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Theorem cvgratnnlemnexp 12291

Description: Lemma for cvgratnn 12298. (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 9958	. . 3 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	eleqtrdi 2331	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		2fveq3 5700	. . . . 5 ⊢ (𝑤 = 1 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘1)))
5		oveq1 6092	. . . . . . 7 ⊢ (𝑤 = 1 → (𝑤 − 1) = (1 − 1))
6	5	oveq2d 6101	. . . . . 6 ⊢ (𝑤 = 1 → (𝐴↑(𝑤 − 1)) = (𝐴↑(1 − 1)))
7	6	oveq2d 6101	. . . . 5 ⊢ (𝑤 = 1 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))))
8	4, 7	breq12d 4143	. . . 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 5700	. . . . 5 ⊢ (𝑤 = 𝑘 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑘)))
11		oveq1 6092	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑤 − 1) = (𝑘 − 1))
12	11	oveq2d 6101	. . . . . 6 ⊢ (𝑤 = 𝑘 → (𝐴↑(𝑤 − 1)) = (𝐴↑(𝑘 − 1)))
13	12	oveq2d 6101	. . . . 5 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))))
14	10, 13	breq12d 4143	. . . 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 5700	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝑘 + 1))))
17		oveq1 6092	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (𝑤 − 1) = ((𝑘 + 1) − 1))
18	17	oveq2d 6101	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (𝐴↑(𝑤 − 1)) = (𝐴↑((𝑘 + 1) − 1)))
19	18	oveq2d 6101	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))
20	16, 19	breq12d 4143	. . . 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 5700	. . . . 5 ⊢ (𝑤 = 𝑁 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑁)))
23		oveq1 6092	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 − 1) = (𝑁 − 1))
24	23	oveq2d 6101	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝐴↑(𝑤 − 1)) = (𝐴↑(𝑁 − 1)))
25	24	oveq2d 6101	. . . . 5 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1))))
26	22, 25	breq12d 4143	. . . 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 5695	. . . . . . . . 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 9315	. . . . . . . . 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 11947	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
36	35	leidd 8842	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘1)) ≤ (abs‘(𝐹‘1)))
37		1m1e0 9373	. . . . . . . . . 10 ⊢ (1 − 1) = 0
38	37	a1i 9	. . . . . . . . 9 ⊢ (𝜑 → (1 − 1) = 0)
39	38	oveq2d 6101	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(1 − 1)) = (𝐴↑0))
40		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
41	40	recnd 8354	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
42	41	exp0d 11105	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑0) = 1)
43	39, 42	eqtrd 2271	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(1 − 1)) = 1)
44	43	oveq2d 6101	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))) = ((abs‘(𝐹‘1)) · 1))
45	35	recnd 8354	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℂ)
46	45	mulridd 8343	. . . . . 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 4158	. . . 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 9959	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
51	30	abscld 11947	. . . . . . . . 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 9604	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
55	54	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 − 1) ∈ ℕ₀)
56	53, 55	reexpcld 11128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑘 − 1)) ∈ ℝ)
57	52, 56	remulcld 8356	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) ∈ ℝ)
58		0red 8327	. . . . . . . . . 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 8445	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
62		lemul2a 9189	. . . . . . . . . 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 8354	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑘 − 1)) ∈ ℂ)
68	65, 66, 67	mul12d 8478	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))) = ((abs‘(𝐹‘1)) · (𝐴 · (𝐴↑(𝑘 − 1)))))
69	65, 55	expp1d 11112	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 − 1) + 1)) = ((𝐴↑(𝑘 − 1)) · 𝐴))
70		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
71	70	nncnd 9318	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
72		1cnd 8342	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
73	71, 72, 72	addsubd 8658	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) − 1) = ((𝑘 − 1) + 1))
74	73	oveq2d 6101	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 + 1) − 1)) = (𝐴↑((𝑘 − 1) + 1)))
75	65, 67	mulcomd 8347	. . . . . . . . . . . 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 6101	. . . . . . . . . 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 4142	. . . . . . . 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 5695	. . . . . . . . . . . 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 5695	. . . . . . . . . . . . . . 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 9316	. . . . . . . . . . . 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 11947	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
93	53, 51	remulcld 8356	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
94		elnnuz 9959	. . . . . . . . . . . . . 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 9954	. . . . . . . . . . . 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 11128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 + 1) − 1)) ∈ ℝ)
100	52, 99	remulcld 8356	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))) ∈ ℝ)
101		letr 8408	. . . . . . . . 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 9988	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∀wral 2528 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 ℝcr 8178 0cc0 8179 1c1 8180 + caddc 8182 · cmul 8184 < clt 8360 ≤ cle 8361 − cmin 8497 ℕcn 9304 ℕ₀cn0 9563 ℤcz 9644 ℤ_≥cuz 9921 ↑cexp 10975 abscabs 11763

This proof depends on 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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof 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 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-n0 9564 df-z 9645 df-uz 9922 df-rp 10055 df-seqfrec 10885 df-exp 10976 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765

This theorem is used by: cvgratnnlemfm 12296

W3C validator