Theorem climxrre 43181

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 722	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
3		climxrre.z	. . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		climxrre.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
5	4	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
6		climxrre.c	. . . . 5 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
7	6	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
8		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ∈ ℂ)
9		climxrre.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
10	9	recnd 10934	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐴 ∈ ℂ)
12	8, 11	subcld 11262	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ∈ ℂ)
13		renepnf 10954	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
14	13	necomd 2998	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → +∞ ≠ 𝐴)
15	9, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → +∞ ≠ 𝐴)
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ≠ 𝐴)
17	8, 11, 16	subne0d 11271	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ≠ 0)
18	12, 17	absrpcld 15088	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
19	18	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
20		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ∈ ℂ)
21	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℂ)
22	20, 21	subcld 11262	. . . . . . 7 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ∈ ℂ)
23	9	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℝ)
24		renemnf 10955	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
25	24	necomd 2998	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -∞ ≠ 𝐴)
26	23, 25	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ≠ 𝐴)
27	20, 21, 26	subne0d 11271	. . . . . . 7 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ≠ 0)
28	22, 27	absrpcld 15088	. . . . . 6 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
29	28	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
30	19, 29	ifcld 4502	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ∈ ℝ+)
31	19	rpred 12701	. . . . . 6 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
32	29	rpred 12701	. . . . . 6 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
33	31, 32	min1d 42902	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
34	33	adantr 480	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
35	31, 32	min2d 42903	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(-∞ − 𝐴)))
36	35	adantr 480	. . . 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 43180	. . 3 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
38	1	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
39	4	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
40	6	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
41	18	adantr 480	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
42	18	rpred 12701	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
43	42	leidd 11471	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
44	43	ad2antrr 722	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
45		pm2.21 123	. . . . . 6 ⊢ (¬ -∞ ∈ ℂ → (-∞ ∈ ℂ → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴))))
46	45	imp 406	. . . . 5 ⊢ ((¬ -∞ ∈ ℂ ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
47	46	adantll 710	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
48	38, 3, 39, 40, 41, 44, 47	climxrrelem 43180	. . 3 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
49	37, 48	pm2.61dan 809	. 2 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
50	1	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
51	4	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
52	6	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
53	28	adantlr 711	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
54		pm2.21 123	. . . . . 6 ⊢ (¬ +∞ ∈ ℂ → (+∞ ∈ ℂ → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴))))
55	54	imp 406	. . . . 5 ⊢ ((¬ +∞ ∈ ℂ ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
56	55	ad4ant24 750	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
57	28	rpred 12701	. . . . . 6 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
58	57	leidd 11471	. . . . 5 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
59	58	ad4ant13 747	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
60	50, 3, 51, 52, 53, 56, 59	climxrrelem 43180	. . 3 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
61		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ)
62		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
63		nfra1 3142	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ
64	62, 63	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑘(𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
65	61, 64	nfan 1903	. . . . . 6 ⊢ Ⅎ𝑘(((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ))
66		simp-4l 779	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝜑)
67	3	uztrn2 12530	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
68	67	adantlr 711	. . . . . . . . 9 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
69	68	adantll 710	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
70		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
71	4	fdmd 6595	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑍)
72	71	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom 𝐹 = 𝑍)
73	70, 72	eleqtrrd 2842	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ dom 𝐹)
74	66, 69, 73	syl2anc 583	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
75	4	ffvelrnda 6943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ*)
76	66, 69, 75	syl2anc 583	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ*)
77		rspa 3130	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
78	77	adantll 710	. . . . . . . . . 10 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
79	78	adantll 710	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
80		simpllr 772	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ¬ -∞ ∈ ℂ)
81		nelne2 3041	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹‘𝑘) ≠ -∞)
82	79, 80, 81	syl2anc 583	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ≠ -∞)
83		simp-4r 780	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ¬ +∞ ∈ ℂ)
84		nelne2 3041	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ ¬ +∞ ∈ ℂ) → (𝐹‘𝑘) ≠ +∞)
85	79, 83, 84	syl2anc 583	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ≠ +∞)
86	76, 82, 85	xrred 42794	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
87	74, 86	jca 511	. . . . . 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 3420	. . . . 5 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
89	4	ffund 6588	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
90		ffvresb 6980	. . . . . . 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 726	. . . . 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 256	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
94		r19.26 3094	. . . . . . . . 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 497	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
96	95	ad2antll 725	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
97		breq2 5074	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))
98	97	anbi2d 628	. . . . . . . . 9 ⊢ (𝑥 = 1 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1)))
99	98	rexralbidv 3229	. . . . . . . 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 6770	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
101	100	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ V)
102	4, 101	fexd 7085	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
103		eqidd 2739	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘))
104	102, 103	clim 15131	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
106	105	simprd 495	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
107		1rp 12663	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
108	107	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ+)
109	99, 106, 108	rspcdva 3554	. . . . . . 7 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))
110	96, 109	reximddv 3203	. . . . . 6 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
111	3	rexuz3 14988	. . . . . . 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 256	. . . . 5 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
114	113	ad2antrr 722	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
115	93, 114	reximddv 3203	. . 3 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
116	60, 115	pm2.61dan 809	. 2 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
117	49, 116	pm2.61dan 809	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422  ifcif 4456   class class class wbr 5070  dom cdm 5580   ↾ cres 5582  Fun wfun 6412  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  ℝcr 10801  1c1 10803  +∞cpnf 10937  -∞cmnf 10938  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   − cmin 11135  ℤcz 12249  ℤ_≥cuz 12511  ℝ⁺crp 12659  abscabs 14873   ⇝ cli 15121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125

This theorem is referenced by:  xlimclim2  43271