climsuse - Mathbox for Glauco Siliprandi

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Theorem climsuse 45613

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 15472	. . 3 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
4		nfv 1914	. . 3 ⊢ Ⅎ𝑥𝜑
5		simpllr 775	. . . . . . 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 732	. . . . . . 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 4527	. . . . . 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 1914	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ)
10		nfra1 3262	. . . . . . . 8 ⊢ Ⅎ𝑖∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)
11	9, 10	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ ℤ) ∧ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))
12		simp-4l 782	. . . . . . . . . 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 775	. . . . . . . . . . . 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 511	. . . . . . . . . . 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 484	. . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → 𝑀 ≤ 𝑗)
17	6	anim1i 615	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ))
18	17	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ))
19		eluz 12814	. . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ (ℤ_≥‘𝑀))
22		simpll 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ ¬ 𝑀 ≤ 𝑗) → 𝜑)
23		uzid 12815	. . . . . . . . . . . . . . . 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 4527	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ∈ (ℤ_≥‘𝑀))
26		uzss 12823	. . . . . . . . . . . . . 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	sseqtrrdi 3991	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) ⊆ 𝑍)
30	29	sseld 3948	. . . . . . . . . . 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 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 ∈ 𝑍
34	32, 33	nfan 1899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑖 ∈ 𝑍)
35		climsuse.2	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐺
36		nfcv 2892	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
37	35, 36	nffv 6871	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐺‘𝑖)
38		climsuse.3	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐹
39		climsuse.4	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝐼
40	39, 36	nffv 6871	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐼‘𝑖)
41	38, 40	nffv 6871	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘(𝐼‘𝑖))
42	37, 41	nfeq 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖))
43	34, 42	nfim 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖)))
44		eleq1 2817	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝑘 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
45	44	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
46		fveq2 6861	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝐺‘𝑘) = (𝐺‘𝑖))
47		2fveq3 6866	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝐹‘(𝐼‘𝑘)) = (𝐹‘(𝐼‘𝑖)))
48	46, 47	eqeq12d 2746	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → ((𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘)) ↔ (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖))))
49	45, 48	imbi12d 344	. . . . . . . . . . . 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 2241	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = (𝐹‘(𝐼‘𝑖)))
52	28	eleq2i 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ 𝑍 ↔ 𝑖 ∈ (ℤ_≥‘𝑀))
53	52	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑀))
54	53	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ (ℤ_≥‘𝑀))
55		uzss 12823	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
56	54, 55	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
57		climsuse.10	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼‘𝑀) ∈ 𝑍)
58		nfcv 2892	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(𝑖 + 1)
59	39, 58	nffv 6871	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝐼‘(𝑖 + 1))
60		nfcv 2892	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘ℤ_≥
61		nfcv 2892	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘 +
62		nfcv 2892	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘1
63	40, 61, 62	nfov 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝐼‘𝑖) + 1)
64	60, 63	nffv 6871	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(ℤ_≥‘((𝐼‘𝑖) + 1))
65	59, 64	nfel 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1))
66	34, 65	nfim 1896	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1)))
67		fvoveq1 7413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (𝐼‘(𝑘 + 1)) = (𝐼‘(𝑖 + 1)))
68		fveq2 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝐼‘𝑘) = (𝐼‘𝑖))
69	68	fvoveq1d 7412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (ℤ_≥‘((𝐼‘𝑘) + 1)) = (ℤ_≥‘((𝐼‘𝑖) + 1)))
70	67, 69	eleq12d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → ((𝐼‘(𝑘 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑘) + 1)) ↔ (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1))))
71	45, 70	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑖 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑘) + 1))) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1)))))
