climsuse - Mathbox for Glauco Siliprandi

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Theorem climsuse 39840

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 14230	. . 3 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
4		nfv 1843	. . 3 ⊢ Ⅎ𝑥𝜑
5		simpllr 799	. . . . . . 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 768	. . . . . . 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 4120	. . . . . 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 1843	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ)
10		nfra1 2941	. . . . . . . 8 ⊢ Ⅎ𝑖∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)
11	9, 10	nfan 1828	. . . . . . 7 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))
12		simp-4l 806	. . . . . . . . . 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 799	. . . . . . . . . . . 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 554	. . . . . . . . . . 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 477	. . . . . . . . . . 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 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → 𝑀 ≤ 𝑗)
17	6	anim1i 592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ))
18	17	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ))
19		eluz 11701	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑗))
20	18, 19	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → (𝑗 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑗))
21	16, 20	mpbird 247	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ (ℤ_≥‘𝑀))
22		simpll 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ ¬ 𝑀 ≤ 𝑗) → 𝜑)
23		uzid 11702	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
24	22, 6, 23	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ ¬ 𝑀 ≤ 𝑗) → 𝑀 ∈ (ℤ_≥‘𝑀))
25	21, 24	ifclda 4120	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ∈ (ℤ_≥‘𝑀))
26		uzss 11708	. . . . . . . . . . . . . 14 ⊢ (if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) ⊆ (ℤ_≥‘𝑀))
27	25, 26	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) ⊆ (ℤ_≥‘𝑀))
28		climsuse.5	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ_≥‘𝑀)
29	27, 28	syl6sseqr 3652	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) ⊆ 𝑍)
30	29	sseld 3602	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) → 𝑖 ∈ 𝑍))
31	14, 15, 30	sylc 65	. . . . . . . . . 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 1843	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 ∈ 𝑍
34	32, 33	nfan 1828	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑖 ∈ 𝑍)
35		climsuse.2	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐺
36		nfcv 2764	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
37	35, 36	nffv 6198	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐺‘𝑖)
38		climsuse.3	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐹
39		climsuse.4	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝐼
40	39, 36	nffv 6198	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐼‘𝑖)
41	38, 40	nffv 6198	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘(𝐼‘𝑖))
42	37, 41	nfeq 2776	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖))
43	34, 42	nfim 1825	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖)))
44		eleq1 2689	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝑘 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
45	44	anbi2d 740	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
46		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝐺‘𝑘) = (𝐺‘𝑖))
47		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑖 → (𝐼‘𝑘) = (𝐼‘𝑖))
48	47	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝐹‘(𝐼‘𝑘)) = (𝐹‘(𝐼‘𝑖)))
49	46, 48	eqeq12d 2637	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → ((𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘)) ↔ (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖))))
50	45, 49	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘))) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖)))))
51		climsuse.13	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘)))
52	43, 50, 51	chvar 2262	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖)))
53	28	eleq2i 2693	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ 𝑍 ↔ 𝑖 ∈ (ℤ_≥‘𝑀))
54	53	biimpi 206	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑀))
55	54	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ (ℤ_≥‘𝑀))
56		uzss 11708	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
57	55, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
58		climsuse.10	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼‘𝑀) ∈ 𝑍)
59		nfcv 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(𝑖 + 1)
60	39, 59	nffv 6198	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝐼‘(𝑖 + 1))
61		nfcv 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘ℤ_≥
62		nfcv 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘 +
63		nfcv 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘1
64	40, 62, 63	nfov 6676	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝐼‘𝑖) + 1)
65	61, 64	nffv 6198	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(ℤ_≥‘((𝐼‘𝑖) + 1))
66	60, 65	nfel 2777	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1))
67	34, 66	nfim 1825	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1)))
68		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝑘 + 1) = (𝑖 + 1))
69	68	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (𝐼‘(𝑘 + 1)) = (𝐼‘(𝑖 + 1)))
