stoweidlem7 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem7		Structured version Visualization version GIF version

Theorem stoweidlem7 45021

Description: This lemma is used to prove that q_n as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91, (at the top of page 91), is such that q_n < ε on 𝑇 ∖ 𝑈, and q_n > 1 - ε on 𝑉. Here it is proven that, for 𝑛 large enough, 1-(k*δ/2)^n > 1 - ε , and 1/(k*δ)^n < ε. The variable 𝐴 is used to represent (k*δ) in the paper, and 𝐵 is used to represent (k*δ/2). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem7.1	⊢ 𝐹 = (𝑖 ∈ ℕ₀ ↦ ((1 / 𝐴)↑𝑖))
stoweidlem7.2	⊢ 𝐺 = (𝑖 ∈ ℕ₀ ↦ (𝐵↑𝑖))
stoweidlem7.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
stoweidlem7.4	⊢ (𝜑 → 1 < 𝐴)
stoweidlem7.5	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
stoweidlem7.6	⊢ (𝜑 → 𝐵 < 1)
stoweidlem7.7	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)

**Assertion**
Ref	Expression
stoweidlem7	⊢ (𝜑 → ∃𝑛 ∈ ℕ ((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ (1 / (𝐴↑𝑛)) < 𝐸))

Distinct variable groups: 𝑖,𝑛,𝐴 𝐵,𝑖,𝑛 𝑖,𝐸,𝑛 𝜑,𝑖,𝑛 𝑛,𝐹 𝑛,𝐺 Allowed substitution hints: 𝐹(𝑖) 𝐺(𝑖)

Proof of Theorem stoweidlem7 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12869	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12597	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
3		stoweidlem7.7	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
4		stoweidlem7.2	. . . . . 6 ⊢ 𝐺 = (𝑖 ∈ ℕ₀ ↦ (𝐵↑𝑖))
5		oveq2 7419	. . . . . 6 ⊢ (𝑖 = 𝑘 → (𝐵↑𝑖) = (𝐵↑𝑘))
6		nnnn0 12483	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
7	6	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ₀)
8		stoweidlem7.5	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
9	8	rpcnd 13022	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
10	9	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℂ)
11	10, 7	expcld 14115	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵↑𝑘) ∈ ℂ)
12	4, 5, 7, 11	fvmptd3 7020	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) = (𝐵↑𝑘))
13		1red 11219	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
14	13	renegcld 11645	. . . . . . . . 9 ⊢ (𝜑 → -1 ∈ ℝ)
15		0red 11221	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
16	8	rpred 13020	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
17		neg1lt0 12333	. . . . . . . . . 10 ⊢ -1 < 0
18	17	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -1 < 0)
19	8	rpgt0d 13023	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝐵)
20	14, 15, 16, 18, 19	lttrd 11379	. . . . . . . 8 ⊢ (𝜑 → -1 < 𝐵)
21		stoweidlem7.6	. . . . . . . 8 ⊢ (𝜑 → 𝐵 < 1)
22	16, 13	absltd 15380	. . . . . . . 8 ⊢ (𝜑 → ((abs‘𝐵) < 1 ↔ (-1 < 𝐵 ∧ 𝐵 < 1)))
23	20, 21, 22	mpbir2and 709	. . . . . . 7 ⊢ (𝜑 → (abs‘𝐵) < 1)
24	9, 23	expcnv 15814	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ ℕ₀ ↦ (𝐵↑𝑖)) ⇝ 0)
25	4, 24	eqbrtrid 5182	. . . . 5 ⊢ (𝜑 → 𝐺 ⇝ 0)
26	1, 2, 3, 12, 25	climi 15458	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸))
27		r19.26 3109	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸) ↔ (∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐵↑𝑘) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)(abs‘((𝐵↑𝑘) − 0)) < 𝐸))
28	27	simprbi 495	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(abs‘((𝐵↑𝑘) − 0)) < 𝐸)
