Theorem climxrre 44111

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 724	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
3		climxrre.z	. . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		climxrre.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
5	4	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
6		climxrre.c	. . . . 5 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
7	6	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
8		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ∈ ℂ)
9		climxrre.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
10	9	recnd 11192	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
11	10	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐴 ∈ ℂ)
12	8, 11	subcld 11521	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ∈ ℂ)
13		renepnf 11212	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
14	13	necomd 2995	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → +∞ ≠ 𝐴)
15	9, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → +∞ ≠ 𝐴)
16	15	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → +∞ ≠ 𝐴)
17	8, 11, 16	subne0d 11530	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (+∞ − 𝐴) ≠ 0)
18	12, 17	absrpcld 15345	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
19	18	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
20		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ∈ ℂ)
21	10	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℂ)
22	20, 21	subcld 11521	. . . . . . 7 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ∈ ℂ)
23	9	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐴 ∈ ℝ)
24		renemnf 11213	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
25	24	necomd 2995	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -∞ ≠ 𝐴)
26	23, 25	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → -∞ ≠ 𝐴)
27	20, 21, 26	subne0d 11530	. . . . . . 7 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (-∞ − 𝐴) ≠ 0)
28	22, 27	absrpcld 15345	. . . . . 6 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
29	28	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
30	19, 29	ifcld 4537	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ∈ ℝ+)
31	19	rpred 12966	. . . . . 6 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
32	29	rpred 12966	. . . . . 6 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
33	31, 32	min1d 43827	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
34	33	adantr 481	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(+∞ − 𝐴)))
35	31, 32	min2d 43828	. . . . 5 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → if((abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)), (abs‘(+∞ − 𝐴)), (abs‘(-∞ − 𝐴))) ≤ (abs‘(-∞ − 𝐴)))
36	35	adantr 481	. . . 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 44110	. . 3 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
38	1	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
39	4	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
40	6	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
41	18	adantr 481	. . . 4 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ+)
42	18	rpred 12966	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
43	42	leidd 11730	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
44	43	ad2antrr 724	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
45		pm2.21 123	. . . . . 6 ⊢ (¬ -∞ ∈ ℂ → (-∞ ∈ ℂ → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴))))
46	45	imp 407	. . . . 5 ⊢ ((¬ -∞ ∈ ℂ ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
47	46	adantll 712	. . . 4 ⊢ ((((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(+∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
48	38, 3, 39, 40, 41, 44, 47	climxrrelem 44110	. . 3 ⊢ (((𝜑 ∧ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
49	37, 48	pm2.61dan 811	. 2 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
50	1	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝑀 ∈ ℤ)
51	4	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹:𝑍⟶ℝ*)
52	6	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → 𝐹 ⇝ 𝐴)
53	28	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ+)
54		pm2.21 123	. . . . . 6 ⊢ (¬ +∞ ∈ ℂ → (+∞ ∈ ℂ → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴))))
55	54	imp 407	. . . . 5 ⊢ ((¬ +∞ ∈ ℂ ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
56	55	ad4ant24 752	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ +∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(+∞ − 𝐴)))
57	28	rpred 12966	. . . . . 6 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
58	57	leidd 11730	. . . . 5 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
59	58	ad4ant13 749	. . . 4 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → (abs‘(-∞ − 𝐴)) ≤ (abs‘(-∞ − 𝐴)))
60	50, 3, 51, 52, 53, 56, 59	climxrrelem 44110	. . 3 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
61		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ)
62		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
63		nfra1 3265	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ
64	62, 63	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑘(𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
65	61, 64	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑘(((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ))
66		simp-4l 781	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝜑)
67	3	uztrn2 12791	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
68	67	adantlr 713	. . . . . . . . 9 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
69	68	adantll 712	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
70		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
71	4	fdmd 6684	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑍)
72	71	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom 𝐹 = 𝑍)
73	70, 72	eleqtrrd 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ dom 𝐹)
74	66, 69, 73	syl2anc 584	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
75	4	ffvelcdmda 7040	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ*)
76	66, 69, 75	syl2anc 584	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ*)
77		rspa 3229	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
78	77	adantll 712	. . . . . . . . . 10 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
79	78	adantll 712	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
80		simpllr 774	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ¬ -∞ ∈ ℂ)
81		nelne2 3039	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹‘𝑘) ≠ -∞)
82	79, 80, 81	syl2anc 584	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ≠ -∞)
83		simp-4r 782	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ¬ +∞ ∈ ℂ)
84		nelne2 3039	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ ¬ +∞ ∈ ℂ) → (𝐹‘𝑘) ≠ +∞)
85	79, 83, 84	syl2anc 584	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ≠ +∞)
86	76, 82, 85	xrred 43720	. . . . . . 7 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
87	74, 86	jca 512	. . . . . 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 3239	. . . . 5 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
89	4	ffund 6677	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
90		ffvresb 7077	. . . . . . 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 728	. . . . 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 3110	. . . . . . . . 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 498	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
96	95	ad2antll 727	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
97		breq2 5114	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))
98	97	anbi2d 629	. . . . . . . . 9 ⊢ (𝑥 = 1 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1)))
99	98	rexralbidv 3210	. . . . . . . 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 6861	. . . . . . . . . . . . 13 ⊢ 𝑍 ∈ V
101	100	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍 ∈ V)
102	4, 101	fexd 7182	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
103		eqidd 2732	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘))
104	102, 103	clim 15388	. . . . . . . . . 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 496	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
107		1rp 12928	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
108	107	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ+)
109	99, 106, 108	rspcdva 3583	. . . . . . 7 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 1))
110	96, 109	reximddv 3164	. . . . . 6 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
111	3	rexuz3 15245	. . . . . . 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 724	. . . 4 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
115	93, 114	reximddv 3164	. . 3 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ℂ) ∧ ¬ -∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
116	60, 115	pm2.61dan 811	. 2 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ℂ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)
117	49, 116	pm2.61dan 811	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3446  ifcif 4491   class class class wbr 5110  dom cdm 5638   ↾ cres 5640  Fun wfun 6495  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362  ℂcc 11058  ℝcr 11059  1c1 11061  +∞cpnf 11195  -∞cmnf 11196  ℝ^*cxr 11197   < clt 11198   ≤ cle 11199   − cmin 11394  ℤcz 12508  ℤ_≥cuz 12772  ℝ⁺crp 12924  abscabs 15131   ⇝ cli 15378

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9387  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-nn 12163  df-2 12225  df-3 12226  df-n0 12423  df-z 12509  df-uz 12773  df-rp 12925  df-seq 13917  df-exp 13978  df-cj 14996  df-re 14997  df-im 14998  df-sqrt 15132  df-abs 15133  df-clim 15382

This theorem is referenced by:  xlimclim2  44201