climxlim2lem - Mathbox for Glauco Siliprandi

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Theorem climxlim2lem 46777

Description: In this lemma for climxlim2 46778 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 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹 ⇝ 𝐴)
3		climxlim2lem.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑀 ∈ ℤ)
5		climxlim2lem.2	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6		climxlim2lem.3	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
8		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
9	4, 5, 7, 8	xlimclim2 46772	. . 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 7072	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
13	12	anim1i 627	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴))
14	13	adantllr 732	. . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) → 𝐹:𝑍⟶ℝ^*)
16	15	ffvelcdmda 7072	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
17		simplr 781	. . . . . . . . . . 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 2848	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ∈ ℂ ↔ (𝐹‘𝑘) ∈ ℂ))
19		neeq1 3017	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ≠ 𝐴 ↔ (𝐹‘𝑘) ≠ 𝐴))
20	18, 19	anbi12d 644	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴)))
21		fvoveq1 7431	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (abs‘(𝑦 − 𝐴)) = (abs‘((𝐹‘𝑘) − 𝐴)))
22	21	breq2d 5114	. . . . . . . . . . . . 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 3574	. . . . . . . . . . 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 596	. . . . . . . . . 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 486	. . . . . . . . 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 418	. . . . . . 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 3154	. . . . . 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 765	. . . . 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 15634	. . . . . . . 8 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
32	1, 31	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
33	32	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 ∈ ℂ)
34		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ¬ 𝐴 ∈ ℝ)
35		prfi 9293	. . . . . . 7 ⊢ {+∞, -∞} ∈ Fin
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → {+∞, -∞} ∈ Fin)
37		df-xr 11319	. . . . . 6 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
38	33, 34, 36, 37	cnrefiisp 46762	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))
39	30, 38	reximddv3 3179	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
40		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ⁺)
41		nfra1 3286	. . . . . . . . . 10 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
42	40, 41	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
43		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
44	42, 43	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍)
45		nfra1 3286	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥
46	44, 45	nfan 1932	. . . . . . 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 779	. . . . . . . . . . . 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 12954	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
49	48	adantll 727	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
50		rspa 3251	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 2963	. . . . . . . . . . 11 ⊢ (¬ (𝐹‘𝑘) = 𝐴 → (𝐹‘𝑘) ≠ 𝐴)
53	51, 52	impel 515	. . . . . . . . . 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 1397	. . . . . . . . 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 732	. . . . . . . 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 3251	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
57	56	adantll 727	. . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℂ)
59	48	adantll 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
60	58, 59	ffvelcdmd 7073	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
61	60	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
62	32	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐴 ∈ ℂ)
63	61, 62	subcld 11641	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) − 𝐴) ∈ ℂ)
64	63	abscld 15574	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
65	64	adantl3r 763	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
66		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
67	66	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ⁺)
68	67	rpred 13134	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ)
69	65, 68	ltnled 11429	. . . . . . . . . . 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 763	. . . . . . . . 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 486	. . . . . . . 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 830	. . . . . . 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 3261	. . . . . 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 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝐹
76	75, 3, 5, 11	climuz 46676	. . . . . . . . . 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 501	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
79	78	r19.21bi 3254	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
80	79	adantr 486	. . . . . 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 3179	. . . . 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 732	. . . 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 46755	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
84		nfv 1947	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍)
85		nfra1 3286	. . . . 5 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴
86	84, 85	nfan 1932	. . . 4 ⊢ Ⅎ𝑘(((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
87	6	ad3antrrr 743	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐹:𝑍⟶ℝ^*)
88		simplr 781	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝑗 ∈ 𝑍)
89	5	uzid3 46367	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ_≥‘𝑗))
90		fveq2 6873	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
91	90	eqeq1d 2762	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘𝑗) = 𝐴))
92	91	rspcva 3574	. . . . . . . 8 ⊢ ((𝑗 ∈ (ℤ_≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
93	89, 92	sylan 592	. . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
94	93	3adant1 1148	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
95	6	ffvelcdmda 7072	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ^*)
96	95	3adant3 1150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) ∈ ℝ^*)
97	94, 96	eqeltrrd 2861	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐴 ∈ ℝ^*)
98	97	ad4ant134 1193	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐴 ∈ ℝ^*)
99		rspa 3251	. . . . 5 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = 𝐴)
100	99	adantll 727	. . . 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 46767	. . 3 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐹~~>*𝐴)
102	83, 101	rexlimddv2 46755	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → 𝐹~~>*𝐴)
103	10, 102	pm2.61dan 825	1 ⊢ (𝜑 → 𝐹~~>*𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 {cpr 4585 class class class wbr 5102 ⟶wf 6523 ‘cfv 6527 (class class class)co 7408 Fincfn 8951 ℂcc 11170 ℝcr 11171 +∞cpnf 11312 -∞cmnf 11313 ℝ^*cxr 11314 < clt 11315 ≤ cle 11316 − cmin 11513 ℤcz 12663 ℤ_≥cuz 12935 ℝ⁺crp 13090 abscabs 15369 ⇝ cli 15619 ~~>*clsxlim 46750

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249 ax-pre-sup 11250

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-pm 8828 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-fi 9381 df-sup 9412 df-inf 9413 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-div 11944 df-nn 12306 df-2 12375 df-3 12376 df-4 12377 df-5 12378 df-6 12379 df-7 12380 df-8 12381 df-9 12382 df-n0 12577 df-z 12664 df-dec 12785 df-uz 12936 df-q 13046 df-rp 13091 df-xneg 13211 df-xadd 13212 df-xmul 13213 df-ioo 13450 df-ioc 13451 df-ico 13452 df-icc 13453 df-fz 13610 df-fl 13901 df-seq 14114 df-exp 14174 df-cj 15234 df-re 15235 df-im 15236 df-sqrt 15370 df-abs 15371 df-clim 15623 df-rlim 15624 df-struct 17287 df-slot 17322 df-ndx 17334 df-base 17350 df-plusg 17403 df-mulr 17404 df-starv 17405 df-tset 17409 df-ple 17410 df-ds 17412 df-unif 17413 df-rest 17555 df-topn 17556 df-topgen 17576 df-ordt 17635 df-ps 18702 df-tsr 18703 df-psmet 21632 df-xmet 21633 df-met 21634 df-bl 21635 df-mopn 21636 df-cnfld 21641 df-top 23174 df-topon 23191 df-topsp 23213 df-bases 23226 df-lm 23509 df-xms 24601 df-ms 24602 df-xlim 46751

This theorem is used by: climxlim2 46778

W3C validator