29	28	ad2antlr 723	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(abs‘((𝐵↑𝑘) − 0)) < 𝐸)
30		oveq2 7419	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑖 → (𝐵↑𝑘) = (𝐵↑𝑖))
31	30	oveq1d 7426	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑖 → ((𝐵↑𝑘) − 0) = ((𝐵↑𝑖) − 0))
32	31	fveq2d 6894	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (abs‘((𝐵↑𝑘) − 0)) = (abs‘((𝐵↑𝑖) − 0)))
33	32	breq1d 5157	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → ((abs‘((𝐵↑𝑘) − 0)) < 𝐸 ↔ (abs‘((𝐵↑𝑖) − 0)) < 𝐸))
34	33	rspccva 3610	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑛)(abs‘((𝐵↑𝑘) − 0)) < 𝐸 ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → (abs‘((𝐵↑𝑖) − 0)) < 𝐸)
35	29, 34	sylancom 586	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → (abs‘((𝐵↑𝑖) − 0)) < 𝐸)
36		simplll 771	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → 𝜑)
37	36, 8	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → 𝐵 ∈ ℝ⁺)
38	37	rpred 13020	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → 𝐵 ∈ ℝ)
39		simpllr 772	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ)
40		nnnn0 12483	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
41	39, 40	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ₀)
42		eluznn0 12905	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → 𝑖 ∈ ℕ₀)
43	41, 42	sylancom 586	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → 𝑖 ∈ ℕ₀)
44	38, 43	reexpcld 14132	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → (𝐵↑𝑖) ∈ ℝ)
45		rpre 12986	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ ℝ⁺ → 𝐸 ∈ ℝ)
46	36, 3, 45	3syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → 𝐸 ∈ ℝ)
47		recn 11202	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵↑𝑖) ∈ ℝ → (𝐵↑𝑖) ∈ ℂ)
48	47	subid1d 11564	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵↑𝑖) ∈ ℝ → ((𝐵↑𝑖) − 0) = (𝐵↑𝑖))
49	48	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝐵↑𝑖) ∈ ℝ ∧ 𝐸 ∈ ℝ) → ((𝐵↑𝑖) − 0) = (𝐵↑𝑖))
50	49	fveq2d 6894	. . . . . . . . . . . . . 14 ⊢ (((𝐵↑𝑖) ∈ ℝ ∧ 𝐸 ∈ ℝ) → (abs‘((𝐵↑𝑖) − 0)) = (abs‘(𝐵↑𝑖)))
51	50	breq1d 5157	. . . . . . . . . . . . 13 ⊢ (((𝐵↑𝑖) ∈ ℝ ∧ 𝐸 ∈ ℝ) → ((abs‘((𝐵↑𝑖) − 0)) < 𝐸 ↔ (abs‘(𝐵↑𝑖)) < 𝐸))
52		abslt 15265	. . . . . . . . . . . . 13 ⊢ (((𝐵↑𝑖) ∈ ℝ ∧ 𝐸 ∈ ℝ) → ((abs‘(𝐵↑𝑖)) < 𝐸 ↔ (-𝐸 < (𝐵↑𝑖) ∧ (𝐵↑𝑖) < 𝐸)))
53	51, 52	bitrd 278	. . . . . . . . . . . 12 ⊢ (((𝐵↑𝑖) ∈ ℝ ∧ 𝐸 ∈ ℝ) → ((abs‘((𝐵↑𝑖) − 0)) < 𝐸 ↔ (-𝐸 < (𝐵↑𝑖) ∧ (𝐵↑𝑖) < 𝐸)))
54	44, 46, 53	syl2anc 582	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → ((abs‘((𝐵↑𝑖) − 0)) < 𝐸 ↔ (-𝐸 < (𝐵↑𝑖) ∧ (𝐵↑𝑖) < 𝐸)))
55	35, 54	mpbid 231	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → (-𝐸 < (𝐵↑𝑖) ∧ (𝐵↑𝑖) < 𝐸))
56	55	simprd 494	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → (𝐵↑𝑖) < 𝐸)
57		eluznn 12906	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → 𝑖 ∈ ℕ)
58	39, 57	sylancom 586	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → 𝑖 ∈ ℕ)
59	16	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝐵 ∈ ℝ)
60		nnnn0 12483	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℕ₀)
61	60	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ₀)
62	59, 61	reexpcld 14132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐵↑𝑖) ∈ ℝ)
63	3	rpred 13020	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℝ)