70	47	oveq1d 6665	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝐼‘𝑘) + 1) = ((𝐼‘𝑖) + 1))
71	70	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (ℤ_≥‘((𝐼‘𝑘) + 1)) = (ℤ_≥‘((𝐼‘𝑖) + 1)))
72	69, 71	eleq12d 2695	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → ((𝐼‘(𝑘 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑘) + 1)) ↔ (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1))))
73	45, 72	imbi12d 334	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑖 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑘) + 1))) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1)))))
74		climsuse.11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑘) + 1)))
75	67, 73, 74	chvar 2262	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1)))
76	28, 6, 58, 75	climsuselem1 39839	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘𝑖) ∈ (ℤ_≥‘𝑖))
77	57, 76	sseldd 3604	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘𝑖) ∈ (ℤ_≥‘𝑀))
78	77, 28	syl6eleqr 2712	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘𝑖) ∈ 𝑍)
79	78	ex 450	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑖 ∈ 𝑍 → (𝐼‘𝑖) ∈ 𝑍))
80	79	imdistani 726	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍))
81	33	nfci 2754	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝑍
82	40, 81	nfel 2777	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐼‘𝑖) ∈ 𝑍
83	32, 82	nfan 1828	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍)
84	41	nfel1 2779	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘(𝐼‘𝑖)) ∈ ℂ
85	83, 84	nfim 1825	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ)
86		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝐼‘𝑖) → (𝑘 ∈ 𝑍 ↔ (𝐼‘𝑖) ∈ 𝑍))
87	86	anbi2d 740	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐼‘𝑖) → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍)))
88		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝐼‘𝑖) → (𝐹‘𝑘) = (𝐹‘(𝐼‘𝑖)))
89	88	eleq1d 2686	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐼‘𝑖) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝐼‘𝑖)) ∈ ℂ))
90	87, 89	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐼‘𝑖) → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ)))
91		climsuse.8	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
92	40, 85, 90, 91	vtoclgf 3264	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝑖) ∈ 𝑍 → ((𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ))
93	78, 80, 92	sylc 65	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ)
94	52, 93	eqeltrd 2701	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) ∈ ℂ)
95	12, 31, 94	syl2anc 693	. . . . . . . . 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)
96	12, 31, 52	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((((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)))
97	96	oveq1d 6665	. . . . . . . . . . 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) − 𝐴) = ((𝐹‘(𝐼‘𝑖)) − 𝐴))
98	97	fveq2d 6195	. . . . . . . . . 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)))
99		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = ℎ → (𝐹‘𝑖) = (𝐹‘ℎ))
100	99	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = ℎ → ((𝐹‘𝑖) ∈ ℂ ↔ (𝐹‘ℎ) ∈ ℂ))
101	99	oveq1d 6665	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = ℎ → ((𝐹‘𝑖) − 𝐴) = ((𝐹‘ℎ) − 𝐴))
102	101	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = ℎ → (abs‘((𝐹‘𝑖) − 𝐴)) = (abs‘((𝐹‘ℎ) − 𝐴)))
103	102	breq1d 4663	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = ℎ → ((abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥))
104	100, 103	anbi12d 747	. . . . . . . . . . . . . 14 ⊢ (𝑖 = ℎ → (((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥) ↔ ((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥)))
105	104	cbvralv 3171	. . . . . . . . . . . . 13 ⊢ (∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥) ↔ ∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥))
106	105	biimpi 206	. . . . . . . . . . . 12 ⊢ (∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥) → ∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥))
107	106	ad2antlr 763	. . . . . . . . . . 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))
108		zre 11381	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ ℝ)
109	108	3ad2ant2 1083	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ∈ ℝ)
110		simp3 1063	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)))
111		eluzelz 11697	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) → 𝑖 ∈ ℤ)
112		zre 11381	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
113	110, 111, 112	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ ℝ)
114		simp1 1061	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝜑)
115	6	zred 11482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℝ)
116	114, 115	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ∈ ℝ)
117		simpl2 1065	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ ℤ)
118	117	zred 11482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ ℝ)
119	116	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) ∧ ¬ 𝑀 ≤ 𝑗) → 𝑀 ∈ ℝ)
120	118, 119	ifclda 4120	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ∈ ℝ)
121		max1 12016	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℝ ∧ 𝑗 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
122	116, 109, 121	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
123		eluzle 11700	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ≤ 𝑖)
124	123	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ≤ 𝑖)
