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Theorem cvgrat 15856

Description: Ratio test for convergence of a complex infinite series. If the ratio 𝐴 of the absolute values of successive terms in an infinite sequence 𝐹 is less than 1 for all terms beyond some index 𝐵, then the infinite sum of the terms of 𝐹 converges to a complex number. Equivalent to first part of Exercise 4 of [Gleason] p. 182. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 27-Apr-2014.)

**Hypotheses**
Ref	Expression
cvgrat.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
cvgrat.2	⊢ 𝑊 = (ℤ_≥‘𝑁)
cvgrat.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
cvgrat.4	⊢ (𝜑 → 𝐴 < 1)
cvgrat.5	⊢ (𝜑 → 𝑁 ∈ 𝑍)
cvgrat.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
cvgrat.7	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))

**Assertion**
Ref	Expression
cvgrat	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Distinct variable groups: 𝐴,𝑘 𝑘,𝐹 𝑘,𝑀 𝑘,𝑁 𝜑,𝑘 𝑘,𝑊 𝑘,𝑍

Proof of Theorem cvgrat Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvgrat.2	. . 3 ⊢ 𝑊 = (ℤ_≥‘𝑁)
2		cvgrat.5	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
3		cvgrat.1	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	2, 3	eleqtrdi 2839	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzelz 12810	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		uzid 12815	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
9	8, 1	eleqtrrdi 2840	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑊)
10		oveq1 7397	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 − 𝑁) = (𝑘 − 𝑁))
11	10	oveq2d 7406	. . . . . 6 ⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
12		eqid 2730	. . . . . 6 ⊢ (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))
13		ovex 7423	. . . . . 6 ⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ V
14	11, 12, 13	fvmpt 6971	. . . . 5 ⊢ (𝑘 ∈ 𝑊 → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
15	14	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
16		0re 11183	. . . . . . 7 ⊢ 0 ∈ ℝ
17		cvgrat.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
18		ifcl 4537	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
19	16, 17, 18	sylancr 587	. . . . . 6 ⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
21		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊)
22	21, 1	eleqtrdi 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ (ℤ_≥‘𝑁))
23		uznn0sub 12839	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → (𝑘 − 𝑁) ∈ ℕ₀)
24	22, 23	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 − 𝑁) ∈ ℕ₀)
25	20, 24	reexpcld 14135	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℝ)
26	15, 25	eqeltrd 2829	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) ∈ ℝ)
27		uzss 12823	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
28	4, 27	syl 17	. . . . . 6 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
29	28, 1, 3	3sstr4g 4003	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
30	29	sselda 3949	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍)
31		cvgrat.6	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
32	30, 31	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ∈ ℂ)
33	23	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝑘 − 𝑁) ∈ ℕ₀)
34		oveq2 7398	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 − 𝑁) → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
35		eqid 2730	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))
36	34, 35, 13	fvmpt 6971	. . . . . . . 8 ⊢ ((𝑘 − 𝑁) ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
37	33, 36	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
38	6	zcnd 12646	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ)
39		eluzelz 12810	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → 𝑘 ∈ ℤ)
40	39	zcnd 12646	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → 𝑘 ∈ ℂ)
41		nn0ex 12455	. . . . . . . . . 10 ⊢ ℕ₀ ∈ V
42	41	mptex 7200	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) ∈ V
43	42	shftval 15047	. . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘) = ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘 − 𝑁)))
44	38, 40, 43	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘) = ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘 − 𝑁)))
45		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ (ℤ_≥‘𝑁))
46	45, 1	eleqtrrdi 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ 𝑊)
47	46, 14	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
48	37, 44, 47	3eqtr4rd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘))
49	6, 48	seqfeq 13999	. . . . 5 ⊢ (𝜑 → seq𝑁( + , (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) = seq𝑁( + , ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)))
50	42	seqshft 15058	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) → seq𝑁( + , ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)) = (seq(𝑁 − 𝑁)( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
51	6, 6, 50	syl2anc 584	. . . . 5 ⊢ (𝜑 → seq𝑁( + , ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)) = (seq(𝑁 − 𝑁)( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
52	38	subidd 11528	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 𝑁) = 0)
53	52	seqeq1d 13979	. . . . . 6 ⊢ (𝜑 → seq(𝑁 − 𝑁)( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) = seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))))
54	53	oveq1d 7405	. . . . 5 ⊢ (𝜑 → (seq(𝑁 − 𝑁)( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) = (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
55	49, 51, 54	3eqtrd 2769	. . . 4 ⊢ (𝜑 → seq𝑁( + , (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) = (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
56	19	recnd 11209	. . . . . . 7 ⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
57		max2 13154	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
58	17, 16, 57	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
59	19, 58	absidd 15396	. . . . . . . 8 ⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) = if(𝐴 ≤ 0, 0, 𝐴))
60		0lt1 11707	. . . . . . . . 9 ⊢ 0 < 1
61		cvgrat.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 1)
