climsuse - Mathbox for Glauco Siliprandi

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Theorem climsuse 46294

Description: A subsequence 𝐺 of a converging sequence 𝐹, converges to the same limit. 𝐼 is the strictly increasing and it is used to index the subsequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

**Hypotheses**
Ref	Expression
climsuse.1	⊢ Ⅎ𝑘𝜑
climsuse.3	⊢ Ⅎ𝑘𝐹
climsuse.2	⊢ Ⅎ𝑘𝐺
climsuse.4	⊢ Ⅎ𝑘𝐼
climsuse.5	⊢ 𝑍 = (ℤ_≥‘𝑀)
climsuse.6	⊢ (𝜑 → 𝑀 ∈ ℤ)
climsuse.7	⊢ (𝜑 → 𝐹 ∈ 𝑋)
climsuse.8	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
climsuse.9	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climsuse.10	⊢ (𝜑 → (𝐼‘𝑀) ∈ 𝑍)
climsuse.11	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑘) + 1)))
climsuse.12	⊢ (𝜑 → 𝐺 ∈ 𝑌)
climsuse.13	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘)))

**Assertion**
Ref	Expression
climsuse	⊢ (𝜑 → 𝐺 ⇝ 𝐴)

Distinct variable group: 𝑘,𝑍 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑘) 𝐹(𝑘) 𝐺(𝑘) 𝐼(𝑘) 𝑀(𝑘) 𝑋(𝑘) 𝑌(𝑘)

