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Theorem cvgratnnlemnexp 12210

Description: Lemma for cvgratnn 12217. (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 9890	. . 3 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	eleqtrdi 2325	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		2fveq3 5675	. . . . 5 ⊢ (𝑤 = 1 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘1)))
5		oveq1 6057	. . . . . . 7 ⊢ (𝑤 = 1 → (𝑤 − 1) = (1 − 1))
6	5	oveq2d 6066	. . . . . 6 ⊢ (𝑤 = 1 → (𝐴↑(𝑤 − 1)) = (𝐴↑(1 − 1)))
7	6	oveq2d 6066	. . . . 5 ⊢ (𝑤 = 1 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))))
8	4, 7	breq12d 4122	. . . 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 5675	. . . . 5 ⊢ (𝑤 = 𝑘 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑘)))
11		oveq1 6057	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑤 − 1) = (𝑘 − 1))
12	11	oveq2d 6066	. . . . . 6 ⊢ (𝑤 = 𝑘 → (𝐴↑(𝑤 − 1)) = (𝐴↑(𝑘 − 1)))
13	12	oveq2d 6066	. . . . 5 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))))
14	10, 13	breq12d 4122	. . . 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 5675	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝑘 + 1))))
17		oveq1 6057	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (𝑤 − 1) = ((𝑘 + 1) − 1))
18	17	oveq2d 6066	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (𝐴↑(𝑤 − 1)) = (𝐴↑((𝑘 + 1) − 1)))
19	18	oveq2d 6066	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))
20	16, 19	breq12d 4122	. . . 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 5675	. . . . 5 ⊢ (𝑤 = 𝑁 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑁)))
23		oveq1 6057	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 − 1) = (𝑁 − 1))
24	23	oveq2d 6066	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝐴↑(𝑤 − 1)) = (𝐴↑(𝑁 − 1)))
25	24	oveq2d 6066	. . . . 5 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘1)) · (𝐴↑(𝑤 − 1))) = ((abs‘(𝐹‘1)) · (𝐴↑(𝑁 − 1))))
26	22, 25	breq12d 4122	. . . 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 5670	. . . . . . . . 9 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
29	28	eleq1d 2301	. . . . . . . 8 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘1) ∈ ℂ))
30		cvgratnn.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
31	30	ralrimiva 2615	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
32		1nn 9248	. . . . . . . . 9 ⊢ 1 ∈ ℕ
33	32	a1i 9	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℕ)
34	29, 31, 33	rspcdva 2926	. . . . . . 7 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
35	34	abscld 11866	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℝ)
36	35	leidd 8788	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘1)) ≤ (abs‘(𝐹‘1)))
37		1m1e0 9306	. . . . . . . . . 10 ⊢ (1 − 1) = 0
38	37	a1i 9	. . . . . . . . 9 ⊢ (𝜑 → (1 − 1) = 0)
39	38	oveq2d 6066	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(1 − 1)) = (𝐴↑0))
40		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
41	40	recnd 8302	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
42	41	exp0d 11029	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑0) = 1)
43	39, 42	eqtrd 2265	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(1 − 1)) = 1)
44	43	oveq2d 6066	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))) = ((abs‘(𝐹‘1)) · 1))
45	35	recnd 8302	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘1)) ∈ ℂ)
46	45	mulridd 8291	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · 1) = (abs‘(𝐹‘1)))
47	44, 46	eqtrd 2265	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘1)) · (𝐴↑(1 − 1))) = (abs‘(𝐹‘1)))
48	36, 47	breqtrrd 4137	. . . 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 9891	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
51	30	abscld 11866	. . . . . . . . 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 9537	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
55	54	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 − 1) ∈ ℕ₀)
56	53, 55	reexpcld 11052	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑘 − 1)) ∈ ℝ)
57	52, 56	remulcld 8304	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1))) ∈ ℝ)
58		0red 8275	. . . . . . . . . 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 8392	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
62		lemul2a 9133	. . . . . . . . . 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 1278	. . . . . . . 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 8302	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑘 − 1)) ∈ ℂ)
68	65, 66, 67	mul12d 8425	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))) = ((abs‘(𝐹‘1)) · (𝐴 · (𝐴↑(𝑘 − 1)))))
69	65, 55	expp1d 11036	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 − 1) + 1)) = ((𝐴↑(𝑘 − 1)) · 𝐴))
70		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
71	70	nncnd 9251	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
72		1cnd 8290	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
73	71, 72, 72	addsubd 8605	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) − 1) = ((𝑘 − 1) + 1))
74	73	oveq2d 6066	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 + 1) − 1)) = (𝐴↑((𝑘 − 1) + 1)))
75	65, 67	mulcomd 8295	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (𝐴↑(𝑘 − 1))) = ((𝐴↑(𝑘 − 1)) · 𝐴))
76	69, 74, 75	3eqtr4rd 2276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (𝐴↑(𝑘 − 1))) = (𝐴↑((𝑘 + 1) − 1)))
77	76	oveq2d 6066	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴 · (𝐴↑(𝑘 − 1)))) = ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))
78	68, 77	eqtrd 2265	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · ((abs‘(𝐹‘1)) · (𝐴↑(𝑘 − 1)))) = ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))))
79	78	breq2d 4121	. . . . . . . 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 5670	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
83	82	eleq1d 2301	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
84		fveq2 5670	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
85	84	eleq1d 2301	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ))
86	85	cbvralv 2778	. . . . . . . . . . . . 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 9249	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
90	89	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
91	83, 88, 90	rspcdva 2926	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
92	91	abscld 11866	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
93	53, 51	remulcld 8304	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
94		elnnuz 9891	. . . . . . . . . . . . . 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 9886	. . . . . . . . . . . 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 11052	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑘 + 1) − 1)) ∈ ℝ)
100	52, 99	remulcld 8304	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(𝐹‘1)) · (𝐴↑((𝑘 + 1) − 1))) ∈ ℝ)
101		letr 8356	. . . . . . . . 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 1274	. . . . . . . 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 429	. . . . . . 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 9920	. 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 1005 = wceq 1398 ∈ wcel 2203 ∀wral 2520 class class class wbr 4109 ‘cfv 5352 (class class class)co 6050 ℂcc 8125 ℝcr 8126 0cc0 8127 1c1 8128 + caddc 8130 · cmul 8132 < clt 8308 ≤ cle 8309 − cmin 8444 ℕcn 9237 ℕ₀cn0 9496 ℤcz 9577 ℤ_≥cuz 9853 ↑cexp 10900 abscabs 11682

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-rp 9987 df-seqfrec 10810 df-exp 10901 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684

This theorem is referenced by: cvgratnnlemfm 12215

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