trilpolemlt1 - Mathbox for Jim Kingdon

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Theorem trilpolemlt1 16595

Description: Lemma for trilpo 16597. 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 7333 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 8187	. . . 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 16591	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	1, 4	resubcld 8553	. . 3 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ)
6		trilpolemlt1.a	. . . 4 ⊢ (𝜑 → 𝐴 < 1)
7	4, 1	posdifd 8705	. . . 4 ⊢ (𝜑 → (𝐴 < 1 ↔ 0 < (1 − 𝐴)))
8	6, 7	mpbid 147	. . 3 ⊢ (𝜑 → 0 < (1 − 𝐴))
9		nnrecl 9393	. . 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 10282	. . . . . . 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 5660	. . . . . . 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 2264	. . . . . 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 2629	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (∃𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 0 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0))
20		2rp 9886	. . . . . . . . . 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 9594	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℤ)
24	21, 23	rpexpcld 10952	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (2↑𝑛) ∈ ℝ⁺)
25	24	rprecred 9936	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / (2↑𝑛)) ∈ ℝ)
26	22	nnrecred 9183	. . . . . . 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 9500	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
29		uzid 9763	. . . . . . . . . 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 9448	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℕ₀)
32		bernneq3 10917	. . . . . . . . 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 9922	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℝ⁺)
35	34, 24	ltrecd 9943	. . . . . . . 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 8298	. . . . . 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 8187	. . . . . . 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 8173	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 0 ∈ ℝ)
45		eqid 2229	. . . . . . . . . . 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 9151	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (𝑛 + 1) ∈ ℕ)
48	47	nnzd 9594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (𝑛 + 1) ∈ ℤ)
49		eluznn 9827	. . . . . . . . . . . . 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 2229	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗))) = (𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗)))
52		oveq2 6021	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (2↑𝑗) = (2↑𝑖))
53	52	oveq2d 6029	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → (1 / (2↑𝑗)) = (1 / (2↑𝑖)))
54		fveq2 5635	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
55	53, 54	oveq12d 6031	. . . . . . . . . . . . 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 9594	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
59	57, 58	rpexpcld 10952	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
60	59	rprecred 9936	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
61		0re 8172	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
62		1re 8171	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
63		prssi 3829	. . . . . . . . . . . . . . . 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 5779	. . . . . . . . . . . . . . 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 3223	. . . . . . . . . . . . . 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 8203	. . . . . . . . . . . . 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 5736	. . . . . . . . . . . 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 16590	. . . . . . . . . . . 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 9785	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
76	69	recnd 8201	. . . . . . . . . . . . . 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 2306	. . . . . . . . . . . . 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 11893	. . . . . . . . . . . 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 11983	. . . . . . . . . 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 9499	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 1 ∈ ℤ)
82	81, 23	fzfigd 10686	. . . . . . . . . . . 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 10253	. . . . . . . . . . . . . 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 10952	. . . . . . . . . . . 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 9936	. . . . . . . . . . 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 11955	. . . . . . . . . 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 9935	. . . . . . . . . . . . . 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 9928	. . . . . . . . . . . 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 9225	. . . . . . . . . . . . . 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 4124	. . . . . . . . . . . . 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 8654	. . . . . . . . . . . . . 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 4124	. . . . . . . . . . . . 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 5779	. . . . . . . . . . . . . 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 3690	. . . . . . . . . . . . . 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 803	. . . . . . . . . . . 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 8794	. . . . . . . . . . 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 11984	. . . . . . . . . 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 8733	. . . . . . . . 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 8201	. . . . . . . . . . 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	addridd 8321	. . . . . . . . . 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 2235	. . . . . . . . 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 12045	. . . . . . . . . . 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 2274	. . . . . . . . . 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 9150	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝑛 ∈ ℂ)
115		1cnd 8188	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 1 ∈ ℂ)
116	114, 115	pncand 8484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → ((𝑛 + 1) − 1) = 𝑛)
117	116	oveq2d 6029	. . . . . . . . . . . 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 5660	. . . . . . . . . . . . . . 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 5644	. . . . . . . . . . . . . . . 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 2913	. . . . . . . . . . . . . . 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 2264	. . . . . . . . . . . . . 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 6029	. . . . . . . . . . . . 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 9935	. . . . . . . . . . . . . . 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 9926	. . . . . . . . . . . . . 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 8189	. . . . . . . . . . . . 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 2262	. . . . . . . . . . . 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 11927	. . . . . . . . . . 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 6028	. . . . . . . . . 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 2262	. . . . . . . . 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 4118	. . . . . . . 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 12068	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 1 ∈ ℂ) → Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) = (1 − (1 / (2↑𝑛))))
134	133	breq1d 4096	. . . . . . . . 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 8721	. . . . . 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 8294	. . . . 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 623	. . . 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 5634	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (𝑓‘𝑥) = ((𝐹 ↾ (1...𝑛))‘𝑥))
142	141	eqeq1d 2238	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → ((𝑓‘𝑥) = 0 ↔ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0))
143	142	rexbidv 2531	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (∃𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 0 ↔ ∃𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 0))
144	141	eqeq1d 2238	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → ((𝑓‘𝑥) = 1 ↔ ((𝐹 ↾ (1...𝑛))‘𝑥) = 1))
145	144	ralbidv 2530	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (∀𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 1 ↔ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1))
146	143, 145	orbi12d 798	. . . 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 7333	. . . . . 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 16585	. . . . . 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 10284	. . . . . . 7 ⊢ (1...𝑛) ⊆ ℕ
153	152	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1...𝑛) ⊆ ℕ)
154	65, 153	fssresd 5510	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶{0, 1})
155		0red 8173	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 0 ∈ ℝ)
156		1red 8187	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 1 ∈ ℝ)
157		prexg 4299	. . . . . . 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 6826	. . . . 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 2913	. . 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 723	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)
163	10, 162	rexlimddv 2653	1 ⊢ (𝜑 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 713 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ∃wrex 2509 Vcvv 2800 ⊆ wss 3198 {cpr 3668 class class class wbr 4086 ↦ cmpt 4148 dom cdm 4723 ↾ cres 4725 ⟶wf 5320 ‘cfv 5324 (class class class)co 6013 ↑_𝑚 cmap 6812 Fincfn 6904 Omnicomni 7327 ℂcc 8023 ℝcr 8024 0cc0 8025 1c1 8026 + caddc 8028 · cmul 8030 < clt 8207 ≤ cle 8208 − cmin 8343 / cdiv 8845 ℕcn 9136 2c2 9187 ℕ₀cn0 9395 ℤcz 9472 ℤ_≥cuz 9748 ℝ⁺crp 9881 ...cfz 10236 seqcseq 10702 ↑cexp 10793 ⇝ cli 11832 Σcsu 11907

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 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145

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 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-2o 6578 df-oadd 6581 df-er 6697 df-map 6814 df-en 6905 df-dom 6906 df-fin 6907 df-omni 7328 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-n0 9396 df-z 9473 df-uz 9749 df-q 9847 df-rp 9882 df-ico 10122 df-fz 10237 df-fzo 10371 df-seqfrec 10703 df-exp 10794 df-ihash 11031 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-clim 11833 df-sumdc 11908

This theorem is referenced by: trilpolemres 16596 neapmkvlem 16621

W3C validator