72		climsuse.11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑘) + 1)))
73	66, 71, 72	chvarfv 2241	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘(𝑖 + 1)) ∈ (ℤ_≥‘((𝐼‘𝑖) + 1)))
74	28, 6, 57, 73	climsuselem1 45612	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘𝑖) ∈ (ℤ_≥‘𝑖))
75	56, 74	sseldd 3950	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘𝑖) ∈ (ℤ_≥‘𝑀))
76	75, 28	eleqtrrdi 2840	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐼‘𝑖) ∈ 𝑍)
77	76	ex 412	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑖 ∈ 𝑍 → (𝐼‘𝑖) ∈ 𝑍))
78	77	imdistani 568	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍))
79	33	nfci 2880	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝑍
80	40, 79	nfel 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐼‘𝑖) ∈ 𝑍
81	32, 80	nfan 1899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍)
82	41	nfel1 2909	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝐹‘(𝐼‘𝑖)) ∈ ℂ
83	81, 82	nfim 1896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ)
84		eleq1 2817	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝐼‘𝑖) → (𝑘 ∈ 𝑍 ↔ (𝐼‘𝑖) ∈ 𝑍))
85	84	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐼‘𝑖) → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍)))
86		fveq2 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝐼‘𝑖) → (𝐹‘𝑘) = (𝐹‘(𝐼‘𝑖)))
87	86	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐼‘𝑖) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝐼‘𝑖)) ∈ ℂ))
88	85, 87	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐼‘𝑖) → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ)))
89		climsuse.8	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
90	40, 83, 88, 89	vtoclgf 3538	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝑖) ∈ 𝑍 → ((𝜑 ∧ (𝐼‘𝑖) ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ))
91	76, 78, 90	sylc 65	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘(𝐼‘𝑖)) ∈ ℂ)
92	51, 91	eqeltrd 2829	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) ∈ ℂ)
93	12, 31, 92	syl2anc 584	. . . . . . . . 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)
94	12, 31, 51	syl2anc 584	. . . . . . . . . . 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)))
95	94	fvoveq1d 7412	. . . . . . . . . 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)))
96		fveq2 6861	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = ℎ → (𝐹‘𝑖) = (𝐹‘ℎ))
97	96	eleq1d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = ℎ → ((𝐹‘𝑖) ∈ ℂ ↔ (𝐹‘ℎ) ∈ ℂ))
98	96	fvoveq1d 7412	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = ℎ → (abs‘((𝐹‘𝑖) − 𝐴)) = (abs‘((𝐹‘ℎ) − 𝐴)))
99	98	breq1d 5120	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = ℎ → ((abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥))
100	97, 99	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (𝑖 = ℎ → (((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥) ↔ ((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥)))
101	100	cbvralvw 3216	. . . . . . . . . . . . 13 ⊢ (∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥) ↔ ∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥))
102	101	biimpi 216	. . . . . . . . . . . 12 ⊢ (∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥) → ∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥))
103	102	ad2antlr 727	. . . . . . . . . . 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))
104		zre 12540	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ ℝ)
105	104	3ad2ant2 1134	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ∈ ℝ)
106		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)))
107		eluzelz 12810	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) → 𝑖 ∈ ℤ)
108		zre 12540	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
109	106, 107, 108	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ ℝ)
110		simp1 1136	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝜑)
111	6	zred 12645	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℝ)
112	110, 111	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ∈ ℝ)
113		simpl2 1193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ ℤ)
114	113	zred 12645	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ ℝ)
115	112	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) ∧ ¬ 𝑀 ≤ 𝑗) → 𝑀 ∈ ℝ)
116	114, 115	ifclda 4527	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ∈ ℝ)
117		max1 13152	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℝ ∧ 𝑗 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
118	112, 105, 117	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
119		eluzle 12813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ≤ 𝑖)
120	119	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → if(𝑀 ≤ 𝑗, 𝑗, 𝑀) ≤ 𝑖)
121	112, 116, 109, 118, 120	letrd 11338	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ≤ 𝑖)
122	110, 6	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑀 ∈ ℤ)
123	107	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ ℤ)
124		eluz 12814	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑖))
125	122, 123, 124	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝑖 ∈ (ℤ_≥‘𝑀) ↔ 𝑀 ≤ 𝑖))
