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Theorem cvgrat 16045

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 2871	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzelz 12968	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
6	4, 5	syl 18	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		uzid 12973	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
8	6, 7	syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
9	8, 1	eleqtrrdi 2872	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑊)
10		oveq1 7425	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 − 𝑁) = (𝑘 − 𝑁))
11	10	oveq2d 7434	. . . . . 6 ⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
12		eqid 2761	. . . . . 6 ⊢ (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))
13		ovex 7451	. . . . . 6 ⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ V
14	11, 12, 13	fvmpt 6991	. . . . 5 ⊢ (𝑘 ∈ 𝑊 → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
15	14	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
16		0re 11303	. . . . . . 7 ⊢ 0 ∈ ℝ
17		cvgrat.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
18		ifcl 4528	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
19	16, 17, 18	sylancr 599	. . . . . 6 ⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
20	19	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
21		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊)
22	21, 1	eleqtrdi 2871	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ (ℤ_≥‘𝑁))
23		uznn0sub 12993	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → (𝑘 − 𝑁) ∈ ℕ₀)
24	22, 23	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 − 𝑁) ∈ ℕ₀)
25	20, 24	reexpcld 14299	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℝ)
26	15, 25	eqeltrd 2861	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) ∈ ℝ)
27		uzss 12981	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
28	4, 27	syl 18	. . . . . 6 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
29	28, 1, 3	3sstr4g 3984	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
30	29	sselda 3931	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍)
31		cvgrat.6	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
32	30, 31	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ∈ ℂ)
33	23	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝑘 − 𝑁) ∈ ℕ₀)
34		oveq2 7426	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 − 𝑁) → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
35		eqid 2761	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))
36	34, 35, 13	fvmpt 6991	. . . . . . . 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 18	. . . . . . 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 12797	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ)
39		eluzelz 12968	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → 𝑘 ∈ ℤ)
40	39	zcnd 12797	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → 𝑘 ∈ ℂ)
41		nn0ex 12605	. . . . . . . . . 10 ⊢ ℕ₀ ∈ V
42	41	mptex 7227	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) ∈ V
43	42	shftval 15220	. . . . . . . 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 608	. . . . . . 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 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ (ℤ_≥‘𝑁))
46	45, 1	eleqtrrdi 2872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝑘 ∈ 𝑊)
47	46, 14	syl 18	. . . . . . 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 2807	. . . . . 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 14163	. . . . 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 15231	. . . . . 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 596	. . . . 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 11650	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 𝑁) = 0)
53	52	seqeq1d 14143	. . . . . 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 7433	. . . . 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 2800	. . . 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 11330	. . . . . . 7 ⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
57		max2 13310	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
58	17, 16, 57	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
59	19, 58	absidd 15583	. . . . . . . 8 ⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) = if(𝐴 ≤ 0, 0, 𝐴))
60		0lt1 11831	. . . . . . . . 9 ⊢ 0 < 1
61		cvgrat.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 1)
62		breq1 5106	. . . . . . . . . 10 ⊢ (0 = if(𝐴 ≤ 0, 0, 𝐴) → (0 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
63		breq1 5106	. . . . . . . . . 10 ⊢ (𝐴 = if(𝐴 ≤ 0, 0, 𝐴) → (𝐴 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
64	62, 63	ifboth 4522	. . . . . . . . 9 ⊢ ((0 < 1 ∧ 𝐴 < 1) → if(𝐴 ≤ 0, 0, 𝐴) < 1)
65	60, 61, 64	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) < 1)
66	59, 65	eqbrtrd 5127	. . . . . . 7 ⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) < 1)
67		oveq2 7426	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
68		ovex 7451	. . . . . . . . 9 ⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘) ∈ V
69	67, 35, 68	fvmpt 6991	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
70	69	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
71	56, 66, 70	geolim 16032	. . . . . 6 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
72		seqex 14139	. . . . . . 7 ⊢ seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V
73		climshft 15736	. . . . . . 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 598	. . . . . 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 260	. . . . 5 ⊢ (𝜑 → (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
76		ovex 7451	. . . . . 6 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ V
77		ovex 7451	. . . . . 6 ⊢ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ∈ V
78	76, 77	breldm 5890	. . . . 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 18	. . . 4 ⊢ (𝜑 → (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ dom ⇝ )
80	55, 79	eqeltrd 2861	. . 3 ⊢ (𝜑 → seq𝑁( + , (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ∈ dom ⇝ )
81		fveq2 6883	. . . . . 6 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
82	81	eleq1d 2846	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑁) ∈ ℂ))
83	31	ralrimiva 3155	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
84	82, 83, 2	rspcdva 3578	. . . 4 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℂ)
85	84	abscld 15599	. . 3 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℝ)
86		2fveq3 6888	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑁)))
87		oveq1 7425	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑁) = (𝑁 − 𝑁))
88	87	oveq2d 7434	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)))
89	88	oveq2d 7434	. . . . . . . 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 5116	. . . . . . 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 343	. . . . . 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 6888	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑘)))
