trilpolemlt1 - Mathbox for Jim Kingdon

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Theorem trilpolemlt1 14792

Description: Lemma for trilpo 14794. The 𝐴 < 1 case. We can use the distance between 𝐴 and one (that is, 1 − 𝐴) to find a position in the sequence 𝑛 where terms after that point will not add up to as much as 1 − 𝐴. By finomni 7138 we know the terms up to 𝑛 either contain a zero or are all one. But if they are all one that contradicts the way we constructed 𝑛, so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.)

**Hypotheses**
Ref	Expression
trilpolemgt1.f	⊢ (𝜑 → 𝐹:ℕ⟶{0, 1})
trilpolemgt1.a	⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))
trilpolemlt1.a	⊢ (𝜑 → 𝐴 < 1)

**Assertion**
Ref	Expression
trilpolemlt1	⊢ (𝜑 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)

Distinct variable groups: 𝐴,𝑖,𝑥 𝑥,𝐹,𝑖 𝜑,𝑖,𝑥

Proof of Theorem trilpolemlt1 Dummy variables 𝑛 𝑓 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 7972	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
2		trilpolemgt1.f	. . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1})
3		trilpolemgt1.a	. . . . 5 ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))
4	2, 3	trilpolemcl 14788	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	1, 4	resubcld 8338	. . 3 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ)
6		trilpolemlt1.a	. . . 4 ⊢ (𝜑 → 𝐴 < 1)
7	4, 1	posdifd 8489	. . . 4 ⊢ (𝜑 → (𝐴 < 1 ↔ 0 < (1 − 𝐴)))
8	6, 7	mpbid 147	. . 3 ⊢ (𝜑 → 0 < (1 − 𝐴))
9		nnrecl 9174	. . 3 ⊢ (((1 − 𝐴) ∈ ℝ ∧ 0 < (1 − 𝐴)) → ∃𝑛 ∈ ℕ (1 / 𝑛) < (1 − 𝐴))
10	5, 8, 9	syl2anc 411	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ (1 / 𝑛) < (1 − 𝐴))
11		elfznn 10054	. . . . . . 7 ⊢ (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
12	11	ad2antrl 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → 𝑥 ∈ ℕ)
13		simprl 529	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → 𝑥 ∈ (1...𝑛))
14	13	fvresd 5541	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → ((𝐹 ↾ (1...𝑛))‘𝑥) = (𝐹‘𝑥))
15		simprr 531	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)
16	14, 15	eqtr3d 2212	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → (𝐹‘𝑥) = 0)
17	12, 16	jca 306	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → (𝑥 ∈ ℕ ∧ (𝐹‘𝑥) = 0))
18	17	ex 115	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → ((𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0) → (𝑥 ∈ ℕ ∧ (𝐹‘𝑥) = 0)))
19	18	reximdv2 2576	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (∃𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 0 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0))
20		2rp 9658	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
21	20	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 2 ∈ ℝ⁺)
22		simprl 529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℕ)
23	22	nnzd 9374	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℤ)
24	21, 23	rpexpcld 10678	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (2↑𝑛) ∈ ℝ⁺)
25	24	rprecred 9708	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / (2↑𝑛)) ∈ ℝ)
26	22	nnrecred 8966	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / 𝑛) ∈ ℝ)
27	5	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 − 𝐴) ∈ ℝ)
28		2z 9281	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
29		uzid 9542	. . . . . . . . . 10 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
30	28, 29	mp1i 10	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 2 ∈ (ℤ_≥‘2))
31	22	nnnn0d 9229	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℕ₀)
32		bernneq3 10643	. . . . . . . . 9 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → 𝑛 < (2↑𝑛))
33	30, 31, 32	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 < (2↑𝑛))
34	22	nnrpd 9694	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℝ⁺)
35	34, 24	ltrecd 9715	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (𝑛 < (2↑𝑛) ↔ (1 / (2↑𝑛)) < (1 / 𝑛)))
36	33, 35	mpbid 147	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / (2↑𝑛)) < (1 / 𝑛))
37		simprr 531	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / 𝑛) < (1 − 𝐴))
38	25, 26, 27, 36, 37	lttrd 8083	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / (2↑𝑛)) < (1 − 𝐴))
39	38	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1 / (2↑𝑛)) < (1 − 𝐴))
40	27	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1 − 𝐴) ∈ ℝ)
41	25	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1 / (2↑𝑛)) ∈ ℝ)
42		1red 7972	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 1 ∈ ℝ)
43	4	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝐴 ∈ ℝ)
