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Theorem cvgrat 15931

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 2854	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzelz 12913	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		uzid 12918	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
9	8, 1	eleqtrrdi 2855	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑊)
10		oveq1 7455	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 − 𝑁) = (𝑘 − 𝑁))
11	10	oveq2d 7464	. . . . . 6 ⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
12		eqid 2740	. . . . . 6 ⊢ (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))
13		ovex 7481	. . . . . 6 ⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ V
14	11, 12, 13	fvmpt 7029	. . . . 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 11292	. . . . . . 7 ⊢ 0 ∈ ℝ
17		cvgrat.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
18		ifcl 4593	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
19	16, 17, 18	sylancr 586	. . . . . 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 2854	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ (ℤ_≥‘𝑁))
23		uznn0sub 12942	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → (𝑘 − 𝑁) ∈ ℕ₀)
24	22, 23	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 − 𝑁) ∈ ℕ₀)
25	20, 24	reexpcld 14213	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℝ)
26	15, 25	eqeltrd 2844	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) ∈ ℝ)
27		uzss 12926	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
28	4, 27	syl 17	. . . . . 6 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
29	28, 1, 3	3sstr4g 4054	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
30	29	sselda 4008	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍)
31		cvgrat.6	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
32	30, 31	syldan 590	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ∈ ℂ)
33	23	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝑘 − 𝑁) ∈ ℕ₀)
34		oveq2 7456	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 − 𝑁) → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
35		eqid 2740	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))
36	34, 35, 13	fvmpt 7029	. . . . . . . 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 12748	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ)
39		eluzelz 12913	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → 𝑘 ∈ ℤ)
40	39	zcnd 12748	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → 𝑘 ∈ ℂ)
41		nn0ex 12559	. . . . . . . . . 10 ⊢ ℕ₀ ∈ V
42	41	mptex 7260	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) ∈ V
43	42	shftval 15123	. . . . . . . 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 595	. . . . . . 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 2855	. . . . . . . 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 2791	. . . . . 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 14078	. . . . 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 15134	. . . . . 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 583	. . . . 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 11635	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 𝑁) = 0)
53	52	seqeq1d 14058	. . . . . 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 7463	. . . . 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 2784	. . . 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 11318	. . . . . . 7 ⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
57		max2 13249	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
58	17, 16, 57	sylancl 585	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
59	19, 58	absidd 15471	. . . . . . . 8 ⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) = if(𝐴 ≤ 0, 0, 𝐴))
60		0lt1 11812	. . . . . . . . 9 ⊢ 0 < 1
61		cvgrat.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 1)
62		breq1 5169	. . . . . . . . . 10 ⊢ (0 = if(𝐴 ≤ 0, 0, 𝐴) → (0 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
63		breq1 5169	. . . . . . . . . 10 ⊢ (𝐴 = if(𝐴 ≤ 0, 0, 𝐴) → (𝐴 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
64	62, 63	ifboth 4587	. . . . . . . . 9 ⊢ ((0 < 1 ∧ 𝐴 < 1) → if(𝐴 ≤ 0, 0, 𝐴) < 1)
65	60, 61, 64	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) < 1)
66	59, 65	eqbrtrd 5188	. . . . . . 7 ⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) < 1)
67		oveq2 7456	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
68		ovex 7481	. . . . . . . . 9 ⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘) ∈ V
69	67, 35, 68	fvmpt 7029	. . . . . . . 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 15918	. . . . . 6 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
72		seqex 14054	. . . . . . 7 ⊢ seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V
73		climshft 15622	. . . . . . 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 585	. . . . . 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 7481	. . . . . 6 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ V
77		ovex 7481	. . . . . 6 ⊢ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ∈ V
78	76, 77	breldm 5933	. . . . 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 2844	. . 3 ⊢ (𝜑 → seq𝑁( + , (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ∈ dom ⇝ )
81		fveq2 6920	. . . . . 6 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
82	81	eleq1d 2829	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑁) ∈ ℂ))
83	31	ralrimiva 3152	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
84	82, 83, 2	rspcdva 3636	. . . 4 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℂ)
85	84	abscld 15485	. . 3 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℝ)
86		2fveq3 6925	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑁)))
87		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑁) = (𝑁 − 𝑁))
88	87	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)))
89	88	oveq2d 7464	. . . . . . . 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 5179	. . . . . . 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 6925	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑘)))
93	11	oveq2d 7464	. . . . . . . 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 5179	. . . . . . 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 6925	. . . . . . . 8 ⊢ (𝑛 = (𝑘 + 1) → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘(𝑘 + 1))))
97		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 𝑁) = ((𝑘 + 1) − 𝑁))
98	97	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
99	98	oveq2d 7464	. . . . . . . 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 5179	. . . . . . 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 11856	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑁)))
103	52	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑0))
104	56	exp0d 14190	. . . . . . . . . 10 ⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑0) = 1)
105	103, 104	eqtrd 2780	. . . . . . . . 9 ⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = 1)
106	105	oveq2d 7464	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · 1))
107	85	recnd 11318	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℂ)
108	107	mulridd 11307	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · 1) = (abs‘(𝐹‘𝑁)))
109	106, 108	eqtrd 2780	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = (abs‘(𝐹‘𝑁)))
110	102, 109	breqtrrd 5194	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))))
111	32	abscld 15485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
112	85	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ∈ ℝ)
113	112, 25	remulcld 11320	. . . . . . . . . . . 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 12149	. . . . . . . . . . . . 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 1374	. . . . . . . . . . 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 11318	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℂ)
121	118, 119, 120	mul12d 11499	. . . . . . . . . . . . 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 14197	. . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑊 → 𝑘 ∈ ℂ)
124		ax-1cn 11242	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
125		addsub 11547	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1))
126	124, 125	mp3an2 1449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1))
127	123, 38, 126	syl2anr 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1))
128	127	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 − 𝑁) + 1)))
129	118, 120	mulcomd 11311	. . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . 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 2780	. . . . . . . . . . . 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 5178	. . . . . . . . . . 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 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
136	135	eleq1d 2829	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
137		fveq2 6920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
138	137	eleq1d 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ))
139	138	cbvralvw 3243	. . . . . . . . . . . . . . . 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 12967	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑊 → (𝑘 + 1) ∈ 𝑊)
143	29	sselda 4008	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍)
144	142, 143	sylan2 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍)
145	136, 141, 144	rspcdva 3636	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
146	145	abscld 15485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
147	17	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℝ)
148	147, 111	remulcld 11320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
149	20, 111	remulcld 11320	. . . . . . . . . . . 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 15493	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 ≤ (abs‘(𝐹‘𝑘)))
152		max1 13247	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
153	17, 16, 152	sylancl 585	. . . . . . . . . . . . . 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 12234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))))
156	146, 148, 149, 150, 155	letrd 11447	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))))
157		peano2uz 12966	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → (𝑘 + 1) ∈ (ℤ_≥‘𝑁))
158	22, 157	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈ (ℤ_≥‘𝑁))
159		uznn0sub 12942	. . . . . . . . . . . . . . 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 14213	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) ∈ ℝ)
162	112, 161	remulcld 11320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ)
163		letr 11384	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 694	. . . . . . . . . 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 590	. . . . . . . 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 12975	. . . . 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 7464	. . . 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 5194	. . 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 15866	. 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
175	3, 2, 31	iserex 15705	. 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 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ⊆ wss 3976 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 seqcseq 14052 ↑cexp 14112 shift cshi 15115 abscabs 15283 ⇝ cli 15530

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-ico 13413 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735

This theorem is referenced by: efcllem 16125 cvgdvgrat 44282

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