7	6	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
8		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ∈ ℂ)
9		climxrre.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
10	9	recnd 10658	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
11	10	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐴 ∈ ℂ)
12	8, 11	subcld 10986	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ∈ ℂ)
13		renepnf 10678	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
14	13	necomd 3042	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → +∞ ≠ 𝐴)
15	9, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → +∞ ≠ 𝐴)
16	15	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ≠ 𝐴)
17	8, 11, 16	subne0d 10995	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ≠ 0)
18	12, 17	absrpcld 14800	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
19	18	adantr 484	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
20		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ∈ ℂ)
21	10	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℂ)
22	20, 21	subcld 10986	. . . . . . 7 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ∈ ℂ)
23	9	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℝ)
24		renemnf 10679	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
25	24	necomd 3042	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -∞ ≠ 𝐴)
26	23, 25	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ≠ 𝐴)
27	20, 21, 26	subne0d 10995	. . . . . . 7 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ≠ 0)
28	22, 27	absrpcld 14800	. . . . . 6 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
29	28	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
30	19, 29	ifcld 4470	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ∈ ℝ+)
31	19	rpred 12419	. . . . . 6 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
32	29	rpred 12419	. . . . . 6 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
33	31, 32	min1d 42111	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
34	33	adantr 484	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
35	31, 32	min2d 42112	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(-∞ − 𝐴)))
36	35	adantr 484	. . . 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 42391	. . 3 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
38	1	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
39	4	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
40	6	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
41	18	adantr 484	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
42	18	rpred 12419	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
43	42	leidd 11195	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
44	43	ad2antrr 725	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
45		pm2.21 123	. . . . . 6 ⊢ (¬ -∞ ∈ ℂ → (-∞ ∈ ℂ → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴))))
46	45	imp 410	. . . . 5 ⊢ ((¬ -∞ ∈ ℂ ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
47	46	adantll 713	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
48	38, 3, 39, 40, 41, 44, 47	climxrrelem 42391	. . 3 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
49	37, 48	pm2.61dan 812	. 2 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
50	1	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
51	4	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
52	6	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
53	28	adantlr 714	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
54		pm2.21 123	. . . . . 6 ⊢ (¬ +∞ ∈ ℂ → (+∞ ∈ ℂ → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴))))
55	54	imp 410	. . . . 5 ⊢ ((¬ +∞ ∈ ℂ ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
56	55	ad4ant24 753	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
57	28	rpred 12419	. . . . . 6 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
58	57	leidd 11195	. . . . 5 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
59	58	ad4ant13 750	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
60	50, 3, 51, 52, 53, 56, 59	climxrrelem 42391	. . 3 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
61		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ)
62		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
63		nfra1 3183	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ
64	62, 63	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑘(𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
65	61, 64	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑘(((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ))
66		simp-4l 782	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝜑)
67	3	uztrn2 12250	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
68	67	adantlr 714	. . . . . . . . 9 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
69	68	adantll 713	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
70		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
71	4	fdmd 6497	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑍)
72	71	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom 𝐹 = 𝑍)
73	70, 72	eleqtrrd 2893	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ dom 𝐹)
74	66, 69, 73	syl2anc 587	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
75	4	ffvelrnda 6828	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ*)
76	66, 69, 75	syl2anc 587	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ*)
77		rspa 3171	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
78	77	adantll 713	. . . . . . . . . 10 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
79	78	adantll 713	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
80		simpllr 775	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ¬ -∞ ∈ ℂ)
81		nelne2 3084	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹‘𝑘) ≠ -∞)
82	79, 80, 81	syl2anc 587	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ≠ -∞)
83		simp-4r 783	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ¬ +∞ ∈ ℂ)
84		nelne2 3084	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ ¬ +∞ ∈ ℂ) → (𝐹‘𝑘) ≠ +∞)
85	79, 83, 84	syl2anc 587	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ≠ +∞)
86	76, 82, 85	xrred 41997	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
87	74, 86	jca 515	. . . . . 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 41767	. . . . 5 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
89	4	ffund 6491	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
90		ffvresb 6865	. . . . . . 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 729	. . . . 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 260	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
94		r19.26 3137	. . . . . . . . 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 501	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
96	95	ad2antll 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
97		breq2 5034	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))
98	97	anbi2d 631	. . . . . . . . 9 ⊢ (𝑥 = 1 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1)))
99	98	rexralbidv 3260	. . . . . . . 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 6659	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
101	100	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ V)
102	4, 101	fexd 6967	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
103		eqidd 2799	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘))
104	102, 103	clim 14843	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
106	105	simprd 499	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
107		1rp 12381	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
108	107	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ+)
109	99, 106, 108	rspcdva 3573	. . . . . . 7 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))
110	96, 109	reximddv 3234	. . . . . 6 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
111	3	rexuz3 14700	. . . . . . 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 260	. . . . 5 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
114	113	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
115	93, 114	reximddv 3234	. . 3 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
116	60, 115	pm2.61dan 812	. 2 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
117	49, 116	pm2.61dan 812	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441  ifcif 4425   class class class wbr 5030  dom cdm 5519   ↾ cres 5521  Fun wfun 6318  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ℂcc 10524  ℝcr 10525  1c1 10527  +∞cpnf 10661  -∞cmnf 10662  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665   − cmin 10859  ℤcz 11969  ℤ_≥cuz 12231  ℝ⁺crp 12377  abscabs 14585   ⇝ cli 14833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  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 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-sup 8890  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837

This theorem is referenced by:  xlimclim2  42482