44		0red 7958	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 0 ∈ ℝ)
45		eqid 2177	. . . . . . . . . . 11 ⊢ (ℤ_≥‘(𝑛 + 1)) = (ℤ_≥‘(𝑛 + 1))
46	22	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝑛 ∈ ℕ)
47	46	peano2nnd 8934	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (𝑛 + 1) ∈ ℕ)
48	47	nnzd 9374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (𝑛 + 1) ∈ ℤ)
49		eluznn 9600	. . . . . . . . . . . . 13 ⊢ (((𝑛 + 1) ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑖 ∈ ℕ)
50	47, 49	sylan 283	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑖 ∈ ℕ)
51		eqid 2177	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗))) = (𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗)))
52		oveq2 5883	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (2↑𝑗) = (2↑𝑖))
53	52	oveq2d 5891	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → (1 / (2↑𝑗)) = (1 / (2↑𝑖)))
54		fveq2 5516	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
55	53, 54	oveq12d 5893	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → ((1 / (2↑𝑗)) · (𝐹‘𝑗)) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
56		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
57	20	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → 2 ∈ ℝ⁺)
58	56	nnzd 9374	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
59	57, 58	rpexpcld 10678	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
60	59	rprecred 9708	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
61		0re 7957	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
62		1re 7956	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
63		prssi 3751	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ)
64	61, 62, 63	mp2an 426	. . . . . . . . . . . . . . 15 ⊢ {0, 1} ⊆ ℝ
65	2	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝐹:ℕ⟶{0, 1})
66	65	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → 𝐹:ℕ⟶{0, 1})
67	66, 56	ffvelcdmd 5653	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ {0, 1})
68	64, 67	sselid 3154	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
69	60, 68	remulcld 7988	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
70	51, 55, 56, 69	fvmptd3 5610	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
71	50, 70	syldan 282	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
72	50, 69	syldan 282	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
73	65	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝐹:ℕ⟶{0, 1})
74	73, 51	trilpolemclim 14787	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → seq1( + , (𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗)))) ∈ dom ⇝ )
75		nnuz 9563	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
76	69	recnd 7986	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℂ)
77	70, 76	eqeltrd 2254	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗)))‘𝑖) ∈ ℂ)
78	75, 47, 77	iserex 11347	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (seq1( + , (𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗)))) ∈ dom ⇝ ↔ seq(𝑛 + 1)( + , (𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗)))) ∈ dom ⇝ ))
79	74, 78	mpbid 147	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → seq(𝑛 + 1)( + , (𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗)))) ∈ dom ⇝ )
80	45, 48, 71, 72, 79	isumrecl 11437	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → Σ𝑖 ∈ (ℤ_≥‘(𝑛 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
81		1zzd 9280	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 1 ∈ ℤ)
82	81, 23	fzfigd 10431	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1...𝑛) ∈ Fin)
83	82	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1...𝑛) ∈ Fin)
84	20	a1i 9	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → 2 ∈ ℝ⁺)
85		elfzelz 10025	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℤ)
86	85	adantl 277	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ ℤ)
87	84, 86	rpexpcld 10678	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → (2↑𝑖) ∈ ℝ⁺)
88	87	rprecred 9708	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → (1 / (2↑𝑖)) ∈ ℝ)
89	83, 88	fsumrecl 11409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) ∈ ℝ)
90	50, 60	syldan 282	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → (1 / (2↑𝑖)) ∈ ℝ)
91	50, 68	syldan 282	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑖) ∈ ℝ)
92	59	rpreccld 9707	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
93	50, 92	syldan 282	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → (1 / (2↑𝑖)) ∈ ℝ⁺)
94	93	rpge0d 9700	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → 0 ≤ (1 / (2↑𝑖)))
95		0le0 9008	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