62		breq1 5113	. . . . . . . . . 10 ⊢ (0 = if(𝐴 ≤ 0, 0, 𝐴) → (0 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
63		breq1 5113	. . . . . . . . . 10 ⊢ (𝐴 = if(𝐴 ≤ 0, 0, 𝐴) → (𝐴 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
64	62, 63	ifboth 4531	. . . . . . . . 9 ⊢ ((0 < 1 ∧ 𝐴 < 1) → if(𝐴 ≤ 0, 0, 𝐴) < 1)
65	60, 61, 64	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) < 1)
66	59, 65	eqbrtrd 5132	. . . . . . 7 ⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) < 1)
67		oveq2 7398	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
68		ovex 7423	. . . . . . . . 9 ⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘) ∈ V
69	67, 35, 68	fvmpt 6971	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
70	69	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
71	56, 66, 70	geolim 15843	. . . . . 6 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
72		seqex 13975	. . . . . . 7 ⊢ seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V
73		climshft 15549	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V) → ((seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ↔ seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴)))))
74	6, 72, 73	sylancl 586	. . . . . 6 ⊢ (𝜑 → ((seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ↔ seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴)))))
75	71, 74	mpbird 257	. . . . 5 ⊢ (𝜑 → (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
76		ovex 7423	. . . . . 6 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ V
77		ovex 7423	. . . . . 6 ⊢ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ∈ V
78	76, 77	breldm 5875	. . . . 5 ⊢ ((seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) → (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ dom ⇝ )
79	75, 78	syl 17	. . . 4 ⊢ (𝜑 → (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ dom ⇝ )
80	55, 79	eqeltrd 2829	. . 3 ⊢ (𝜑 → seq𝑁( + , (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ∈ dom ⇝ )
81		fveq2 6861	. . . . . 6 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
82	81	eleq1d 2814	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑁) ∈ ℂ))
83	31	ralrimiva 3126	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
84	82, 83, 2	rspcdva 3592	. . . 4 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℂ)
85	84	abscld 15412	. . 3 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℝ)
86		2fveq3 6866	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑁)))
87		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑁) = (𝑁 − 𝑁))
88	87	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)))
89	88	oveq2d 7406	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))))
90	86, 89	breq12d 5123	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) ↔ (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)))))
91	90	imbi2d 340	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝜑 → (abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ↔ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))))))
92		2fveq3 6866	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑘)))
93	11	oveq2d 7406	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))
94	92, 93	breq12d 5123	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) ↔ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))))
95	94	imbi2d 340	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝜑 → (abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ↔ (𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))))
96		2fveq3 6866	. . . . . . . 8 ⊢ (𝑛 = (𝑘 + 1) → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘(𝑘 + 1))))
97		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 𝑁) = ((𝑘 + 1) − 𝑁))
98	97	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
99	98	oveq2d 7406	. . . . . . . 8 ⊢ (𝑛 = (𝑘 + 1) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))
100	96, 99	breq12d 5123	. . . . . . 7 ⊢ (𝑛 = (𝑘 + 1) → ((abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) ↔ (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
101	100	imbi2d 340	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → ((𝜑 → (abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ↔ (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))))
102	85	leidd 11751	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑁)))
103	52	oveq2d 7406	. . . . . . . . . 10 ⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑0))
104	56	exp0d 14112	. . . . . . . . . 10 ⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑0) = 1)
105	103, 104	eqtrd 2765	. . . . . . . . 9 ⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = 1)
106	105	oveq2d 7406	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · 1))
107	85	recnd 11209	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℂ)
108	107	mulridd 11198	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · 1) = (abs‘(𝐹‘𝑁)))
109	106, 108	eqtrd 2765	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = (abs‘(𝐹‘𝑁)))
110	102, 109	breqtrrd 5138	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))))
111	32	abscld 15412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
112	85	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ∈ ℝ)
113	112, 25	remulcld 11211	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) ∈ ℝ)
114	58	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
115		lemul2a 12044	. . . . . . . . . . . . 13 ⊢ ((((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) ∈ ℝ ∧ (if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ ∧ 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))) ∧ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))))
116	115	ex 412	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) ∈ ℝ ∧ (if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ ∧ 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))))
117	111, 113, 20, 114, 116	syl112anc 1376	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))))
118	56	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
119	107	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ∈ ℂ)
120	25	recnd 11209	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℂ)
121	118, 119, 120	mul12d 11390	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))))
