climxlim2lem - Mathbox for Glauco Siliprandi

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Theorem climxlim2lem 43276

Description: In this lemma for climxlim2 43277 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 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹 ⇝ 𝐴)
3		climxlim2lem.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑀 ∈ ℤ)
5		climxlim2lem.2	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6		climxlim2lem.3	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
8		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
9	4, 5, 7, 8	xlimclim2 43271	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴))
10	2, 9	mpbird 256	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹~~>*𝐴)
11		climxlim2lem.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
12	11	ffvelrnda 6943	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
13	12	anim1i 614	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴))
14	13	adantllr 715	. . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) → 𝐹:𝑍⟶ℝ^*)
16	15	ffvelrnda 6943	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
17		simplr 765	. . . . . . . . . . 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 2826	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ∈ ℂ ↔ (𝐹‘𝑘) ∈ ℂ))
19		neeq1 3005	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ≠ 𝐴 ↔ (𝐹‘𝑘) ≠ 𝐴))
20	18, 19	anbi12d 630	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴)))
21		fvoveq1 7278	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (abs‘(𝑦 − 𝐴)) = (abs‘((𝐹‘𝑘) − 𝐴)))
22	21	breq2d 5082	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑥 ≤ (abs‘(𝑦 − 𝐴)) ↔ 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
23	20, 22	imbi12d 344	. . . . . . . . . . . 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 3550	. . . . . . . . . . 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 583	. . . . . . . . . 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 480	. . . . . . . . 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 412	. . . . . . 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 3107	. . . . . 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 748	. . . . 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 15136	. . . . . . . 8 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
32	1, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
33	32	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 ∈ ℂ)
34		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ¬ 𝐴 ∈ ℝ)
35		prfi 9019	. . . . . . 7 ⊢ {+∞, -∞} ∈ Fin
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → {+∞, -∞} ∈ Fin)
37		df-xr 10944	. . . . . 6 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
38	33, 34, 36, 37	cnrefiisp 43261	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))
39	30, 38	reximddv3 42589	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
40		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ⁺)
41		nfra1 3142	. . . . . . . . . 10 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
42	40, 41	nfan 1903	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
43		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
44	42, 43	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍)
45		nfra1 3142	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥
46	44, 45	nfan 1903	. . . . . . 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 763	. . . . . . . . . . . 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 12530	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
49	48	adantll 710	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
50		rspa 3130	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 2950	. . . . . . . . . . 11 ⊢ (¬ (𝐹‘𝑘) = 𝐴 → (𝐹‘𝑘) ≠ 𝐴)
53	51, 52	impel 505	. . . . . . . . . 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 1368	. . . . . . . . 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 715	. . . . . . . 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 3130	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
57	56	adantll 710	. . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℂ)
59	48	adantll 710	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
60	58, 59	ffvelrnd 6944	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
61	60	adantlr 711	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
62	32	ad3antrrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐴 ∈ ℂ)
63	61, 62	subcld 11262	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) − 𝐴) ∈ ℂ)
64	63	abscld 15076	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
65	64	adantl3r 746	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
66		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
67	66	ad3antrrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ⁺)
68	67	rpred 12701	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ)
69	65, 68	ltnled 11052	. . . . . . . . . . 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 231	. . . . . . . . . 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 746	. . . . . . . . 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 480	. . . . . . . 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 814	. . . . . . 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 3420	. . . . . 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 2906	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝐹
76	75, 3, 5, 11	climuz 43175	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
77	1, 76	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
78	77	simprd 495	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
79	78	r19.21bi 3132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
80	79	adantr 480	. . . . . 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 42589	. . . . 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 715	. . . 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 43254	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
84		nfv 1918	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍)
85		nfra1 3142	. . . . 5 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴
86	84, 85	nfan 1903	. . . 4 ⊢ Ⅎ𝑘(((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
87	6	ad3antrrr 726	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐹:𝑍⟶ℝ^*)
88		simplr 765	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝑗 ∈ 𝑍)
89	5	uzid3 42865	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ_≥‘𝑗))
90		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
91	90	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘𝑗) = 𝐴))
92	91	rspcva 3550	. . . . . . . 8 ⊢ ((𝑗 ∈ (ℤ_≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
93	89, 92	sylan 579	. . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
94	93	3adant1 1128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
95	6	ffvelrnda 6943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ^*)
96	95	3adant3 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) ∈ ℝ^*)
97	94, 96	eqeltrrd 2840	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐴 ∈ ℝ^*)
98	97	ad4ant134 1172	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐴 ∈ ℝ^*)
99		rspa 3130	. . . . 5 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = 𝐴)
100	99	adantll 710	. . . 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 43266	. . 3 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐹~~>*𝐴)
102	83, 101	rexlimddv2 43254	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → 𝐹~~>*𝐴)
103	10, 102	pm2.61dan 809	1 ⊢ (𝜑 → 𝐹~~>*𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {cpr 4560 class class class wbr 5070 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℂcc 10800 ℝcr 10801 +∞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 ~~>*clsxlim 43249

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-int 4877 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-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 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-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fl 13440 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-topn 17051 df-topgen 17071 df-ordt 17129 df-ps 18199 df-tsr 18200 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-lm 22288 df-xms 23381 df-ms 23382 df-xlim 43250

This theorem is referenced by: climxlim2 43277

W3C validator