trilpolemlt1 - Mathbox for Jim Kingdon

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Theorem trilpolemlt1 13409

Description: Lemma for trilpo 13411. 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 7020 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 7805	. . . 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 13405	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	1, 4	resubcld 8167	. . 3 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ)
6		trilpolemlt1.a	. . . 4 ⊢ (𝜑 → 𝐴 < 1)
7	4, 1	posdifd 8318	. . . 4 ⊢ (𝜑 → (𝐴 < 1 ↔ 0 < (1 − 𝐴)))
8	6, 7	mpbid 146	. . 3 ⊢ (𝜑 → 0 < (1 − 𝐴))
9		nnrecl 8999	. . 3 ⊢ (((1 − 𝐴) ∈ ℝ ∧ 0 < (1 − 𝐴)) → ∃𝑛 ∈ ℕ (1 / 𝑛) < (1 − 𝐴))
10	5, 8, 9	syl2anc 409	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ (1 / 𝑛) < (1 − 𝐴))
11		elfznn 9865	. . . . . . 7 ⊢ (𝑥 ∈ (1...𝑛) → 𝑥 ∈ ℕ)
12	11	ad2antrl 482	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → 𝑥 ∈ ℕ)
13		simprl 521	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → 𝑥 ∈ (1...𝑛))
14	13	fvresd 5454	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → ((𝐹 ↾ (1...𝑛))‘𝑥) = (𝐹‘𝑥))
15		simprr 522	. . . . . . 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 2175	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → (𝐹‘𝑥) = 0)
17	12, 16	jca 304	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ (𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0)) → (𝑥 ∈ ℕ ∧ (𝐹‘𝑥) = 0))
18	17	ex 114	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → ((𝑥 ∈ (1...𝑛) ∧ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0) → (𝑥 ∈ ℕ ∧ (𝐹‘𝑥) = 0)))
19	18	reximdv2 2534	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (∃𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 0 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0))
20		2rp 9475	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
21	20	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 2 ∈ ℝ⁺)
22		simprl 521	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℕ)
23	22	nnzd 9196	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℤ)
24	21, 23	rpexpcld 10479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (2↑𝑛) ∈ ℝ⁺)
25	24	rprecred 9525	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / (2↑𝑛)) ∈ ℝ)
26	22	nnrecred 8791	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / 𝑛) ∈ ℝ)
27	5	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 − 𝐴) ∈ ℝ)
28		2z 9106	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
29		uzid 9364	. . . . . . . . . 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 9054	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℕ₀)
32		bernneq3 10445	. . . . . . . . 9 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → 𝑛 < (2↑𝑛))
33	30, 31, 32	syl2anc 409	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 < (2↑𝑛))
34	22	nnrpd 9511	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝑛 ∈ ℝ⁺)
35	34, 24	ltrecd 9532	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (𝑛 < (2↑𝑛) ↔ (1 / (2↑𝑛)) < (1 / 𝑛)))
36	33, 35	mpbid 146	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / (2↑𝑛)) < (1 / 𝑛))
37		simprr 522	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / 𝑛) < (1 − 𝐴))
38	25, 26, 27, 36, 37	lttrd 7912	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1 / (2↑𝑛)) < (1 − 𝐴))
39	38	adantr 274	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1 / (2↑𝑛)) < (1 − 𝐴))
40	27	adantr 274	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1 − 𝐴) ∈ ℝ)
41	25	adantr 274	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1 / (2↑𝑛)) ∈ ℝ)
42		1red 7805	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 1 ∈ ℝ)
43	4	ad2antrr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝐴 ∈ ℝ)
44		0red 7791	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 0 ∈ ℝ)
45		eqid 2140	. . . . . . . . . . 11 ⊢ (ℤ_≥‘(𝑛 + 1)) = (ℤ_≥‘(𝑛 + 1))
46	22	adantr 274	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝑛 ∈ ℕ)
47	46	peano2nnd 8759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (𝑛 + 1) ∈ ℕ)
48	47	nnzd 9196	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (𝑛 + 1) ∈ ℤ)
49		eluznn 9421	. . . . . . . . . . . . 13 ⊢ (((𝑛 + 1) ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑖 ∈ ℕ)
50	47, 49	sylan 281	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑖 ∈ ℕ)
51		eqid 2140	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗))) = (𝑗 ∈ ℕ ↦ ((1 / (2↑𝑗)) · (𝐹‘𝑗)))
52		oveq2 5790	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (2↑𝑗) = (2↑𝑖))
53	52	oveq2d 5798	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → (1 / (2↑𝑗)) = (1 / (2↑𝑖)))
54		fveq2 5429	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
55	53, 54	oveq12d 5800	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → ((1 / (2↑𝑗)) · (𝐹‘𝑗)) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
56		simpr 109	. . . . . . . . . . . . 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 9196	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
59	57, 58	rpexpcld 10479	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
60	59	rprecred 9525	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
61		0re 7790	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
62		1re 7789	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
63		prssi 3686	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ)
64	61, 62, 63	mp2an 423	. . . . . . . . . . . . . . 15 ⊢ {0, 1} ⊆ ℝ
65	2	adantr 274	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 𝐹:ℕ⟶{0, 1})
66	65	ad2antrr 480	. . . . . . . . . . . . . . . 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	ffvelrnd 5564	. . . . . . . . . . . . . . 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	sseldi 3100	. . . . . . . . . . . . . 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 7820	. . . . . . . . . . . . 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 5522	. . . . . . . . . . . 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 280	. . . . . . . . . . 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 280	. . . . . . . . . . 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 274	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝐹:ℕ⟶{0, 1})
