climxlim2lem - Mathbox for Glauco Siliprandi

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Theorem climxlim2lem 42119

Description: In this lemma for climxlim2 42120 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 483	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹 ⇝ 𝐴)
3		climxlim2lem.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑀 ∈ ℤ)
5		climxlim2lem.2	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6		climxlim2lem.3	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
7	6	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
8		simpr 487	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
9	4, 5, 7, 8	xlimclim2 42114	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴))
10	2, 9	mpbird 259	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹~~>*𝐴)
11		climxlim2lem.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
12	11	ffvelrnda 6845	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
13	12	anim1i 616	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴))
14	13	adantllr 717	. . . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) → 𝐹:𝑍⟶ℝ^*)
16	15	ffvelrnda 6845	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
17		simplr 767	. . . . . . . . . . 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 2900	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ∈ ℂ ↔ (𝐹‘𝑘) ∈ ℂ))
19		neeq1 3078	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ≠ 𝐴 ↔ (𝐹‘𝑘) ≠ 𝐴))
20	18, 19	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) ≠ 𝐴)))
21		fvoveq1 7173	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑘) → (abs‘(𝑦 − 𝐴)) = (abs‘((𝐹‘𝑘) − 𝐴)))
22	21	breq2d 5070	. . . . . . . . . . . . 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 3620	. . . . . . . . . . 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 586	. . . . . . . . . 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 483	. . . . . . . . 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 415	. . . . . . 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 3182	. . . . . 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 750	. . . . 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 14850	. . . . . . . 8 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
32	1, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
33	32	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 ∈ ℂ)
34		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ¬ 𝐴 ∈ ℝ)
35		prfi 8787	. . . . . . 7 ⊢ {+∞, -∞} ∈ Fin
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → {+∞, -∞} ∈ Fin)
37		df-xr 10673	. . . . . 6 ⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
38	33, 34, 36, 37	cnrefiisp 42104	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ ℝ^* ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴))))
39	30, 38	reximddv3 41413	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
40		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ⁺)
41		nfra1 3219	. . . . . . . . . 10 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
42	40, 41	nfan 1896	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))))
43		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
44	42, 43	nfan 1896	. . . . . . . 8 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))) ∧ 𝑗 ∈ 𝑍)
45		nfra1 3219	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥
46	44, 45	nfan 1896	. . . . . . 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 765	. . . . . . . . . . . 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 12256	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
49	48	adantll 712	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ≠ 𝐴 → 𝑥 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
50		rspa 3206	. . . . . . . . . . . 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 586	. . . . . . . . . . 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 3024	. . . . . . . . . . 11 ⊢ (¬ (𝐹‘𝑘) = 𝐴 → (𝐹‘𝑘) ≠ 𝐴)
53	51, 52	impel 508	. . . . . . . . . 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 1366	. . . . . . . . 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 717	. . . . . . . 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 3206	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
57	56	adantll 712	. . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℂ)
59	48	adantll 712	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
60	58, 59	ffvelrnd 6846	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
61	60	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
62	32	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐴 ∈ ℂ)
63	61, 62	subcld 10991	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) − 𝐴) ∈ ℂ)
64	63	abscld 14790	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
65	64	adantl3r 748	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
66		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
67	66	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ⁺)
68	67	rpred 12425	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ)
69	65, 68	ltnled 10781	. . . . . . . . . . 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 234	. . . . . . . . . 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 748	. . . . . . . . 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 483	. . . . . . . 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 816	. . . . . . 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 41391	. . . . . 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 2977	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝐹
76	75, 3, 5, 11	climuz 42018	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
77	1, 76	mpbid 234	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
78	77	simprd 498	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
79	78	r19.21bi 3208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)
80	79	adantr 483	. . . . . 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 41413	. . . . 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 717	. . . 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 42097	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
84		nfv 1911	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍)
85		nfra1 3219	. . . . 5 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴
86	84, 85	nfan 1896	. . . 4 ⊢ Ⅎ𝑘(((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
87	6	ad3antrrr 728	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐹:𝑍⟶ℝ^*)
88		simplr 767	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝑗 ∈ 𝑍)
89	5	uzid3 41702	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ_≥‘𝑗))
90		fveq2 6664	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
91	90	eqeq1d 2823	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘𝑗) = 𝐴))
92	91	rspcva 3620	. . . . . . . 8 ⊢ ((𝑗 ∈ (ℤ_≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
93	89, 92	sylan 582	. . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
94	93	3adant1 1126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) = 𝐴)
95	6	ffvelrnda 6845	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ^*)
96	95	3adant3 1128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → (𝐹‘𝑗) ∈ ℝ^*)
97	94, 96	eqeltrrd 2914	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐴 ∈ ℝ^*)
98	97	ad4ant134 1170	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐴 ∈ ℝ^*)
99		rspa 3206	. . . . 5 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = 𝐴)
100	99	adantll 712	. . . 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 42109	. . 3 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴) → 𝐹~~>*𝐴)
102	83, 101	rexlimddv2 42097	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ) → 𝐹~~>*𝐴)
103	10, 102	pm2.61dan 811	1 ⊢ (𝜑 → 𝐹~~>*𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {cpr 4562 class class class wbr 5058 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 ℂcc 10529 ℝcr 10530 +∞cpnf 10666 -∞cmnf 10667 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 − cmin 10864 ℤcz 11975 ℤ_≥cuz 12237 ℝ⁺crp 12383 abscabs 14587 ⇝ cli 14835 ~~>*clsxlim 42092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fl 13156 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-rest 16690 df-topn 16691 df-topgen 16711 df-ordt 16768 df-ps 17804 df-tsr 17805 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-lm 21831 df-xms 22924 df-ms 22925 df-xlim 42093

This theorem is referenced by: climxlim2 42120

W3C validator