122	118, 24	expp1d 14119	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 − 𝑁) + 1)) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) · if(𝐴 ≤ 0, 0, 𝐴)))
123	40, 1	eleq2s 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑊 → 𝑘 ∈ ℂ)
124		ax-1cn 11133	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
125		addsub 11439	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1))
126	124, 125	mp3an2 1451	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1))
127	123, 38, 126	syl2anr 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1))
128	127	oveq2d 7406	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 − 𝑁) + 1)))
129	118, 120	mulcomd 11202	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) · if(𝐴 ≤ 0, 0, 𝐴)))
130	122, 128, 129	3eqtr4rd 2776	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
131	130	oveq2d 7406	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))
132	121, 131	eqtrd 2765	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))
133	132	breq2d 5122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) ↔ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
134	117, 133	sylibd 239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
135		fveq2 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
136	135	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
137		fveq2 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
138	137	eleq1d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ))
139	138	cbvralvw 3216	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ)
140	83, 139	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ)
141	140	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ)
142	1	peano2uzs 12868	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑊 → (𝑘 + 1) ∈ 𝑊)
143	29	sselda 3949	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍)
144	142, 143	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍)
145	136, 141, 144	rspcdva 3592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
146	145	abscld 15412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
147	17	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℝ)
148	147, 111	remulcld 11211	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
149	20, 111	remulcld 11211	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ∈ ℝ)
150		cvgrat.7	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
151	32	absge0d 15420	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 ≤ (abs‘(𝐹‘𝑘)))
152		max1 13152	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
153	17, 16, 152	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
154	153	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
155	147, 20, 111, 151, 154	lemul1ad 12129	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))))
156	146, 148, 149, 150, 155	letrd 11338	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))))
157		peano2uz 12867	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → (𝑘 + 1) ∈ (ℤ_≥‘𝑁))
158	22, 157	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈ (ℤ_≥‘𝑁))
159		uznn0sub 12839	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ (ℤ_≥‘𝑁) → ((𝑘 + 1) − 𝑁) ∈ ℕ₀)
160	158, 159	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 + 1) − 𝑁) ∈ ℕ₀)
161	20, 160	reexpcld 14135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) ∈ ℝ)
162	112, 161	remulcld 11211	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ)
163		letr 11275	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ ∧ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ∈ ℝ ∧ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ) → (((abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ∧ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
164	146, 149, 162, 163	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (((abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ∧ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
165	156, 164	mpand 695	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
166	134, 165	syld 47	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
167	46, 166	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
168	167	expcom 413	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → (𝜑 → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))))
169	168	a2d 29	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → ((𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) → (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))))
170	91, 95, 101, 95, 110, 169	uzind4i 12876	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → (𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))))
171	170	impcom 407	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))
172	47	oveq2d 7406	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → ((abs‘(𝐹‘𝑁)) · ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘)) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))
173	171, 172	breqtrrd 5138	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘)))
174	1, 9, 26, 32, 80, 85, 173	cvgcmpce 15791	. 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
175	3, 2, 31	iserex 15630	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ ))
176	174, 175	mpbird 257	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 Vcvv 3450 ⊆ wss 3917 ifcif 4491 class class class wbr 5110 ↦ cmpt 5191 dom cdm 5641 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 < clt 11215 ≤ cle 11216 − cmin 11412 / cdiv 11842 ℕ₀cn0 12449 ℤcz 12536 ℤ_≥cuz 12800 seqcseq 13973 ↑cexp 14033 shift cshi 15039 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-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 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 ax-pre-sup 11153

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-rmo 3356 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-int 4914 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-se 5595 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-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-ico 13319 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660

This theorem is referenced by: efcllem 16050 cvgdvgrat 44309

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