64	63	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝐸 ∈ ℝ)
65		1red 11219	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 1 ∈ ℝ)
66	62, 64, 65	ltsub2d 11828	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐵↑𝑖) < 𝐸 ↔ (1 − 𝐸) < (1 − (𝐵↑𝑖))))
67	36, 58, 66	syl2anc 582	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → ((𝐵↑𝑖) < 𝐸 ↔ (1 − 𝐸) < (1 − (𝐵↑𝑖))))
68	56, 67	mpbid 231	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) ∧ 𝑖 ∈ (ℤ_≥‘𝑛)) → (1 − 𝐸) < (1 − (𝐵↑𝑖)))
69	68	ralrimiva 3144	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) → ∀𝑖 ∈ (ℤ_≥‘𝑛)(1 − 𝐸) < (1 − (𝐵↑𝑖)))
70	30	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑘 = 𝑖 → (1 − (𝐵↑𝑘)) = (1 − (𝐵↑𝑖)))
71	70	breq2d 5159	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → ((1 − 𝐸) < (1 − (𝐵↑𝑘)) ↔ (1 − 𝐸) < (1 − (𝐵↑𝑖))))
72	71	cbvralvw 3232	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)(1 − 𝐸) < (1 − (𝐵↑𝑘)) ↔ ∀𝑖 ∈ (ℤ_≥‘𝑛)(1 − 𝐸) < (1 − (𝐵↑𝑖)))
73	69, 72	sylibr 233	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸)) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(1 − 𝐸) < (1 − (𝐵↑𝑘)))
74	73	ex 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(1 − 𝐸) < (1 − (𝐵↑𝑘))))
75	74	reximdva 3166	. . . 4 ⊢ (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐵↑𝑘) ∈ ℂ ∧ (abs‘((𝐵↑𝑘) − 0)) < 𝐸) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(1 − 𝐸) < (1 − (𝐵↑𝑘))))
76	26, 75	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(1 − 𝐸) < (1 − (𝐵↑𝑘)))
77		stoweidlem7.1	. . . . . 6 ⊢ 𝐹 = (𝑖 ∈ ℕ₀ ↦ ((1 / 𝐴)↑𝑖))
78		oveq2 7419	. . . . . 6 ⊢ (𝑖 = 𝑘 → ((1 / 𝐴)↑𝑖) = ((1 / 𝐴)↑𝑘))
79		stoweidlem7.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
80	79	recnd 11246	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
81		0lt1 11740	. . . . . . . . . . . 12 ⊢ 0 < 1
82	81	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 1)
83		stoweidlem7.4	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 𝐴)
84	15, 13, 79, 82, 83	lttrd 11379	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐴)
85	84	gt0ne0d 11782	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 0)
86	80, 85	reccld 11987	. . . . . . . 8 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ)
87	86	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝐴) ∈ ℂ)
88	87, 7	expcld 14115	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1 / 𝐴)↑𝑘) ∈ ℂ)
89	77, 78, 7, 88	fvmptd3 7020	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘))
90	79, 85	rereccld 12045	. . . . . . . . 9 ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
91	79, 84	recgt0d 12152	. . . . . . . . 9 ⊢ (𝜑 → 0 < (1 / 𝐴))
92	14, 15, 90, 18, 91	lttrd 11379	. . . . . . . 8 ⊢ (𝜑 → -1 < (1 / 𝐴))
93		ltdiv23 12109	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (1 ∈ ℝ ∧ 0 < 1)) → ((1 / 𝐴) < 1 ↔ (1 / 1) < 𝐴))
94	13, 79, 84, 13, 82, 93	syl122anc 1377	. . . . . . . . . 10 ⊢ (𝜑 → ((1 / 𝐴) < 1 ↔ (1 / 1) < 𝐴))
95		1cnd 11213	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
96	95	div1d 11986	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 1) = 1)
97	96	breq1d 5157	. . . . . . . . . 10 ⊢ (𝜑 → ((1 / 1) < 𝐴 ↔ 1 < 𝐴))
98	94, 97	bitrd 278	. . . . . . . . 9 ⊢ (𝜑 → ((1 / 𝐴) < 1 ↔ 1 < 𝐴))
99	83, 98	mpbird 256	. . . . . . . 8 ⊢ (𝜑 → (1 / 𝐴) < 1)