96		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ (𝐹‘𝑖) = 0) → (𝐹‘𝑖) = 0)
97	95, 96	breqtrrid 4042	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ (𝐹‘𝑖) = 0) → 0 ≤ (𝐹‘𝑖))
98		0le1 8438	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 1
99		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ (𝐹‘𝑖) = 1) → (𝐹‘𝑖) = 1)
100	98, 99	breqtrrid 4042	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) ∧ (𝐹‘𝑖) = 1) → 0 ≤ (𝐹‘𝑖))
101	73	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝐹:ℕ⟶{0, 1})
102	101, 50	ffvelcdmd 5653	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑖) ∈ {0, 1})
103		elpri 3616	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑖) ∈ {0, 1} → ((𝐹‘𝑖) = 0 ∨ (𝐹‘𝑖) = 1))
104	102, 103	syl 14	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝐹‘𝑖) = 0 ∨ (𝐹‘𝑖) = 1))
105	97, 100, 104	mpjaodan 798	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → 0 ≤ (𝐹‘𝑖))
106	90, 91, 94, 105	mulge0d 8578	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → 0 ≤ ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
107	45, 48, 71, 72, 79, 106	isumge0 11438	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 0 ≤ Σ𝑖 ∈ (ℤ_≥‘(𝑛 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)))
108	44, 80, 89, 107	leadd2dd 8517	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) + 0) ≤ (Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑛 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))))
109	89	recnd 7986	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) ∈ ℂ)
110	109	addid1d 8106	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) + 0) = Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)))
111	110	eqcomd 2183	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) = (Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) + 0))
112	75, 45, 47, 70, 76, 74	isumsplit 11499	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑛 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))))
113	3, 112	eqtrid 2222	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝐴 = (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑛 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))))
114	46	nncnd 8933	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝑛 ∈ ℂ)
115		1cnd 7973	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 1 ∈ ℂ)
116	114, 115	pncand 8269	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → ((𝑛 + 1) − 1) = 𝑛)
117	116	oveq2d 5891	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
118		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → 𝑖 ∈ (1...𝑛))
119	118	fvresd 5541	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → ((𝐹 ↾ (1...𝑛))‘𝑖) = (𝐹‘𝑖))
120		fveqeq2 5525	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑖 → (((𝐹 ↾ (1...𝑛))‘𝑥) = 1 ↔ ((𝐹 ↾ (1...𝑛))‘𝑖) = 1))
121		simplr 528	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1)
122	120, 121, 118	rspcdva 2847	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → ((𝐹 ↾ (1...𝑛))‘𝑖) = 1)
123	119, 122	eqtr3d 2212	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → (𝐹‘𝑖) = 1)
124	123	oveq2d 5891	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 1))
125	87	rpreccld 9707	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → (1 / (2↑𝑖)) ∈ ℝ⁺)
126	125	rpcnd 9698	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → (1 / (2↑𝑖)) ∈ ℂ)
127	126	mulridd 7974	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → ((1 / (2↑𝑖)) · 1) = (1 / (2↑𝑖)))
128	124, 127	eqtrd 2210	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (1...𝑛)) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (1 / (2↑𝑖)))
129	117, 128	sumeq12rdv 11381	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → Σ𝑖 ∈ (1...((𝑛 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) = Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)))
130	129	oveq1d 5890	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (Σ𝑖 ∈ (1...((𝑛 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑛 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))) = (Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑛 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))))
131	113, 130	eqtrd 2210	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝐴 = (Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑛 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))))
132	108, 111, 131	3brtr4d 4036	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) ≤ 𝐴)
133		geo2sum 11522	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 1 ∈ ℂ) → Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) = (1 − (1 / (2↑𝑛))))
