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Theorem cvgratnnlemmn 11707

Description: Lemma for cvgratnn 11713. (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 5566	. . . . 5 ⊢ (𝑤 = 𝑀 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑀)))
3		oveq1 5932	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑤 − 𝑀) = (𝑀 − 𝑀))
4	3	oveq2d 5941	. . . . . 6 ⊢ (𝑤 = 𝑀 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑀 − 𝑀)))
5	4	oveq2d 5941	. . . . 5 ⊢ (𝑤 = 𝑀 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))))
6	2, 5	breq12d 4047	. . . 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 5566	. . . . 5 ⊢ (𝑤 = 𝑘 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑘)))
9		oveq1 5932	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑤 − 𝑀) = (𝑘 − 𝑀))
10	9	oveq2d 5941	. . . . . 6 ⊢ (𝑤 = 𝑘 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑘 − 𝑀)))
11	10	oveq2d 5941	. . . . 5 ⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))))
12	8, 11	breq12d 4047	. . . 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 5566	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝑘 + 1))))
15		oveq1 5932	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (𝑤 − 𝑀) = ((𝑘 + 1) − 𝑀))
16	15	oveq2d 5941	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑((𝑘 + 1) − 𝑀)))
17	16	oveq2d 5941	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))
18	14, 17	breq12d 4047	. . . 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 5566	. . . . 5 ⊢ (𝑤 = 𝑁 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑁)))
21		oveq1 5932	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑤 − 𝑀) = (𝑁 − 𝑀))
22	21	oveq2d 5941	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑁 − 𝑀)))
23	22	oveq2d 5941	. . . . 5 ⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀))))
24	20, 23	breq12d 4047	. . . 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 5561	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
27	26	eleq1d 2265	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
28		cvgratnn.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
29	28	ralrimiva 2570	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ)
30		cvgratnn.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
31	27, 29, 30	rspcdva 2873	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
32	31	abscld 11363	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ)
33	32	leidd 8558	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ (abs‘(𝐹‘𝑀)))
34	30	nncnd 9021	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
35	34	subidd 8342	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 − 𝑀) = 0)
36	35	oveq2d 5941	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑(𝑀 − 𝑀)) = (𝐴↑0))
37		cvgratnn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
38	37	recnd 8072	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
39	38	exp0d 10776	. . . . . . . 8 ⊢ (𝜑 → (𝐴↑0) = 1)
40	36, 39	eqtrd 2229	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(𝑀 − 𝑀)) = 1)
41	40	oveq2d 5941	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · 1))
42	32	recnd 8072	. . . . . . 7 ⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℂ)
43	42	mulridd 8060	. . . . . 6 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · 1) = (abs‘(𝐹‘𝑀)))
44	41, 43	eqtrd 2229	. . . . 5 ⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))) = (abs‘(𝐹‘𝑀)))
45	33, 44	breqtrrd 4062	. . . 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 9691	. . . . . . . . . . 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 11363	. . . . . . . 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 9650	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 − 𝑀) ∈ ℕ₀)
54	53	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝑘 − 𝑀) ∈ ℕ₀)
55	52, 54	reexpcld 10799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑(𝑘 − 𝑀)) ∈ ℝ)
56	51, 55	remulcld 8074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) ∈ ℝ)
57		0red 8044	. . . . . . . . 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 8162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 ≤ 𝐴)
61		lemul2a 8903	. . . . . . . . 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 1253	. . . . . . 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 8072	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑(𝑘 − 𝑀)) ∈ ℂ)
67	64, 65, 66	mul12d 8195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴 · (𝐴↑(𝑘 − 𝑀)))))
68	64, 54	expp1d 10783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑((𝑘 − 𝑀) + 1)) = ((𝐴↑(𝑘 − 𝑀)) · 𝐴))
69	48	nncnd 9021	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℂ)
70		1cnd 8059	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℂ)
71		eluzel2 9623	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
72	71	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ ℤ)
73	72	zcnd 9466	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ ℂ)
74	69, 70, 73	addsubd 8375	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝑘 + 1) − 𝑀) = ((𝑘 − 𝑀) + 1))
75	74	oveq2d 5941	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑((𝑘 + 1) − 𝑀)) = (𝐴↑((𝑘 − 𝑀) + 1)))
76	64, 66	mulcomd 8065	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · (𝐴↑(𝑘 − 𝑀))) = ((𝐴↑(𝑘 − 𝑀)) · 𝐴))
77	68, 75, 76	3eqtr4rd 2240	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · (𝐴↑(𝑘 − 𝑀))) = (𝐴↑((𝑘 + 1) − 𝑀)))
78	77	oveq2d 5941	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴 · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))
79	67, 78	eqtrd 2229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))
80	79	breq2d 4046	. . . . . . 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 5561	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
85	84	eleq1d 2265	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
86		fveq2 5561	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
87	86	eleq1d 2265	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ))
88	87	cbvralv 2729	. . . . . . . . . . . 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 9022	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝑘 + 1) ∈ ℕ)
92	85, 90, 91	rspcdva 2873	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
93	92	abscld 11363	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
94	52, 50	remulcld 8074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ)
95		peano2uz 9674	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
96	95	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
97		uznn0sub 9650	. . . . . . . . . . 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 10799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐴↑((𝑘 + 1) − 𝑀)) ∈ ℝ)
100	51, 99	remulcld 8074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))) ∈ ℝ)
101		letr 8126	. . . . . . . 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 1249	. . . . . . 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 9679	. 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 980 = wceq 1364 ∈ wcel 2167 ∀wral 2475 class class class wbr 4034 ‘cfv 5259 (class class class)co 5925 ℂcc 7894 ℝcr 7895 0cc0 7896 1c1 7897 + caddc 7899 · cmul 7901 < clt 8078 ≤ cle 8079 − cmin 8214 ℕcn 9007 ℕ₀cn0 9266 ℤcz 9343 ℤ_≥cuz 9618 ↑cexp 10647 abscabs 11179

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-rp 9746 df-seqfrec 10557 df-exp 10648 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181

This theorem is referenced by: cvgratnnlemabsle 11709

W3C validator