Proof of Theorem climsuse Dummy variables ℎ 𝑖 𝑗 𝑥 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climsuse.9	. . 3 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
2		climcl 15550	. . 3 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
4		nfv 1942	. . 3 ⊢ Ⅎ𝑥𝜑
5		simpllr 787	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ ℤ)
6		climsuse.6	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7	6	ad4antr 744	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ ¬ 𝑀 ≤ 𝑗) → 𝑀 ∈ ℤ)
8	5, 7	ifclda 4522	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ∈ ℤ)
9		nfv 1942	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ)
10		nfra1 3287	. . . . . . . 8 ⊢ Ⅎ𝑖∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)
11	9, 10	nfan 1927	. . . . . . 7 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))
12		simp-4l 794	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝜑)
13		simpllr 787	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ∈ ℤ)
14	12, 13	jca 520	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝜑 ∧ 𝑗 ∈ ℤ))
15		simpr 489	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)))
16		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → 𝑀 ≤ 𝑗)
17	6	anim1i 626	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ))
18	17	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ))
19		eluz 12875	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑗))
20	18, 19	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → (𝑗 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑗))
21	16, 20	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ (ℤ_≥‘𝑀))
22		simpll 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ ¬ 𝑀 ≤ 𝑗) → 𝜑)
23		uzid 12876	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
24	22, 6, 23	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ ¬ 𝑀 ≤ 𝑗) → 𝑀 ∈ (ℤ_≥‘𝑀))
25	21, 24	ifclda 4522	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ∈ (ℤ_≥‘𝑀))
26		uzss 12884	. . . . . . . . . . . . . 14 ⊢ (if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) ⊆ (ℤ_≥‘𝑀))
27	25, 26	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) ⊆ (ℤ_≥‘𝑀))
28		climsuse.5	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ_≥‘𝑀)
29	27, 28	sseqtrrdi 3977	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) ⊆ 𝑍)
30	29	sseld 3935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) → 𝑖 ∈ 𝑍))
31	14, 15, 30	sylc 66	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ 𝑍)
32		climsuse.1	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘𝜑
33		nfv 1942	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 ∈ 𝑍
34	32, 33	nfan 1927	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑖 ∈ 𝑍)
35		climsuse.2	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐺
36		nfcv 2923	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
37	35, 36	nffv 6891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐺‘𝑖)
38		climsuse.3	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐹
39		climsuse.4	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝐼
40	39, 36	nffv 6891	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐼‘𝑖)
41	38, 40	nffv 6891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘(𝐼‘𝑖))
42	37, 41	nfeq 2936	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖))
43	34, 42	nfim 1924	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖)))
44		eleq1 2849	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝑘 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
45	44	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
46		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝐺‘𝑘) = (𝐺‘𝑖))
47		2fveq3 6886	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝐹‘(𝐼‘𝑘)) = (𝐹‘(𝐼‘𝑖)))
48	46, 47	eqeq12d 2777	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → ((𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘)) ↔ (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖))))
49	45, 48	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘))) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖)))))
50		climsuse.13	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘)))
51	43, 49, 50	chvarfv 2274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖)))
52	28	eleq2i 2853	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑍 ↔ 𝑖 ∈ (ℤ_≥‘𝑀))
53	52	bilani 509	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ (ℤ_≥‘𝑀))
54		uzss 12884	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
55	53, 54	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
56		climsuse.10	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼‘𝑀) ∈ 𝑍)
57		nfcv 2923	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(𝑖 + 1)
58	39, 57	nffv 6891	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝐼‘(𝑖 + 1))
59		nfcv 2923	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘ℤ_≥
60		nfcv 2923	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘 +
61		nfcv 2923	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘1
62	40, 60, 61	nfov 7440	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝐼‘𝑖) + 1)
63	59, 62	nffv 6891	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(ℤ_≥‘((𝐼‘𝑖) + 1))
64	58, 63	nfel 2937	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1))
65	34, 64	nfim 1924	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1)))
66		fvoveq1 7433	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (𝐼‘(𝑘 + 1)) = (𝐼‘(𝑖 + 1)))
67		fveq2 6881	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝐼‘𝑘) = (𝐼‘𝑖))
68	67	fvoveq1d 7432	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (ℤ_≥‘((𝐼‘𝑘) + 1)) = (ℤ_≥‘((𝐼‘𝑖) + 1)))
69	66, 68	eleq12d 2855	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → ((𝐼‘(𝑘 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑘) + 1)) ↔ (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1))))
70	45, 69	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑖 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑘) + 1))) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1)))))
71		climsuse.11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑘) + 1)))
72	65, 70, 71	chvarfv 2274	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1)))
73	28, 6, 56, 72	climsuselem1 46293	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘𝑖) ∈ (ℤ_≥‘𝑖))
74	55, 73	sseldd 3937	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘𝑖) ∈ (ℤ_≥‘𝑀))
75	74, 28	eleqtrrdi 2872	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘𝑖) ∈ 𝑍)
76	75	ex 417	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑖 ∈ 𝑍 → (𝐼‘𝑖) ∈ 𝑍))
77	76	imdistani 578	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍))
78	33	nfci 2911	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝑍
79	40, 78	nfel 2937	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐼‘𝑖) ∈ 𝑍
80	32, 79	nfan 1927	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍)
81	41	nfel1 2939	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘(𝐼‘𝑖)) ∈ ℂ
82	80, 81	nfim 1924	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ)
83		eleq1 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝐼‘𝑖) → (𝑘 ∈ 𝑍 ↔ (𝐼‘𝑖) ∈ 𝑍))
84	83	anbi2d 641	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐼‘𝑖) → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍)))
85		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝐼‘𝑖) → (𝐹‘𝑘) = (𝐹‘(𝐼‘𝑖)))
86	85	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐼‘𝑖) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝐼‘𝑖)) ∈ ℂ))
87	84, 86	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐼‘𝑖) → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ)))
88		climsuse.8	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
89	40, 82, 87, 88	vtoclgf 3533	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝑖) ∈ 𝑍 → ((𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ))
90	75, 77, 89	sylc 66	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ)
91	51, 90	eqeltrd 2861	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) ∈ ℂ)
92	12, 31, 91	syl2anc 595	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐺‘𝑖) ∈ ℂ)
93	12, 31, 51	syl2anc 595	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖)))
94	93	fvoveq1d 7432	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (abs‘((𝐺‘𝑖) − 𝐴)) = (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)))
95		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = ℎ → (𝐹‘𝑖) = (𝐹‘ℎ))
