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Theorem cvgratnnlemmn 12031

Description: Lemma for cvgratnn 12037. (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 5631	. . . . 5 ⊢ (𝑤 = 𝑀 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑀)))
3		oveq1 6007	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑤 − 𝑀) = (𝑀 − 𝑀))
4	3	oveq2d 6016	. . . . . 6 ⊢ (𝑤 = 𝑀 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑀 − 𝑀)))
5	4	oveq2d 6016	. . . . 5 ⊢ (𝑤 = 𝑀 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))))
6	2, 5	breq12d 4095	. . . 4 ⊢ (𝑤 = 𝑀 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀)))))
7	6	imbi2d 230	. . 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 5631	. . . . 5 ⊢ (𝑤 = 𝑘 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑘)))
9		oveq1 6007	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑤 − 𝑀) = (𝑘 − 𝑀))
10	9	oveq2d 6016	. . . . . 6 ⊢ (𝑤 = 𝑘 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑘 − 𝑀)))
11	10	oveq2d 6016	. . . . 5 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))))
12	8, 11	breq12d 4095	. . . 4 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))))
13	12	imbi2d 230	. . 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 5631	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝑘 + 1))))
15		oveq1 6007	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (𝑤 − 𝑀) = ((𝑘 + 1) − 𝑀))
16	15	oveq2d 6016	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑((𝑘 + 1) − 𝑀)))
17	16	oveq2d 6016	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))
18	14, 17	breq12d 4095	. . . 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 230	. . 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 5631	. . . . 5 ⊢ (𝑤 = 𝑁 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑁)))
21		oveq1 6007	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 − 𝑀) = (𝑁 − 𝑀))
22	21	oveq2d 6016	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑁 − 𝑀)))
23	22	oveq2d 6016	. . . . 5 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀))))
24	20, 23	breq12d 4095	. . . 4 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀)))))
25	24	imbi2d 230	. . 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 5626	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
27	26	eleq1d 2298	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
28		cvgratnn.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
29	28	ralrimiva 2603	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
30		cvgratnn.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
31	27, 29, 30	rspcdva 2912	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
32	31	abscld 11687	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
33	32	leidd 8657	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ (abs‘(𝐹‘𝑀)))
34	30	nncnd 9120	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
35	34	subidd 8441	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 − 𝑀) = 0)
36	35	oveq2d 6016	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑀 − 𝑀)) = (𝐴↑0))
37		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
38	37	recnd 8171	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
39	38	exp0d 10884	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑0) = 1)
40	36, 39	eqtrd 2262	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(𝑀 − 𝑀)) = 1)
41	40	oveq2d 6016	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · 1))
42	32	recnd 8171	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℂ)
43	42	mulridd 8159	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · 1) = (abs‘(𝐹‘𝑀)))
44	41, 43	eqtrd 2262	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))) = (abs‘(𝐹‘𝑀)))
45	33, 44	breqtrrd 4110	. . . 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 9791	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
48	30, 47	sylan 283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
49	48, 28	syldan 282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
50	49	abscld 11687	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
51	32	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑀)) ∈ ℝ)
52	37	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝐴 ∈ ℝ)
53		uznn0sub 9750	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 − 𝑀) ∈ ℕ₀)
54	53	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝑘 − 𝑀) ∈ ℕ₀)
55	52, 54	reexpcld 10907	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑(𝑘 − 𝑀)) ∈ ℝ)
56	51, 55	remulcld 8173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) ∈ ℝ)
57		0red 8143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 ∈ ℝ)
58		cvgratnn.gt0	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐴)
59	58	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 < 𝐴)
60	57, 52, 59	ltled 8261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 ≤ 𝐴)
61		lemul2a 9002	. . . . . . . . 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 115	. . . . . . . 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 1275	. . . . . . 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 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝐴 ∈ ℂ)
65	42	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑀)) ∈ ℂ)
66	55	recnd 8171	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑(𝑘 − 𝑀)) ∈ ℂ)
67	64, 65, 66	mul12d 8294	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴 · (𝐴↑(𝑘 − 𝑀)))))
68	64, 54	expp1d 10891	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑((𝑘 − 𝑀) + 1)) = ((𝐴↑(𝑘 − 𝑀)) · 𝐴))
69	48	nncnd 9120	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℂ)
70		1cnd 8158	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℂ)
71		eluzel2 9723	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
72	71	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ ℤ)
73	72	zcnd 9566	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ ℂ)
74	69, 70, 73	addsubd 8474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝑘 + 1) − 𝑀) = ((𝑘 − 𝑀) + 1))
75	74	oveq2d 6016	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑((𝑘 + 1) − 𝑀)) = (𝐴↑((𝑘 − 𝑀) + 1)))
76	64, 66	mulcomd 8164	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · (𝐴↑(𝑘 − 𝑀))) = ((𝐴↑(𝑘 − 𝑀)) · 𝐴))
77	68, 75, 76	3eqtr4rd 2273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · (𝐴↑(𝑘 − 𝑀))) = (𝐴↑((𝑘 + 1) − 𝑀)))
78	77	oveq2d 6016	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴 · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))
79	67, 78	eqtrd 2262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))
80	79	breq2d 4094	. . . . . . 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 149	. . . . . 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 282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
84		fveq2 5626	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
85	84	eleq1d 2298	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
86		fveq2 5626	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
87	86	eleq1d 2298	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ))
88	87	cbvralv 2765	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
89	29, 88	sylib 122	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
90	89	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ)
91	48	peano2nnd 9121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝑘 + 1) ∈ ℕ)
92	85, 90, 91	rspcdva 2912	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
93	92	abscld 11687	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
94	52, 50	remulcld 8173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
95		peano2uz 9774	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
96	95	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
97		uznn0sub 9750	. . . . . . . . . . 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 10907	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑((𝑘 + 1) − 𝑀)) ∈ ℝ)
100	51, 99	remulcld 8173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))) ∈ ℝ)
101		letr 8225	. . . . . . . 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 1271	. . . . . . 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 429	. . . . . 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 116	. . . 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 9779	. 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 104 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ∀wral 2508 class class class wbr 4082 ‘cfv 5317 (class class class)co 6000 ℂcc 7993 ℝcr 7994 0cc0 7995 1c1 7996 + caddc 7998 · cmul 8000 < clt 8177 ≤ cle 8178 − cmin 8313 ℕcn 9106 ℕ₀cn0 9365 ℤcz 9442 ℤ_≥cuz 9718 ↑cexp 10755 abscabs 11503

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-frec 6535 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-n0 9366 df-z 9443 df-uz 9719 df-rp 9846 df-seqfrec 10665 df-exp 10756 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505

This theorem is referenced by: cvgratnnlemabsle 12033

W3C validator