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Theorem cvgratnnlemnexp 12269

Description: Lemma for cvgratnn 12276. (Contributed by Jim Kingdon, 15-Nov-2022.)

**Hypotheses**
Ref	Expression
cvgratnn.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
cvgratnn.4	⊢ (𝜑 → 𝐴 < 1)
cvgratnn.gt0	⊢ (𝜑 → 0 < 𝐴)
cvgratnn.6	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
cvgratnn.7	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
cvgratnnlemnexp.n	⊢ (𝜑 → 𝑁 ∈ ℕ)

**Assertion**
Ref	Expression
cvgratnnlemnexp	⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1))))

Distinct variable groups: 𝐴,𝑘 𝑘,𝐹 𝑘,𝑁 𝜑,𝑘

Proof of Theorem cvgratnnlemnexp Dummy variables 𝑛 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvgratnnlemnexp.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		nnuz 9937	. . 3 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	eleqtrdi 2331	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		2fveq3 5695	. . . . 5 ⊢ (𝑤 = 1 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘1)))
5		oveq1 6082	. . . . . . 7 ⊢ (𝑤 = 1 → (𝑤 − 1) = (1 − 1))
6	5	oveq2d 6091	. . . . . 6 ⊢ (𝑤 = 1 → (𝐴↑(𝑤 − 1)) = (𝐴↑(1 − 1)))
7	6	oveq2d 6091	. . . . 5 ⊢ (𝑤 = 1 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))))
8	4, 7	breq12d 4138	. . . 4 ⊢ (𝑤 = 1 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) ↔ (abs‘(𝐹‘1)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1)))))
9	8	imbi2d 230	. . 3 ⊢ (𝑤 = 1 → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1)))) ↔ (𝜑 → (abs‘(𝐹‘1)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))))))
10		2fveq3 5695	. . . . 5 ⊢ (𝑤 = 𝑘 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑘)))
11		oveq1 6082	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑤 − 1) = (𝑘 − 1))
12	11	oveq2d 6091	. . . . . 6 ⊢ (𝑤 = 𝑘 → (𝐴↑(𝑤 − 1)) = (𝐴↑(𝑘 − 1)))
13	12	oveq2d 6091	. . . . 5 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))))
14	10, 13	breq12d 4138	. . . 4 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) ↔ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))))
15	14	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1)))) ↔ (𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))))))
16		2fveq3 5695	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝑘 + 1))))
17		oveq1 6082	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (𝑤 − 1) = ((𝑘 + 1) − 1))
18	17	oveq2d 6091	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (𝐴↑(𝑤 − 1)) = (𝐴↑((𝑘 + 1) − 1)))
19	18	oveq2d 6091	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))
20	16, 19	breq12d 4138	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) ↔ (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1)))))
21	20	imbi2d 230	. . 3 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1)))) ↔ (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))))
22		2fveq3 5695	. . . . 5 ⊢ (𝑤 = 𝑁 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑁)))
23		oveq1 6082	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 − 1) = (𝑁 − 1))
24	23	oveq2d 6091	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝐴↑(𝑤 − 1)) = (𝐴↑(𝑁 − 1)))
25	24	oveq2d 6091	. . . . 5 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1))))
26	22, 25	breq12d 4138	. . . 4 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) ↔ (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1)))))
27	26	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1)))) ↔ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1))))))
28		fveq2 5690	. . . . . . . . 9 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
29	28	eleq1d 2307	. . . . . . . 8 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
30		cvgratnn.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
31	30	ralrimiva 2623	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
32		1nn 9294	. . . . . . . . 9 ⊢ 1 ∈ ℕ
33	32	a1i 9	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℕ)
34	29, 31, 33	rspcdva 2934	. . . . . . 7 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
35	34	abscld 11925	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
36	35	leidd 8832	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘1)) ≤ (abs‘(𝐹‘1)))
37		1m1e0 9352	. . . . . . . . . 10 ⊢ (1 − 1) = 0
38	37	a1i 9	. . . . . . . . 9 ⊢ (𝜑 → (1 − 1) = 0)
39	38	oveq2d 6091	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(1 − 1)) = (𝐴↑0))
40		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
41	40	recnd 8344	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
42	41	exp0d 11083	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑0) = 1)
43	39, 42	eqtrd 2271	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(1 − 1)) = 1)
44	43	oveq2d 6091	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))) = ((abs‘(𝐹‘1)) · 1))
45	35	recnd 8344	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℂ)
46	45	mulridd 8333	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · 1) = (abs‘(𝐹‘1)))
47	44, 46	eqtrd 2271	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))) = (abs‘(𝐹‘1)))
48	36, 47	breqtrrd 4153	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘1)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))))
49	48	a1i 9	. . 3 ⊢ (1 ∈ ℤ → (𝜑 → (abs‘(𝐹‘1)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1)))))
50		elnnuz 9938	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
51	30	abscld 11925	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
52	35	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘1)) ∈ ℝ)
53	40	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
54		nnm1nn0 9583	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
55	54	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 − 1) ∈ ℕ₀)