96	95	eleq1d 2846	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = ℎ → ((𝐹‘𝑖) ∈ ℂ ↔ (𝐹‘ℎ) ∈ ℂ))
97	95	fvoveq1d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = ℎ → (abs‘((𝐹‘𝑖) − 𝐴)) = (abs‘((𝐹‘ℎ) − 𝐴)))
98	97	breq1d 5118	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = ℎ → ((abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥))
99	96, 98	anbi12d 643	. . . . . . . . . . . . . 14 ⊢ (𝑖 = ℎ → (((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥) ↔ ((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥)))
100	99	cbvralvw 3241	. . . . . . . . . . . . 13 ⊢ (∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥) ↔ ∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥))
101	100	biimpi 219	. . . . . . . . . . . 12 ⊢ (∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥) → ∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥))
102	101	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → ∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥))
103		zre 12594	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ ℝ)
104	103	3ad2ant2 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ∈ ℝ)
105		simp3 1154	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)))
106		eluzelz 12871	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) → 𝑖 ∈ ℤ)
107		zre 12594	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
108	105, 106, 107	3syl 19	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ ℝ)
109		simp1 1152	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝜑)
110	6	zred 12699	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℝ)
111	109, 110	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ∈ ℝ)
112		simpl2 1209	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ ℤ)
113	112	zred 12699	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ ℝ)
114	111	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) ∧ ¬ 𝑀 ≤ 𝑗) → 𝑀 ∈ ℝ)
115	113, 114	ifclda 4522	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ∈ ℝ)
116		max1 13210	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℝ ∧ 𝑗 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
117	111, 104, 116	syl2anc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
118		eluzle 12874	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ≤ 𝑖)
119	118	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ≤ 𝑖)
120	111, 115, 108, 117, 119	letrd 11366	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ≤ 𝑖)
121	109, 6	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ∈ ℤ)
122	106	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ ℤ)
123		eluz 12875	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑖))
124	121, 122, 123	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝑖 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑖))
125	120, 124	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ (ℤ_≥‘𝑀))
126	125, 28	eleqtrrdi 2872	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ 𝑍)
127	109, 126	jca 520	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝜑 ∧ 𝑖 ∈ 𝑍))
128		eluzelre 12872	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑀) → (𝐼‘𝑖) ∈ ℝ)
129	127, 74, 128	3syl 19	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐼‘𝑖) ∈ ℝ)
130		max2 13212	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℝ ∧ 𝑗 ∈ ℝ) → 𝑗 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
131	111, 104, 130	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
132	104, 115, 108, 131, 119	letrd 11366	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ≤ 𝑖)
133		eluzle 12874	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑖) → 𝑖 ≤ (𝐼‘𝑖))
134	127, 73, 133	3syl 19	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ≤ (𝐼‘𝑖))
135	104, 108, 129, 132, 134	letrd 11366	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ≤ (𝐼‘𝑖))
136		simp2 1153	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ∈ ℤ)
137		eluzelz 12871	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑖) → (𝐼‘𝑖) ∈ ℤ)
138	127, 73, 137	3syl 19	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐼‘𝑖) ∈ ℤ)
139		eluz 12875	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℤ ∧ (𝐼‘𝑖) ∈ ℤ) → ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐼‘𝑖)))
140	136, 138, 139	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐼‘𝑖)))
141	135, 140	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐼‘𝑖) ∈ (ℤ_≥‘𝑗))
142	12, 13, 15, 141	syl3anc 1396	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐼‘𝑖) ∈ (ℤ_≥‘𝑗))
143		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑖) → (𝐹‘ℎ) = (𝐹‘(𝐼‘𝑖)))
144	143	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑖) → ((𝐹‘ℎ) ∈ ℂ ↔ (𝐹‘(𝐼‘𝑖)) ∈ ℂ))
145	143	fvoveq1d 7432	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑖) → (abs‘((𝐹‘ℎ) − 𝐴)) = (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)))
146	145	breq1d 5118	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑖) → ((abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥))
147	144, 146	anbi12d 643	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐼‘𝑖) → (((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥) ↔ ((𝐹‘(𝐼‘𝑖)) ∈ ℂ ∧ (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥)))
148	147	rspccva 3579	. . . . . . . . . . . 12 ⊢ ((∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥) ∧ (𝐼‘𝑖) ∈ (ℤ_≥‘𝑗)) → ((𝐹‘(𝐼‘𝑖)) ∈ ℂ ∧ (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥))
149	148	simprd 500	. . . . . . . . . . 11 ⊢ ((∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥) ∧ (𝐼‘𝑖) ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥)
150	102, 142, 149	syl2anc 595	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥)
151	94, 150	eqbrtrd 5132	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥)
152	92, 151	jca 520	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → ((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))
153	152	ex 417	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) → (𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) → ((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥)))
154	11, 153	ralrimi 3261	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) → ∀𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))
155		fveq2 6881	. . . . . . . 8 ⊢ (𝑙 = if(𝑀 ≤ 𝑗, 𝑗, 𝑀) → (ℤ_≥‘𝑙) = (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)))
156	155	raleqdv 3321	. . . . . . 7 ⊢ (𝑙 = if(𝑀 ≤ 𝑗, 𝑗, 𝑀) → (∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥) ↔ ∀𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥)))
157	156	rspcev 3580	. . . . . 6 ⊢ ((if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ∈ ℤ ∧ ∀𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥)) → ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))
158	8, 154, 157	syl2anc 595	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)) → ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))
159		climsuse.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝑋)
160		eqidd 2762	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐹‘𝑖) = (𝐹‘𝑖))
161	159, 160	clim 15545	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))))
162	1, 161	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)))
163	162	simprd 500	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))
164	163	r19.21bi 3255	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))
165	158, 164	r19.29a 3171	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))
166	165	ex 417	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ → ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥)))
167	4, 166	ralrimi 3261	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))
168		climsuse.12	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑌)
169		eqidd 2762	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐺‘𝑖) = (𝐺‘𝑖))
170	168, 169	clim 15545	. 2 ⊢ (𝜑 → (𝐺 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))))
171	3, 167, 170	mpbir2and 725	1 ⊢ (𝜑 → 𝐺 ⇝ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1568 Ⅎwnf 1811 ∈ wcel 2141 Ⅎwnfc 2908 ∀wral 3077 ∃wrex 3087 ⊆ wss 3904 ifcif 4486 class class class wbr 5108 ‘cfv 6536 (class class class)co 7410 ℂcc 11097 ℝcr 11098 1c1 11100 + caddc 11102 < clt 11242 ≤ cle 11243 − cmin 11440 ℤcz 12590 ℤ_≥cuz 12861 ℝ⁺crp 13015 abscabs 15285 ⇝ cli 15535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-n0 12504 df-z 12591 df-uz 12862 df-clim 15539

This theorem is referenced by: sumnnodd 46316 stirlinglem8 46765

W3C validator