126	121, 125	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ (ℤ_≥‘𝑀))
127	126, 28	eleqtrrdi 2840	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ∈ 𝑍)
128	110, 127	jca 511	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝜑 ∧ 𝑖 ∈ 𝑍))
129		eluzelre 12811	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑀) → (𝐼‘𝑖) ∈ ℝ)
130	128, 75, 129	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐼‘𝑖) ∈ ℝ)
131		max2 13154	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℝ ∧ 𝑗 ∈ ℝ) → 𝑗 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
132	112, 105, 131	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ≤ if(𝑀 ≤ 𝑗, 𝑗, 𝑀))
133	105, 116, 109, 132, 120	letrd 11338	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ≤ 𝑖)
134		eluzle 12813	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑖) → 𝑖 ≤ (𝐼‘𝑖))
135	128, 74, 134	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑖 ≤ (𝐼‘𝑖))
136	105, 109, 130, 133, 135	letrd 11338	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ≤ (𝐼‘𝑖))
137		simp2 1137	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → 𝑗 ∈ ℤ)
138		eluzelz 12810	. . . . . . . . . . . . . . 15 ⊢ ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑖) → (𝐼‘𝑖) ∈ ℤ)
139	128, 74, 138	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐼‘𝑖) ∈ ℤ)
140		eluz 12814	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℤ ∧ (𝐼‘𝑖) ∈ ℤ) → ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐼‘𝑖)))
141	137, 139, 140	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → ((𝐼‘𝑖) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐼‘𝑖)))
142	136, 141	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀))) → (𝐼‘𝑖) ∈ (ℤ_≥‘𝑗))
143	12, 13, 15, 142	syl3anc 1373	. . . . . . . . . . 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))
144		fveq2 6861	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑖) → (𝐹‘ℎ) = (𝐹‘(𝐼‘𝑖)))
145	144	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑖) → ((𝐹‘ℎ) ∈ ℂ ↔ (𝐹‘(𝐼‘𝑖)) ∈ ℂ))
146	144	fvoveq1d 7412	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝐼‘𝑖) → (abs‘((𝐹‘ℎ) − 𝐴)) = (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)))
147	146	breq1d 5120	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐼‘𝑖) → ((abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥))
148	145, 147	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐼‘𝑖) → (((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥) ↔ ((𝐹‘(𝐼‘𝑖)) ∈ ℂ ∧ (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥)))
149	148	rspccva 3590	. . . . . . . . . . . 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))
150	149	simprd 495	. . . . . . . . . . 11 ⊢ ((∀ℎ ∈ (ℤ_≥‘𝑗)((𝐹‘ℎ) ∈ ℂ ∧ (abs‘((𝐹‘ℎ) − 𝐴)) < 𝑥) ∧ (𝐼‘𝑖) ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘(𝐼‘𝑖)) − 𝐴)) < 𝑥)
151	103, 143, 150	syl2anc 584	. . . . . . . . . 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)
152	95, 151	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)
153	93, 152	jca 511	. . . . . . . 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))
154	153	ex 412	. . . . . . 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)))
155	11, 154	ralrimi 3236	. . . . . 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))
156		fveq2 6861	. . . . . . . 8 ⊢ (𝑙 = if(𝑀 ≤ 𝑗, 𝑗, 𝑀) → (ℤ_≥‘𝑙) = (ℤ_≥‘if(𝑀 ≤ 𝑗, 𝑗, 𝑀)))
157	156	raleqdv 3301	. . . . . . 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)))
158	157	rspcev 3591	. . . . . 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))
159	8, 155, 158	syl2anc 584	. . . . 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))
160		climsuse.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝑋)
161		eqidd 2731	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐹‘𝑖) = (𝐹‘𝑖))
162	160, 161	clim 15467	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))))
163	1, 162	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥)))
164	163	simprd 495	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))
165	164	r19.21bi 3230	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − 𝐴)) < 𝑥))
166	159, 165	r19.29a 3142	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))
167	166	ex 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ → ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥)))
168	4, 167	ralrimi 3236	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))
169		climsuse.12	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑌)
170		eqidd 2731	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐺‘𝑖) = (𝐺‘𝑖))
171	169, 170	clim 15467	. 2 ⊢ (𝜑 → (𝐺 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑙 ∈ ℤ ∀𝑖 ∈ (ℤ_≥‘𝑙)((𝐺‘𝑖) ∈ ℂ ∧ (abs‘((𝐺‘𝑖) − 𝐴)) < 𝑥))))
172	3, 168, 171	mpbir2and 713	1 ⊢ (𝜑 → 𝐺 ⇝ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 Ⅎwnfc 2877 ∀wral 3045 ∃wrex 3054 ⊆ wss 3917 ifcif 4491 class class class wbr 5110 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 1c1 11076 + caddc 11078 < clt 11215 ≤ cle 11216 − cmin 11412 ℤcz 12536 ℤ_≥cuz 12800 ℝ⁺crp 12958 abscabs 15207 ⇝ cli 15457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-n0 12450 df-z 12537 df-uz 12801 df-clim 15461

This theorem is referenced by: sumnnodd 45635 stirlinglem8 46086

W3C validator