climxlim2lem - Mathbox for Glauco Siliprandi

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Theorem climxlim2lem 46289

Description: In this lemma for climxlim2 46290 there is the additional assumption that the converging function is complex-valued on the whole domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
climxlim2lem.1	⊢ (𝜑 → 𝑀 ∈ ℤ)
climxlim2lem.2	⊢ 𝑍 = (ℤ_≥‘𝑀)
climxlim2lem.3	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
climxlim2lem.4	⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
climxlim2lem.5	⊢ (𝜑 → 𝐹 ⇝ 𝐴)

**Assertion**
Ref	Expression
climxlim2lem	⊢ (𝜑 → 𝐹~~>*𝐴)

Proof of Theorem climxlim2lem Dummy variables 𝑗 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climxlim2lem.5	. . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
2	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹 ⇝ 𝐴)
3		climxlim2lem.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑀 ∈ ℤ)
5		climxlim2lem.2	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6		climxlim2lem.3	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
8		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
9	4, 5, 7, 8	xlimclim2 46284	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴))
10	2, 9	mpbird 258	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹~~>*𝐴)
11		climxlim2lem.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
12	11	ffvelcdmda 7032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
13	12	anim1i 621	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴))
14	13	adantllr 725	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴))
15	6	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) → 𝐹:𝑍⟶ℝ^*)
16	15	ffvelcdmda 7032	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
17		simplr 774	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) → ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))
18		eleq1 2828	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ∈ ℂ ↔ (𝐹‘𝑘) ∈ ℂ))
19		neeq1 2997	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ≠ 𝐴 ↔ (𝐹‘𝑘) ≠ 𝐴))
20	18, 19	anbi12d 638	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴)))
21		fvoveq1 7386	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (abs‘(𝑦 − 𝐴)) = (abs‘((𝐹‘𝑘) − 𝐴)))
22	21	breq2d 5091	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑥 ≤ (abs‘(𝑦 − 𝐴)) ↔ 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
23	20, 22	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑘) → (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))) ↔ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))))
24	23	rspcva 3565	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘) ∈ ℝ^* ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) → (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
25	16, 17, 24	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
26	25	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
27	14, 26	mpd 15	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
28	27	ex 413	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
29	28	ralrimiva 3132	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) → ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
30	29	ad4ant14 758	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) → ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
31		climcl 15459	. . . . . . . 8 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
32	1, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
33	32	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 ∈ ℂ)
34		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ¬ 𝐴 ∈ ℝ)
35		prfi 9231	. . . . . . 7 ⊢ {+∞, -∞} ∈ Fin
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → {+∞, -∞} ∈ Fin)
37		df-xr 11181	. . . . . 6 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
38	33, 34, 36, 37	cnrefiisp 46274	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))
39	30, 38	reximddv3 3157	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
40		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ⁺)
41		nfra1 3264	. . . . . . . . . 10 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
42	40, 41	nfan 1906	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
43		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
44	42, 43	nfan 1906	. . . . . . . 8 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍)
45		nfra1 3264	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥
46	44, 45	nfan 1906	. . . . . . 7 ⊢ Ⅎ𝑘((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
47		simpll 772	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
48	5	uztrn2 12805	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
49	48	adantll 720	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
50		rspa 3229	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
51	47, 49, 50	syl2anc 590	. . . . . . . . . . 11 ⊢ (((∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
52		neqne 2943	. . . . . . . . . . 11 ⊢ (¬ (𝐹‘𝑘) = 𝐴 → (𝐹‘𝑘) ≠ 𝐴)
53	51, 52	impel 510	. . . . . . . . . 10 ⊢ ((((∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ¬ (𝐹‘𝑘) = 𝐴) → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
54	53	ad5ant2345 1378	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ¬ (𝐹‘𝑘) = 𝐴) → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
55	54	adantllr 725	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ¬ (𝐹‘𝑘) = 𝐴) → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
56		rspa 3229	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
57	56	adantll 720	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
58	11	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℂ)
59	48	adantll 720	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
60	58, 59	ffvelcdmd 7033	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
61	60	adantlr 721	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
62	32	ad3antrrr 736	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐴 ∈ ℂ)
63	61, 62	subcld 11503	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) − 𝐴) ∈ ℂ)
64	63	abscld 15399	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
65	64	adantl3r 756	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
66		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
67	66	ad3antrrr 736	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ⁺)
68	67	rpred 12984	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ)
69	65, 68	ltnled 11291	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ ¬ 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
70	57, 69	mpbid 233	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ¬ 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
71	70	adantl3r 756	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ¬ 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
72	71	adantr 481	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ¬ (𝐹‘𝑘) = 𝐴) → ¬ 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
73	55, 72	condan 823	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = 𝐴)
74	46, 73	ralrimia 3239	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
75		nfcv 2902	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝐹
76	75, 3, 5, 11	climuz 46188	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
77	1, 76	mpbid 233	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
78	77	simprd 496	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
79	78	r19.21bi 3232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
80	79	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
81	74, 80	reximddv3 3157	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
82	81	adantllr 725	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
83	39, 82	rexlimddv2 46267	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
84		nfv 1921	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍)
85		nfra1 3264	. . . . 5 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴
86	84, 85	nfan 1906	. . . 4 ⊢ Ⅎ𝑘(((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
87	6	ad3antrrr 736	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐹:𝑍⟶ℝ^*)
88		simplr 774	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝑗 ∈ 𝑍)
89	5	uzid3 45879	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ_≥‘𝑗))
90		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
91	90	eqeq1d 2742	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘𝑗) = 𝐴))
92	91	rspcva 3565	. . . . . . . 8 ⊢ ((𝑗 ∈ (ℤ_≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
93	89, 92	sylan 586	. . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
94	93	3adant1 1136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
95	6	ffvelcdmda 7032	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ^*)
96	95	3adant3 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) ∈ ℝ^*)
97	94, 96	eqeltrrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐴 ∈ ℝ^*)
98	97	ad4ant134 1181	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐴 ∈ ℝ^*)
99		rspa 3229	. . . . 5 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = 𝐴)
100	99	adantll 720	. . . 4 ⊢ (((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = 𝐴)
101	86, 75, 5, 87, 88, 98, 100	xlimconst2 46279	. . 3 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐹~~>*𝐴)
102	83, 101	rexlimddv2 46267	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → 𝐹~~>*𝐴)
103	10, 102	pm2.61dan 818	1 ⊢ (𝜑 → 𝐹~~>*𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∀wral 3054 ∃wrex 3064 {cpr 4564 class class class wbr 5079 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 Fincfn 8890 ℂcc 11034 ℝcr 11035 +∞cpnf 11174 -∞cmnf 11175 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 − cmin 11375 ℤcz 12522 ℤ_≥cuz 12786 ℝ⁺crp 12940 abscabs 15194 ⇝ cli 15444 ~~>*clsxlim 46262

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9321 df-sup 9352 df-inf 9353 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ioc 13301 df-ico 13302 df-icc 13303 df-fz 13460 df-fl 13749 df-seq 13962 df-exp 14022 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-clim 15448 df-rlim 15449 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17178 df-plusg 17231 df-mulr 17232 df-starv 17233 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-rest 17383 df-topn 17384 df-topgen 17404 df-ordt 17463 df-ps 18530 df-tsr 18531 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-lm 23219 df-xms 24310 df-ms 24311 df-xlim 46263

This theorem is referenced by: climxlim2 46290

W3C validator