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Theorem cvgratnnlemmn 11294

Description: Lemma for cvgratnn 11300. (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))))
cvgratnn.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
cvgratnn.n	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))

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

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

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

Step	Hyp	Ref	Expression
1		cvgratnn.n	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		2fveq3 5426	. . . . 5 ⊢ (𝑤 = 𝑀 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑀)))
3		oveq1 5781	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑤 − 𝑀) = (𝑀 − 𝑀))
4	3	oveq2d 5790	. . . . . 6 ⊢ (𝑤 = 𝑀 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑀 − 𝑀)))
5	4	oveq2d 5790	. . . . 5 ⊢ (𝑤 = 𝑀 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))))
6	2, 5	breq12d 3942	. . . 4 ⊢ (𝑤 = 𝑀 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀)))))
7	6	imbi2d 229	. . 3 ⊢ (𝑤 = 𝑀 → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀)))) ↔ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))))))
8		2fveq3 5426	. . . . 5 ⊢ (𝑤 = 𝑘 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑘)))
9		oveq1 5781	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑤 − 𝑀) = (𝑘 − 𝑀))
10	9	oveq2d 5790	. . . . . 6 ⊢ (𝑤 = 𝑘 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑘 − 𝑀)))
11	10	oveq2d 5790	. . . . 5 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))))
12	8, 11	breq12d 3942	. . . 4 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))))
13	12	imbi2d 229	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀)))) ↔ (𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))))))
14		2fveq3 5426	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝑘 + 1))))
15		oveq1 5781	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (𝑤 − 𝑀) = ((𝑘 + 1) − 𝑀))
16	15	oveq2d 5790	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑((𝑘 + 1) − 𝑀)))
17	16	oveq2d 5790	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))
18	14, 17	breq12d 3942	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))))
19	18	imbi2d 229	. . 3 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀)))) ↔ (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))))
20		2fveq3 5426	. . . . 5 ⊢ (𝑤 = 𝑁 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑁)))
21		oveq1 5781	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 − 𝑀) = (𝑁 − 𝑀))
22	21	oveq2d 5790	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑁 − 𝑀)))
23	22	oveq2d 5790	. . . . 5 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀))))
24	20, 23	breq12d 3942	. . . 4 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀)))))
25	24	imbi2d 229	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀)))) ↔ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀))))))
26		fveq2 5421	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
27	26	eleq1d 2208	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
28		cvgratnn.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
29	28	ralrimiva 2505	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
30		cvgratnn.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
31	27, 29, 30	rspcdva 2794	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
32	31	abscld 10953	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
33	32	leidd 8276	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ (abs‘(𝐹‘𝑀)))
34	30	nncnd 8734	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
35	34	subidd 8061	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 − 𝑀) = 0)
36	35	oveq2d 5790	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑀 − 𝑀)) = (𝐴↑0))
37		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
38	37	recnd 7794	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
39	38	exp0d 10418	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑0) = 1)
40	36, 39	eqtrd 2172	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(𝑀 − 𝑀)) = 1)
41	40	oveq2d 5790	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · 1))
42	32	recnd 7794	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℂ)
43	42	mulid1d 7783	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · 1) = (abs‘(𝐹‘𝑀)))
44	41, 43	eqtrd 2172	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))) = (abs‘(𝐹‘𝑀)))
45	33, 44	breqtrrd 3956	. . . 4 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))))
46	45	a1i 9	. . 3 ⊢ (𝑀 ∈ ℤ → (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀)))))
47		eluznn 9394	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
48	30, 47	sylan 281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
49	48, 28	syldan 280	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
50	49	abscld 10953	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
51	32	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑀)) ∈ ℝ)
52	37	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝐴 ∈ ℝ)
53		uznn0sub 9357	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 − 𝑀) ∈ ℕ₀)
54	53	adantl 275	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝑘 − 𝑀) ∈ ℕ₀)
55	52, 54	reexpcld 10441	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑(𝑘 − 𝑀)) ∈ ℝ)