56	53, 55	reexpcld 11106	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑘 − 1)) ∈ ℝ)
57	52, 56	remulcld 8346	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) ∈ ℝ)
58		0red 8317	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ∈ ℝ)
59		cvgratnn.gt0	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝐴)
60	59	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝐴)
61	58, 53, 60	ltled 8435	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
62		lemul2a 9179	. . . . . . . . . 10 ⊢ ((((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))))
63	62	ex 115	. . . . . . . . 9 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))))))
64	51, 57, 53, 61, 63	syl112anc 1282	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))))))
65	41	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
66	45	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘1)) ∈ ℂ)
67	56	recnd 8344	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑘 − 1)) ∈ ℂ)
68	65, 66, 67	mul12d 8468	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))) = ((abs‘(𝐹‘1)) · (𝐴 · (𝐴↑(𝑘 − 1)))))
69	65, 55	expp1d 11090	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 − 1) + 1)) = ((𝐴↑(𝑘 − 1)) · 𝐴))
70		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
71	70	nncnd 9297	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
72		1cnd 8332	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
73	71, 72, 72	addsubd 8648	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) − 1) = ((𝑘 − 1) + 1))
74	73	oveq2d 6091	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 + 1) − 1)) = (𝐴↑((𝑘 − 1) + 1)))
75	65, 67	mulcomd 8337	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (𝐴↑(𝑘 − 1))) = ((𝐴↑(𝑘 − 1)) · 𝐴))
76	69, 74, 75	3eqtr4rd 2282	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (𝐴↑(𝑘 − 1))) = (𝐴↑((𝑘 + 1) − 1)))
77	76	oveq2d 6091	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴 · (𝐴↑(𝑘 − 1)))) = ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))
78	68, 77	eqtrd 2271	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))) = ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))
79	78	breq2d 4137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))) ↔ (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1)))))
80	64, 79	sylibd 149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1)))))
81		cvgratnn.7	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
82		fveq2 5690	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
83	82	eleq1d 2307	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
84		fveq2 5690	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
85	84	eleq1d 2307	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ))
86	85	cbvralv 2786	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
87	31, 86	sylib 122	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
88	87	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
89		peano2nn 9295	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
90	89	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
91	83, 88, 90	rspcdva 2934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
92	91	abscld 11925	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
93	53, 51	remulcld 8346	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
94		elnnuz 9938	. . . . . . . . . . . . . 14 ⊢ ((𝑘 + 1) ∈ ℕ ↔ (𝑘 + 1) ∈ (ℤ_≥‘1))
95	89, 94	sylib 122	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ (ℤ_≥‘1))
96	95	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ_≥‘1))
97		uznn0sub 9933	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ (ℤ_≥‘1) → ((𝑘 + 1) − 1) ∈ ℕ₀)
98	96, 97	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) − 1) ∈ ℕ₀)
99	53, 98	reexpcld 11106	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 + 1) − 1)) ∈ ℝ)
100	52, 99	remulcld 8346	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))) ∈ ℝ)
101		letr 8398	. . . . . . . . 9 ⊢ (((abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ ∧ (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ ∧ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))) ∈ ℝ) → (((abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))) ∧ (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1)))))
102	92, 93, 100, 101	syl3anc 1278	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))) ∧ (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1)))))
103	81, 102	mpand 433	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1)))))
104	80, 103	syld 45	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1)))))
105	50, 104	sylan2br 288	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1)))))
106	105	expcom 116	. . . 4 ⊢ (𝑘 ∈ (ℤ_≥‘1) → (𝜑 → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))))
107	106	a2d 26	. . 3 ⊢ (𝑘 ∈ (ℤ_≥‘1) → ((𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))) → (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))))
108	9, 15, 21, 27, 49, 107	uzind4 9967	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘1) → (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1)))))
109	3, 108	mpcom 36	1 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∀wral 2528 class class class wbr 4125 ‘cfv 5372 (class class class)co 6075 ℂcc 8167 ℝcr 8168 0cc0 8169 1c1 8170 + caddc 8172 · cmul 8174 < clt 8350 ≤ cle 8351 − cmin 8487 ℕcn 9283 ℕ₀cn0 9542 ℤcz 9623 ℤ_≥cuz 9900 ↑cexp 10953 abscabs 11741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-z 9624 df-uz 9901 df-rp 10034 df-seqfrec 10863 df-exp 10954 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743

This theorem is referenced by: cvgratnnlemfm 12274

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