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Theorem cvgrat 15836

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 2842	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzelz 12839	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		uzid 12844	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
9	8, 1	eleqtrrdi 2843	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑊)
10		oveq1 7419	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 − 𝑁) = (𝑘 − 𝑁))
11	10	oveq2d 7428	. . . . . 6 ⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
12		eqid 2731	. . . . . 6 ⊢ (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))
13		ovex 7445	. . . . . 6 ⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ V
14	11, 12, 13	fvmpt 6998	. . . . 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 11223	. . . . . . 7 ⊢ 0 ∈ ℝ
17		cvgrat.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
18		ifcl 4573	. . . . . . 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 2842	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ (ℤ_≥‘𝑁))
23		uznn0sub 12868	. . . . . 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 2832	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) ∈ ℝ)
27		uzss 12852	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
28	4, 27	syl 17	. . . . . 6 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
29	28, 1, 3	3sstr4g 4027	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
30	29	sselda 3982	. . . 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 7420	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 − 𝑁) → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))
35		eqid 2731	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))
36	34, 35, 13	fvmpt 6998	. . . . . . . 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 12674	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ)
39		eluzelz 12839	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → 𝑘 ∈ ℤ)
40	39	zcnd 12674	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → 𝑘 ∈ ℂ)
41		nn0ex 12485	. . . . . . . . . 10 ⊢ ℕ₀ ∈ V
42	41	mptex 7227	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) ∈ V
43	42	shftval 15028	. . . . . . . 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 2843	. . . . . . . 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 2782	. . . . . 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 14000	. . . . 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 15039	. . . . . 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 11566	. . . . . . 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 7427	. . . . 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 2775	. . . 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 11249	. . . . . . 7 ⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
57		max2 13173	. . . . . . . . . 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 15376	. . . . . . . 8 ⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) = if(𝐴 ≤ 0, 0, 𝐴))
60		0lt1 11743	. . . . . . . . 9 ⊢ 0 < 1
61		cvgrat.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 1)
62		breq1 5151	. . . . . . . . . 10 ⊢ (0 = if(𝐴 ≤ 0, 0, 𝐴) → (0 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
63		breq1 5151	. . . . . . . . . 10 ⊢ (𝐴 = if(𝐴 ≤ 0, 0, 𝐴) → (𝐴 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
64	62, 63	ifboth 4567	. . . . . . . . 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 5170	. . . . . . 7 ⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) < 1)
67		oveq2 7420	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
68		ovex 7445	. . . . . . . . 9 ⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘) ∈ V
69	67, 35, 68	fvmpt 6998	. . . . . . . 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 15823	. . . . . 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 15527	. . . . . . 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 7445	. . . . . 6 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ V
77		ovex 7445	. . . . . 6 ⊢ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ∈ V
78	76, 77	breldm 5908	. . . . 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 2832	. . 3 ⊢ (𝜑 → seq𝑁( + , (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ∈ dom ⇝ )
81		fveq2 6891	. . . . . 6 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
82	81	eleq1d 2817	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑁) ∈ ℂ))
83	31	ralrimiva 3145	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
84	82, 83, 2	rspcdva 3613	. . . 4 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℂ)
85	84	abscld 15390	. . 3 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℝ)
86		2fveq3 6896	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑁)))
87		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑁) = (𝑁 − 𝑁))
88	87	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)))
89	88	oveq2d 7428	. . . . . . . 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 5161	. . . . . . 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 6896	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑘)))
93	11	oveq2d 7428	. . . . . . . 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 5161	. . . . . . 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 6896	. . . . . . . 8 ⊢ (𝑛 = (𝑘 + 1) → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘(𝑘 + 1))))
97		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 𝑁) = ((𝑘 + 1) − 𝑁))
98	97	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
99	98	oveq2d 7428	. . . . . . . 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 5161	. . . . . . 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 11787	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑁)))
103	52	oveq2d 7428	. . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = 1)
106	105	oveq2d 7428	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · 1))
107	85	recnd 11249	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℂ)
108	107	mulridd 11238	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · 1) = (abs‘(𝐹‘𝑁)))
109	106, 108	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = (abs‘(𝐹‘𝑁)))
110	102, 109	breqtrrd 5176	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))))
111	32	abscld 15390	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
112	85	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ∈ ℝ)
113	112, 25	remulcld 11251	. . . . . . . . . . . 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 12076	. . . . . . . . . . . . 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 1373	. . . . . . . . . . 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 11249	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℂ)
121	118, 119, 120	mul12d 11430	. . . . . . . . . . . . 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 2850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑊 → 𝑘 ∈ ℂ)
124		ax-1cn 11174	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
125		addsub 11478	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1))
126	124, 125	mp3an2 1448	. . . . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 − 𝑁) + 1)))
129	118, 120	mulcomd 11242	. . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 5160	. . . . . . . . . . 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 238	. . . . . . . . . 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 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
136	135	eleq1d 2817	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
137		fveq2 6891	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
138	137	eleq1d 2817	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ))
139	138	cbvralvw 3233	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ)
140	83, 139	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ)
141	140	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ)
142	1	peano2uzs 12893	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑊 → (𝑘 + 1) ∈ 𝑊)
143	29	sselda 3982	. . . . . . . . . . . . . . 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 3613	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
146	145	abscld 15390	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
147	17	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℝ)
148	147, 111	remulcld 11251	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
149	20, 111	remulcld 11251	. . . . . . . . . . . 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 15398	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 ≤ (abs‘(𝐹‘𝑘)))
152		max1 13171	. . . . . . . . . . . . . . 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 12160	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))))
156	146, 148, 149, 150, 155	letrd 11378	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))))
157		peano2uz 12892	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘𝑁) → (𝑘 + 1) ∈ (ℤ_≥‘𝑁))
158	22, 157	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈ (ℤ_≥‘𝑁))
159		uznn0sub 12868	. . . . . . . . . . . . . . 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 11251	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ)
163		letr 11315	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 692	. . . . . . . . . 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 12901	. . . . 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 7428	. . . 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 5176	. . 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 15771	. 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
175	3, 2, 31	iserex 15610	. 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 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 ⊆ wss 3948 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 < clt 11255 ≤ cle 11256 − cmin 11451 / cdiv 11878 ℕ₀cn0 12479 ℤcz 12565 ℤ_≥cuz 12829 seqcseq 13973 ↑cexp 14034 shift cshi 15020 abscabs 15188 ⇝ cli 15435

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-ico 13337 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15640

This theorem is referenced by: efcllem 16028 cvgdvgrat 43534

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