93	11	oveq2d 7434	. . . . . . . 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 5116	. . . . . . 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 343	. . . . . 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 6888	. . . . . . . 8 ⊢ (𝑛 = (𝑘 + 1) → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘(𝑘 + 1))))
97		oveq1 7425	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 𝑁) = ((𝑘 + 1) − 𝑁))
98	97	oveq2d 7434	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
99	98	oveq2d 7434	. . . . . . . 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 5116	. . . . . . 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 343	. . . . . 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 11875	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑁)))
103	52	oveq2d 7434	. . . . . . . . . 10 ⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑0))
104	56	exp0d 14276	. . . . . . . . . 10 ⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑0) = 1)
105	103, 104	eqtrd 2796	. . . . . . . . 9 ⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = 1)
106	105	oveq2d 7434	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · 1))
107	85	recnd 11330	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℂ)
108	107	mulridd 11319	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · 1) = (abs‘(𝐹‘𝑁)))
109	106, 108	eqtrd 2796	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = (abs‘(𝐹‘𝑁)))
110	102, 109	breqtrrd 5133	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))))
111	32	abscld 15599	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
112	85	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ∈ ℝ)
113	112, 25	remulcld 11332	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) ∈ ℝ)
114	58	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
115		lemul2a 12165	. . . . . . . . . . . . 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 418	. . . . . . . . . . . 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 1401	. . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
119	107	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ∈ ℂ)
120	25	recnd 11330	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℂ)
121	118, 119, 120	mul12d 11512	. . . . . . . . . . . . 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 14283	. . . . . . . . . . . . . . 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 2879	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑊 → 𝑘 ∈ ℂ)
124		ax-1cn 11251	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
125		addsub 11561	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1))
126	124, 125	mp3an2 1478	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1))
127	123, 38, 126	syl2anr 609	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1))
128	127	oveq2d 7434	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 − 𝑁) + 1)))
129	118, 120	mulcomd 11323	. . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . 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 2796	. . . . . . . . . . . 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 5115	. . . . . . . . . . 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 242	. . . . . . . . . 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 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
136	135	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
137		fveq2 6883	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
138	137	eleq1d 2846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ))
139	138	cbvralvw 3241	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ)
140	83, 139	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ)
141	140	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ)
142	1	peano2uzs 13022	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑊 → (𝑘 + 1) ∈ 𝑊)
143	29	sselda 3931	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍)
144	142, 143	sylan2 605	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍)
145	136, 141, 144	rspcdva 3578	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
146	145	abscld 15599	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
147	17	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℝ)
148	147, 111	remulcld 11332	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
149	20, 111	remulcld 11332	. . . . . . . . . . . 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 15607	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 ≤ (abs‘(𝐹‘𝑘)))
152		max1 13308	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
153	17, 16, 152	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
154	153	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
155	147, 20, 111, 151, 154	lemul1ad 12249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))))
156	146, 148, 149, 150, 155	letrd 11460	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))))
157		peano2uz 13021	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → (𝑘 + 1) ∈ (ℤ_≥‘𝑁))
158	22, 157	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈ (ℤ_≥‘𝑁))
159		uznn0sub 12993	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ (ℤ_≥‘𝑁) → ((𝑘 + 1) − 𝑁) ∈ ℕ₀)
160	158, 159	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 + 1) − 𝑁) ∈ ℕ₀)
161	20, 160	reexpcld 14299	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) ∈ ℝ)
162	112, 161	remulcld 11332	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ)
163		letr 11397	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 708	. . . . . . . . . 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 48	. . . . . . . . 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 603	. . . . . . . 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 419	. . . . . . 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 30	. . . . . 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 13030	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → (𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))))
171	170	impcom 413	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))
172	47	oveq2d 7434	. . . 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 5133	. . 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 15978	. 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
175	3, 2, 31	iserex 15817	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ ))
176	174, 175	mpbird 260	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 Vcvv 3451 ⊆ wss 3899 ifcif 4482 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 ℝcr 11192 0cc0 11193 1c1 11194 + caddc 11196 · cmul 11198 < clt 11336 ≤ cle 11337 − cmin 11534 / cdiv 11966 ℕ₀cn0 12599 ℤcz 12686 ℤ_≥cuz 12958 seqcseq 14137 ↑cexp 14197 shift cshi 15212 abscabs 15394 ⇝ cli 15644

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-pm 8843 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9427 df-inf 9428 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-n0 12600 df-z 12687 df-uz 12959 df-rp 13114 df-ico 13475 df-fz 13633 df-fzo 13782 df-fl 13925 df-seq 14138 df-exp 14198 df-hash 14468 df-shft 15213 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-limsup 15631 df-clim 15648 df-rlim 15649 df-sum 15847

This theorem is used by: efcllem 16236 cvgdvgrat 45282

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