134	133	breq1d 4014	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 1 ∈ ℂ) → (Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) ≤ 𝐴 ↔ (1 − (1 / (2↑𝑛))) ≤ 𝐴))
135	46, 115, 134	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) ≤ 𝐴 ↔ (1 − (1 / (2↑𝑛))) ≤ 𝐴))
136	132, 135	mpbid 147	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1 − (1 / (2↑𝑛))) ≤ 𝐴)
137	42, 41, 43, 136	subled 8505	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1 − 𝐴) ≤ (1 / (2↑𝑛)))
138	40, 41, 137	lensymd 8079	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → ¬ (1 / (2↑𝑛)) < (1 − 𝐴))
139	39, 138	pm2.21dd 620	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)
140	139	ex 115	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0))
141		fveq1 5515	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (𝑓‘𝑥) = ((𝐹 ↾ (1...𝑛))‘𝑥))
142	141	eqeq1d 2186	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → ((𝑓‘𝑥) = 0 ↔ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0))
143	142	rexbidv 2478	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (∃𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 0 ↔ ∃𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 0))
144	141	eqeq1d 2186	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → ((𝑓‘𝑥) = 1 ↔ ((𝐹 ↾ (1...𝑛))‘𝑥) = 1))
145	144	ralbidv 2477	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (∀𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 1 ↔ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1))
146	143, 145	orbi12d 793	. . . 4 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → ((∃𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 1) ↔ (∃𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 0 ∨ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1)))
147		finomni 7138	. . . . . 6 ⊢ ((1...𝑛) ∈ Fin → (1...𝑛) ∈ Omni)
148	82, 147	syl 14	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1...𝑛) ∈ Omni)
149		isomninn 14782	. . . . . 6 ⊢ ((1...𝑛) ∈ Omni → ((1...𝑛) ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑_𝑚 (1...𝑛))(∃𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 1)))
150	148, 149	syl 14	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → ((1...𝑛) ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑_𝑚 (1...𝑛))(∃𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 1)))
151	148, 150	mpbid 147	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → ∀𝑓 ∈ ({0, 1} ↑_𝑚 (1...𝑛))(∃𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 1))
152		fz1ssnn 10056	. . . . . . 7 ⊢ (1...𝑛) ⊆ ℕ
153	152	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1...𝑛) ⊆ ℕ)
154	65, 153	fssresd 5393	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶{0, 1})
155		0red 7958	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 0 ∈ ℝ)
156		1red 7972	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 1 ∈ ℝ)
157		prexg 4212	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ∈ V)
158	155, 156, 157	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → {0, 1} ∈ V)
159	158, 82	elmapd 6662	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → ((𝐹 ↾ (1...𝑛)) ∈ ({0, 1} ↑_𝑚 (1...𝑛)) ↔ (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶{0, 1}))
160	154, 159	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (𝐹 ↾ (1...𝑛)) ∈ ({0, 1} ↑_𝑚 (1...𝑛)))
161	146, 151, 160	rspcdva 2847	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (∃𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 0 ∨ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1))
162	19, 140, 161	mpjaod 718	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)
163	10, 162	rexlimddv 2599	1 ⊢ (𝜑 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 Vcvv 2738 ⊆ wss 3130 {cpr 3594 class class class wbr 4004 ↦ cmpt 4065 dom cdm 4627 ↾ cres 4629 ⟶wf 5213 ‘cfv 5217 (class class class)co 5875 ↑_𝑚 cmap 6648 Fincfn 6740 Omnicomni 7132 ℂcc 7809 ℝcr 7810 0cc0 7811 1c1 7812 + caddc 7814 · cmul 7816 < clt 7992 ≤ cle 7993 − cmin 8128 / cdiv 8629 ℕcn 8919 2c2 8970 ℕ₀cn0 9176 ℤcz 9253 ℤ_≥cuz 9528 ℝ⁺crp 9653 ...cfz 10008 seqcseq 10445 ↑cexp 10519 ⇝ cli 11286 Σcsu 11361

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-irdg 6371 df-frec 6392 df-1o 6417 df-2o 6418 df-oadd 6421 df-er 6535 df-map 6650 df-en 6741 df-dom 6742 df-fin 6743 df-omni 7133 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-q 9620 df-rp 9654 df-ico 9894 df-fz 10009 df-fzo 10143 df-seqfrec 10446 df-exp 10520 df-ihash 10756 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-clim 11287 df-sumdc 11362

This theorem is referenced by: trilpolemres 14793 neapmkvlem 14817

W3C validator