56	51, 55	remulcld 7796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) ∈ ℝ)
57		0red 7767	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 ∈ ℝ)
58		cvgratnn.gt0	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐴)
59	58	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 < 𝐴)
60	57, 52, 59	ltled 7881	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 ≤ 𝐴)
61		lemul2a 8617	. . . . . . . . 9 ⊢ ((((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))))
62	61	ex 114	. . . . . . . 8 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))))))
63	50, 56, 52, 60, 62	syl112anc 1220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))))))
64	38	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝐴 ∈ ℂ)
65	42	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑀)) ∈ ℂ)
66	55	recnd 7794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑(𝑘 − 𝑀)) ∈ ℂ)
67	64, 65, 66	mul12d 7914	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴 · (𝐴↑(𝑘 − 𝑀)))))
68	64, 54	expp1d 10425	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑((𝑘 − 𝑀) + 1)) = ((𝐴↑(𝑘 − 𝑀)) · 𝐴))
69	48	nncnd 8734	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℂ)
70		1cnd 7782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℂ)
71		eluzel2 9331	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
72	71	adantl 275	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ ℤ)
73	72	zcnd 9174	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ ℂ)
74	69, 70, 73	addsubd 8094	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝑘 + 1) − 𝑀) = ((𝑘 − 𝑀) + 1))
75	74	oveq2d 5790	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑((𝑘 + 1) − 𝑀)) = (𝐴↑((𝑘 − 𝑀) + 1)))
76	64, 66	mulcomd 7787	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · (𝐴↑(𝑘 − 𝑀))) = ((𝐴↑(𝑘 − 𝑀)) · 𝐴))
77	68, 75, 76	3eqtr4rd 2183	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · (𝐴↑(𝑘 − 𝑀))) = (𝐴↑((𝑘 + 1) − 𝑀)))
78	77	oveq2d 5790	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴 · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))
79	67, 78	eqtrd 2172	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))
80	79	breq2d 3941	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) ↔ (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))))
81	63, 80	sylibd 148	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))))
82		cvgratnn.7	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
83	48, 82	syldan 280	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
84		fveq2 5421	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
85	84	eleq1d 2208	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
86		fveq2 5421	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
87	86	eleq1d 2208	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ))
88	87	cbvralv 2654	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
89	29, 88	sylib 121	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
90	89	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
91	48	peano2nnd 8735	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝑘 + 1) ∈ ℕ)
92	85, 90, 91	rspcdva 2794	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
93	92	abscld 10953	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
94	52, 50	remulcld 7796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
95		peano2uz 9378	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
96	95	adantl 275	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
97		uznn0sub 9357	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ (ℤ_≥‘𝑀) → ((𝑘 + 1) − 𝑀) ∈ ℕ₀)
98	96, 97	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝑘 + 1) − 𝑀) ∈ ℕ₀)
99	52, 98	reexpcld 10441	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑((𝑘 + 1) − 𝑀)) ∈ ℝ)
100	51, 99	remulcld 7796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))) ∈ ℝ)
101		letr 7847	. . . . . . . 8 ⊢ (((abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ ∧ (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ ∧ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))) ∈ ℝ) → (((abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))) ∧ (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))))
102	93, 94, 100, 101	syl3anc 1216	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))) ∧ (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))))
103	83, 102	mpand 425	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))))
104	81, 103	syld 45	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))))
105	104	expcom 115	. . . 4 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝜑 → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))))
106	105	a2d 26	. . 3 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → ((𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) → (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))))
107	7, 13, 19, 25, 46, 106	uzind4 9383	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀)))))
108	1, 107	mpcom 36	1 ⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∀wral 2416 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 ℝcr 7619 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 < clt 7800 ≤ cle 7801 − cmin 7933 ℕcn 8720 ℕ₀cn0 8977 ℤcz 9054 ℤ_≥cuz 9326 ↑cexp 10292 abscabs 10769

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771

This theorem is referenced by: cvgratnnlemabsle 11296

W3C validator