climxlim2lem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > climxlim2lem		Structured version Visualization version GIF version

Theorem climxlim2lem 46417

Description: In this lemma for climxlim2 46418 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 485	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹 ⇝ 𝐴)
3		climxlim2lem.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑀 ∈ ℤ)
5		climxlim2lem.2	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6		climxlim2lem.3	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
7	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
8		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
9	4, 5, 7, 8	xlimclim2 46412	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴))
10	2, 9	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹~~>*𝐴)
11		climxlim2lem.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
12	11	ffvelcdmda 7069	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
13	12	anim1i 626	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴))
14	13	adantllr 731	. . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) → 𝐹:𝑍⟶ℝ^*)
16	15	ffvelcdmda 7069	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
17		simplr 780	. . . . . . . . . . 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 2853	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ∈ ℂ ↔ (𝐹‘𝑘) ∈ ℂ))
19		neeq1 3022	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ≠ 𝐴 ↔ (𝐹‘𝑘) ≠ 𝐴))
20	18, 19	anbi12d 643	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴)))
21		fvoveq1 7423	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (abs‘(𝑦 − 𝐴)) = (abs‘((𝐹‘𝑘) − 𝐴)))
22	21	breq2d 5117	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑥 ≤ (abs‘(𝑦 − 𝐴)) ↔ 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
23	20, 22	imbi12d 347	. . . . . . . . . . . 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 3582	. . . . . . . . . . 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 595	. . . . . . . . . 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 485	. . . . . . . . 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 16	. . . . . . . 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 417	. . . . . . 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 3157	. . . . . 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 764	. . . . 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 15540	. . . . . . . 8 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
32	1, 31	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
33	32	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 ∈ ℂ)
34		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ¬ 𝐴 ∈ ℝ)
35		prfi 9271	. . . . . . 7 ⊢ {+∞, -∞} ∈ Fin
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → {+∞, -∞} ∈ Fin)
37		df-xr 11235	. . . . . 6 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
38	33, 34, 36, 37	cnrefiisp 46402	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))
39	30, 38	reximddv3 3182	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
40		nfv 1937	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ⁺)
41		nfra1 3289	. . . . . . . . . 10 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
42	40, 41	nfan 1922	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
43		nfv 1937	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
44	42, 43	nfan 1922	. . . . . . . 8 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍)
45		nfra1 3289	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥
46	44, 45	nfan 1922	. . . . . . 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 778	. . . . . . . . . . . 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 12872	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
49	48	adantll 726	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
50		rspa 3254	. . . . . . . . . . . 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 595	. . . . . . . . . . 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 2968	. . . . . . . . . . 11 ⊢ (¬ (𝐹‘𝑘) = 𝐴 → (𝐹‘𝑘) ≠ 𝐴)
53	51, 52	impel 514	. . . . . . . . . 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 1393	. . . . . . . . 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 731	. . . . . . . 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 3254	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
57	56	adantll 726	. . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℂ)
59	48	adantll 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
60	58, 59	ffvelcdmd 7070	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
61	60	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
62	32	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐴 ∈ ℂ)
63	61, 62	subcld 11557	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) − 𝐴) ∈ ℂ)
64	63	abscld 15480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
65	64	adantl3r 762	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
66		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
67	66	ad3antrrr 742	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ⁺)
68	67	rpred 13051	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ)
69	65, 68	ltnled 11345	. . . . . . . . . . 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 235	. . . . . . . . . 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 762	. . . . . . . . 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 485	. . . . . . . 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 829	. . . . . . 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 3264	. . . . . 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 2927	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝐹
76	75, 3, 5, 11	climuz 46316	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
77	1, 76	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
78	77	simprd 500	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
79	78	r19.21bi 3257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
80	79	adantr 485	. . . . . 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 3182	. . . . 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 731	. . . 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 46395	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
84		nfv 1937	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍)
85		nfra1 3289	. . . . 5 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴
86	84, 85	nfan 1922	. . . 4 ⊢ Ⅎ𝑘(((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
87	6	ad3antrrr 742	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐹:𝑍⟶ℝ^*)
88		simplr 780	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝑗 ∈ 𝑍)
89	5	uzid3 46007	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ_≥‘𝑗))
90		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
91	90	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘𝑗) = 𝐴))
92	91	rspcva 3582	. . . . . . . 8 ⊢ ((𝑗 ∈ (ℤ_≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
93	89, 92	sylan 591	. . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
94	93	3adant1 1146	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
95	6	ffvelcdmda 7069	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ^*)
96	95	3adant3 1148	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) ∈ ℝ^*)
97	94, 96	eqeltrrd 2866	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐴 ∈ ℝ^*)
98	97	ad4ant134 1191	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐴 ∈ ℝ^*)
99		rspa 3254	. . . . 5 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = 𝐴)
100	99	adantll 726	. . . 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 46407	. . 3 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐹~~>*𝐴)
102	83, 101	rexlimddv2 46395	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → 𝐹~~>*𝐴)
103	10, 102	pm2.61dan 824	1 ⊢ (𝜑 → 𝐹~~>*𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∀wral 3079 ∃wrex 3089 {cpr 4587 class class class wbr 5105 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 Fincfn 8931 ℂcc 11086 ℝcr 11087 +∞cpnf 11228 -∞cmnf 11229 ℝ^*cxr 11230 < clt 11231 ≤ cle 11232 − cmin 11429 ℤcz 12582 ℤ_≥cuz 12853 ℝ⁺crp 13007 abscabs 15275 ⇝ cli 15525 ~~>*clsxlim 46390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fi 9359 df-sup 9390 df-inf 9391 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fl 13816 df-seq 14029 df-exp 14089 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-clim 15529 df-rlim 15530 df-struct 17197 df-slot 17232 df-ndx 17244 df-base 17260 df-plusg 17313 df-mulr 17314 df-starv 17315 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-rest 17465 df-topn 17466 df-topgen 17486 df-ordt 17545 df-ps 18612 df-tsr 18613 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-lm 23347 df-xms 24438 df-ms 24439 df-xlim 46391

This theorem is referenced by: climxlim2 46418

W3C validator