125	116, 120, 113, 122, 124	letrd 10194	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ≤ 𝑖)
126	114, 6	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ∈ ℤ)
127	111	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ ℤ)
128		eluz 11701	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑖))
129	126, 127, 128	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝑖 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑖))
130	125, 129	mpbird 247	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ (ℤ_≥‘𝑀))
131	130, 28	syl6eleqr 2712	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ 𝑍)
132	114, 131	jca 554	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝜑 ∧ 𝑖 ∈ 𝑍))
133		eluzelre 11698	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑀) → (𝐼‘𝑖) ∈ ℝ)
134	132, 77, 133	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐼‘𝑖) ∈ ℝ)
135		max2 12018	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℝ ∧ 𝑗 ∈ ℝ) → 𝑗 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
136	116, 109, 135	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
137	109, 120, 113, 136, 124	letrd 10194	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ≤ 𝑖)
138		eluzle 11700	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑖) → 𝑖 ≤ (𝐼‘𝑖))
139	132, 76, 138	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ≤ (𝐼‘𝑖))
140	109, 113, 134, 137, 139	letrd 10194	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ≤ (𝐼‘𝑖))
141		simp2 1062	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ∈ ℤ)
142		eluzelz 11697	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑖) → (𝐼‘𝑖) ∈ ℤ)
143	132, 76, 142	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐼‘𝑖) ∈ ℤ)
144		eluz 11701	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℤ ∧ (𝐼‘𝑖) ∈ ℤ) → ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐼‘𝑖)))
145	141, 143, 144	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐼‘𝑖)))
146	140, 145	mpbird 247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐼‘𝑖) ∈ (ℤ_≥‘𝑗))
147	12, 13, 15, 146	syl3anc 1326	. . . . . . . . . . 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))
148		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑖) → (𝐹‘ℎ) = (𝐹‘(𝐼‘𝑖)))
149	148	eleq1d 2686	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑖) → ((𝐹‘ℎ) ∈ ℂ ↔ (𝐹‘(𝐼‘𝑖)) ∈ ℂ))
150	148	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝐼‘𝑖) → ((𝐹‘ℎ) − 𝐴) = ((𝐹‘(𝐼‘𝑖)) − 𝐴))
151	150	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑖) → (abs‘((𝐹‘ℎ) − 𝐴)) = (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)))
152	151	breq1d 4663	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑖) → ((abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥))
153	149, 152	anbi12d 747	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐼‘𝑖) → (((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥) ↔ ((𝐹‘(𝐼‘𝑖)) ∈ ℂ ∧ (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥)))
154	153	rspccva 3308	. . . . . . . . . . . 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))
155	154	simprd 479	. . . . . . . . . . 11 ⊢ ((∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥) ∧ (𝐼‘𝑖) ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥)
156	107, 147, 155	syl2anc 693	. . . . . . . . . 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)
157	98, 156	eqbrtrd 4675	. . . . . . . . 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)
158	95, 157	jca 554	. . . . . . . 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))
159	158	ex 450	. . . . . . 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)))
160	11, 159	ralrimi 2957	. . . . . 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))
161		fveq2 6191	. . . . . . . 8 ⊢ (𝑙 = if(𝑀 ≤ 𝑗, 𝑗, 𝑀) → (ℤ_≥‘𝑙) = (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)))
162	161	raleqdv 3144	. . . . . . 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)))
163	162	rspcev 3309	. . . . . 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))
164	8, 160, 163	syl2anc 693	. . . . 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))
165		climsuse.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝑋)
166		eqidd 2623	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐹‘𝑖) = (𝐹‘𝑖))
167	165, 166	clim 14225	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))))
168	1, 167	mpbid 222	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)))
169	168	simprd 479	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))
170	169	r19.21bi 2932	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))
171	164, 170	r19.29a 3078	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))
172	171	ex 450	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ → ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥)))
173	4, 172	ralrimi 2957	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))
174		climsuse.12	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑌)
175		eqidd 2623	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐺‘𝑖) = (𝐺‘𝑖))
176	174, 175	clim 14225	. 2 ⊢ (𝜑 → (𝐺 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))))
177	3, 173, 176	mpbir2and 957	1 ⊢ (𝜑 → 𝐺 ⇝ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 ifcif 4086 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 abscabs 13974 ⇝ cli 14215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-clim 14219

This theorem is referenced by: sumnnodd 39862 stirlinglem8 40298

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