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Theorem cvgratnnlemnexp 11325

Description: Lemma for cvgratnn 11332. (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 9385	. . 3 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	eleqtrdi 2233	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		2fveq3 5434	. . . . 5 ⊢ (𝑤 = 1 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘1)))
5		oveq1 5789	. . . . . . 7 ⊢ (𝑤 = 1 → (𝑤 − 1) = (1 − 1))
6	5	oveq2d 5798	. . . . . 6 ⊢ (𝑤 = 1 → (𝐴↑(𝑤 − 1)) = (𝐴↑(1 − 1)))
7	6	oveq2d 5798	. . . . 5 ⊢ (𝑤 = 1 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))))
8	4, 7	breq12d 3950	. . . 4 ⊢ (𝑤 = 1 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) ↔ (abs‘(𝐹‘1)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1)))))
9	8	imbi2d 229	. . 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 5434	. . . . 5 ⊢ (𝑤 = 𝑘 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑘)))
11		oveq1 5789	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑤 − 1) = (𝑘 − 1))
12	11	oveq2d 5798	. . . . . 6 ⊢ (𝑤 = 𝑘 → (𝐴↑(𝑤 − 1)) = (𝐴↑(𝑘 − 1)))
13	12	oveq2d 5798	. . . . 5 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))))
14	10, 13	breq12d 3950	. . . 4 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) ↔ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))))
15	14	imbi2d 229	. . 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 5434	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝑘 + 1))))
17		oveq1 5789	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (𝑤 − 1) = ((𝑘 + 1) − 1))
18	17	oveq2d 5798	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (𝐴↑(𝑤 − 1)) = (𝐴↑((𝑘 + 1) − 1)))
19	18	oveq2d 5798	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))
20	16, 19	breq12d 3950	. . . 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 229	. . 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 5434	. . . . 5 ⊢ (𝑤 = 𝑁 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑁)))
23		oveq1 5789	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 − 1) = (𝑁 − 1))
24	23	oveq2d 5798	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝐴↑(𝑤 − 1)) = (𝐴↑(𝑁 − 1)))
25	24	oveq2d 5798	. . . . 5 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1))))
26	22, 25	breq12d 3950	. . . 4 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) ↔ (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1)))))
27	26	imbi2d 229	. . 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 5429	. . . . . . . . 9 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
29	28	eleq1d 2209	. . . . . . . 8 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
30		cvgratnn.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
31	30	ralrimiva 2508	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
32		1nn 8755	. . . . . . . . 9 ⊢ 1 ∈ ℕ
33	32	a1i 9	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℕ)
34	29, 31, 33	rspcdva 2798	. . . . . . 7 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
35	34	abscld 10985	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
36	35	leidd 8300	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘1)) ≤ (abs‘(𝐹‘1)))
37		1m1e0 8813	. . . . . . . . . 10 ⊢ (1 − 1) = 0
38	37	a1i 9	. . . . . . . . 9 ⊢ (𝜑 → (1 − 1) = 0)
39	38	oveq2d 5798	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(1 − 1)) = (𝐴↑0))
40		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
41	40	recnd 7818	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
42	41	exp0d 10449	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑0) = 1)
43	39, 42	eqtrd 2173	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(1 − 1)) = 1)
44	43	oveq2d 5798	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))) = ((abs‘(𝐹‘1)) · 1))
45	35	recnd 7818	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℂ)
46	45	mulid1d 7807	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · 1) = (abs‘(𝐹‘1)))
47	44, 46	eqtrd 2173	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))) = (abs‘(𝐹‘1)))
48	36, 47	breqtrrd 3964	. . . 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 9386	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
51	30	abscld 10985	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
52	35	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘1)) ∈ ℝ)
53	40	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
54		nnm1nn0 9042	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
55	54	adantl 275	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 − 1) ∈ ℕ₀)
56	53, 55	reexpcld 10472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑘 − 1)) ∈ ℝ)
57	52, 56	remulcld 7820	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) ∈ ℝ)
58		0red 7791	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ∈ ℝ)
59		cvgratnn.gt0	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝐴)
60	59	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝐴)
61	58, 53, 60	ltled 7905	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
62		lemul2a 8641	. . . . . . . . . 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 114	. . . . . . . . 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 1221	. . . . . . . 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 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
66	45	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘1)) ∈ ℂ)
67	56	recnd 7818	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑘 − 1)) ∈ ℂ)
68	65, 66, 67	mul12d 7938	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))) = ((abs‘(𝐹‘1)) · (𝐴 · (𝐴↑(𝑘 − 1)))))
69	65, 55	expp1d 10456	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 − 1) + 1)) = ((𝐴↑(𝑘 − 1)) · 𝐴))
70		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
71	70	nncnd 8758	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
72		1cnd 7806	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
73	71, 72, 72	addsubd 8118	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) − 1) = ((𝑘 − 1) + 1))
74	73	oveq2d 5798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 + 1) − 1)) = (𝐴↑((𝑘 − 1) + 1)))
75	65, 67	mulcomd 7811	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (𝐴↑(𝑘 − 1))) = ((𝐴↑(𝑘 − 1)) · 𝐴))
76	69, 74, 75	3eqtr4rd 2184	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (𝐴↑(𝑘 − 1))) = (𝐴↑((𝑘 + 1) − 1)))
77	76	oveq2d 5798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴 · (𝐴↑(𝑘 − 1)))) = ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))
78	68, 77	eqtrd 2173	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))) = ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))
79	78	breq2d 3949	. . . . . . . 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 148	. . . . . . 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 5429	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
83	82	eleq1d 2209	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
84		fveq2 5429	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
85	84	eleq1d 2209	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ))
86	85	cbvralv 2657	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
87	31, 86	sylib 121	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
88	87	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
89		peano2nn 8756	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
90	89	adantl 275	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
91	83, 88, 90	rspcdva 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
92	91	abscld 10985	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
93	53, 51	remulcld 7820	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
94		elnnuz 9386	. . . . . . . . . . . . . 14 ⊢ ((𝑘 + 1) ∈ ℕ ↔ (𝑘 + 1) ∈ (ℤ_≥‘1))
95	89, 94	sylib 121	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ (ℤ_≥‘1))
96	95	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ_≥‘1))
97		uznn0sub 9381	. . . . . . . . . . . 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 10472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 + 1) − 1)) ∈ ℝ)
100	52, 99	remulcld 7820	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))) ∈ ℝ)
101		letr 7871	. . . . . . . . 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 1217	. . . . . . . 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 426	. . . . . . 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 286	. . . . 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 115	. . . 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 9410	. 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 103 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 ∀wral 2417 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 1c1 7645 + caddc 7647 · cmul 7649 < clt 7824 ≤ cle 7825 − cmin 7957 ℕcn 8744 ℕ₀cn0 9001 ℤcz 9078 ℤ_≥cuz 9350 ↑cexp 10323 abscabs 10801

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-rp 9471 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803

This theorem is referenced by: cvgratnnlemfm 11330

W3C validator