100	90, 13	absltd 15380	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(1 / 𝐴)) < 1 ↔ (-1 < (1 / 𝐴) ∧ (1 / 𝐴) < 1)))
101	92, 99, 100	mpbir2and 709	. . . . . . 7 ⊢ (𝜑 → (abs‘(1 / 𝐴)) < 1)
102	86, 101	expcnv 15814	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ ℕ₀ ↦ ((1 / 𝐴)↑𝑖)) ⇝ 0)
103	77, 102	eqbrtrid 5182	. . . . 5 ⊢ (𝜑 → 𝐹 ⇝ 0)
104	1, 2, 3, 89, 103	climi2 15459	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(abs‘(((1 / 𝐴)↑𝑘) − 0)) < 𝐸)
105		simpll 763	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝜑)
106		uznnssnn 12883	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (ℤ_≥‘𝑛) ⊆ ℕ)
107	106	ad2antlr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (ℤ_≥‘𝑛) ⊆ ℕ)
108		simpr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ (ℤ_≥‘𝑛))
109	107, 108	sseldd 3982	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
110	88	subid1d 11564	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((1 / 𝐴)↑𝑘) − 0) = ((1 / 𝐴)↑𝑘))
111	110	fveq2d 6894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(((1 / 𝐴)↑𝑘) − 0)) = (abs‘((1 / 𝐴)↑𝑘)))
112	90	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝐴) ∈ ℝ)
113	112, 7	reexpcld 14132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1 / 𝐴)↑𝑘) ∈ ℝ)
114	15, 90, 91	ltled 11366	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (1 / 𝐴))
115	114	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / 𝐴))
116	112, 7, 115	expge0d 14133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ ((1 / 𝐴)↑𝑘))
117	113, 116	absidd 15373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘((1 / 𝐴)↑𝑘)) = ((1 / 𝐴)↑𝑘))
118	111, 117	eqtrd 2770	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(((1 / 𝐴)↑𝑘) − 0)) = ((1 / 𝐴)↑𝑘))
119	118	breq1d 5157	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(((1 / 𝐴)↑𝑘) − 0)) < 𝐸 ↔ ((1 / 𝐴)↑𝑘) < 𝐸))
120	119	biimpd 228	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs‘(((1 / 𝐴)↑𝑘) − 0)) < 𝐸 → ((1 / 𝐴)↑𝑘) < 𝐸))
121	105, 109, 120	syl2anc 582	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((abs‘(((1 / 𝐴)↑𝑘) − 0)) < 𝐸 → ((1 / 𝐴)↑𝑘) < 𝐸))
122	121	ralimdva 3165	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑘 ∈ (ℤ_≥‘𝑛)(abs‘(((1 / 𝐴)↑𝑘) − 0)) < 𝐸 → ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 / 𝐴)↑𝑘) < 𝐸))
123	122	reximdva 3166	. . . 4 ⊢ (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(abs‘(((1 / 𝐴)↑𝑘) − 0)) < 𝐸 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 / 𝐴)↑𝑘) < 𝐸))
124	104, 123	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 / 𝐴)↑𝑘) < 𝐸)
125	1	rexanuz2 15300	. . 3 ⊢ (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸) ↔ (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 / 𝐴)↑𝑘) < 𝐸))
126	76, 124, 125	sylanbrc 581	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸))
127		simpr 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸)) → ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸))
128		nnz 12583	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
129		uzid 12841	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
130	128, 129	syl 17	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘𝑛))
131	130	ad2antlr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸)) → 𝑛 ∈ (ℤ_≥‘𝑛))
132		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝐵↑𝑘) = (𝐵↑𝑛))
133	132	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (1 − (𝐵↑𝑘)) = (1 − (𝐵↑𝑛)))
134	133	breq2d 5159	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((1 − 𝐸) < (1 − (𝐵↑𝑘)) ↔ (1 − 𝐸) < (1 − (𝐵↑𝑛))))
135		oveq2 7419	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → ((1 / 𝐴)↑𝑘) = ((1 / 𝐴)↑𝑛))
136	135	breq1d 5157	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (((1 / 𝐴)↑𝑘) < 𝐸 ↔ ((1 / 𝐴)↑𝑛) < 𝐸))
137	134, 136	anbi12d 629	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸) ↔ ((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ ((1 / 𝐴)↑𝑛) < 𝐸)))
138	137	rspccva 3610	. . . . . 6 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸) ∧ 𝑛 ∈ (ℤ_≥‘𝑛)) → ((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ ((1 / 𝐴)↑𝑛) < 𝐸))
139	127, 131, 138	syl2anc 582	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸)) → ((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ ((1 / 𝐴)↑𝑛) < 𝐸))
140		1cnd 11213	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
141	80, 85	jca 510	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
142	141	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
143	40	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
144		expdiv 14083	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑛 ∈ ℕ₀) → ((1 / 𝐴)↑𝑛) = ((1↑𝑛) / (𝐴↑𝑛)))
145	140, 142, 143, 144	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1 / 𝐴)↑𝑛) = ((1↑𝑛) / (𝐴↑𝑛)))
146	128	adantl 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
147		1exp 14061	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1)
148	146, 147	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1↑𝑛) = 1)
149	148	oveq1d 7426	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1↑𝑛) / (𝐴↑𝑛)) = (1 / (𝐴↑𝑛)))
150	145, 149	eqtrd 2770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1 / 𝐴)↑𝑛) = (1 / (𝐴↑𝑛)))
151	150	breq1d 5157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((1 / 𝐴)↑𝑛) < 𝐸 ↔ (1 / (𝐴↑𝑛)) < 𝐸))
152	151	adantr 479	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸)) → (((1 / 𝐴)↑𝑛) < 𝐸 ↔ (1 / (𝐴↑𝑛)) < 𝐸))
153	152	anbi2d 627	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸)) → (((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ ((1 / 𝐴)↑𝑛) < 𝐸) ↔ ((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ (1 / (𝐴↑𝑛)) < 𝐸)))
154	139, 153	mpbid 231	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸)) → ((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ (1 / (𝐴↑𝑛)) < 𝐸))
155	154	ex 411	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸) → ((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ (1 / (𝐴↑𝑛)) < 𝐸)))
156	155	reximdva 3166	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((1 − 𝐸) < (1 − (𝐵↑𝑘)) ∧ ((1 / 𝐴)↑𝑘) < 𝐸) → ∃𝑛 ∈ ℕ ((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ (1 / (𝐴↑𝑛)) < 𝐸)))
157	126, 156	mpd 15	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ (1 / (𝐴↑𝑛)) < 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 < clt 11252 ≤ cle 11253 − cmin 11448 -cneg 11449 / cdiv 11875 ℕcn 12216 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12978 ↑cexp 14031 abscabs 15185 ⇝ cli 15432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fl 13761 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437

This theorem is referenced by: stoweidlem49 45063

W3C validator