74	73, 51	trilpolemclim 13404	. . . . . . . . . . . 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 9385	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
76	69	recnd 7818	. . . . . . . . . . . . . 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 2217	. . . . . . . . . . . . 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 11140	. . . . . . . . . . . 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 146	. . . . . . . . . . 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 11230	. . . . . . . . . 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 9105	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 1 ∈ ℤ)
82	81, 23	fzfigd 10235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1...𝑛) ∈ Fin)
83	82	adantr 274	. . . . . . . . . . 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 9837	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑛) → 𝑖 ∈ ℤ)
86	85	adantl 275	. . . . . . . . . . . . 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 10479	. . . . . . . . . . . 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 9525	. . . . . . . . . . 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 11202	. . . . . . . . . 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 280	. . . . . . . . . . . 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 280	. . . . . . . . . . . 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 9524	. . . . . . . . . . . . . 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 280	. . . . . . . . . . . . 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 9517	. . . . . . . . . . . 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 8833	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
96		simpr 109	. . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . 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 8267	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 1
99		simpr 109	. . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . 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	ffvelrnd 5564	. . . . . . . . . . . . . 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 3555	. . . . . . . . . . . . . 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 788	. . . . . . . . . . . 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 8407	. . . . . . . . . . 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 11231	. . . . . . . . . 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 8346	. . . . . . . . 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 7818	. . . . . . . . . . 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 7935	. . . . . . . . . 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 2146	. . . . . . . . 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 11292	. . . . . . . . . . 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	syl5eq 2185	. . . . . . . . . 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 8758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 𝑛 ∈ ℂ)
115		1cnd 7806	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → 1 ∈ ℂ)
116	114, 115	pncand 8098	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → ((𝑛 + 1) − 1) = 𝑛)
117	116	oveq2d 5798	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
118		simpr 109	. . . . . . . . . . . . . . . 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 5454	. . . . . . . . . . . . . . 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 5438	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑖 → (((𝐹 ↾ (1...𝑛))‘𝑥) = 1 ↔ ((𝐹 ↾ (1...𝑛))‘𝑖) = 1))
121		simplr 520	. . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . 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 2175	. . . . . . . . . . . . . 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 5798	. . . . . . . . . . . . 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 9524	. . . . . . . . . . . . . . 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 9515	. . . . . . . . . . . . . 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	mulid1d 7807	. . . . . . . . . . . . 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 2173	. . . . . . . . . . . 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 11174	. . . . . . . . . . 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 5797	. . . . . . . . . 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 2173	. . . . . . . . 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 3968	. . . . . . . 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 11315	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 1 ∈ ℂ) → Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) = (1 − (1 / (2↑𝑛))))
134	133	breq1d 3947	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 1 ∈ ℂ) → (Σ𝑖 ∈ (1...𝑛)(1 / (2↑𝑖)) ≤ 𝐴 ↔ (1 − (1 / (2↑𝑛))) ≤ 𝐴))
135	46, 115, 134	syl2anc 409	. . . . . . . 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 146	. . . . . . 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 8334	. . . . . 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 7908	. . . . 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 610	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) ∧ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1) → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)
140	139	ex 114	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0))
141		fveq1 5428	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (𝑓‘𝑥) = ((𝐹 ↾ (1...𝑛))‘𝑥))
142	141	eqeq1d 2149	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → ((𝑓‘𝑥) = 0 ↔ ((𝐹 ↾ (1...𝑛))‘𝑥) = 0))
143	142	rexbidv 2439	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (∃𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 0 ↔ ∃𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 0))
144	141	eqeq1d 2149	. . . . . 6 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → ((𝑓‘𝑥) = 1 ↔ ((𝐹 ↾ (1...𝑛))‘𝑥) = 1))
145	144	ralbidv 2438	. . . . 5 ⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (∀𝑥 ∈ (1...𝑛)(𝑓‘𝑥) = 1 ↔ ∀𝑥 ∈ (1...𝑛)((𝐹 ↾ (1...𝑛))‘𝑥) = 1))
146	143, 145	orbi12d 783	. . . 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 7020	. . . . . 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 13401	. . . . . 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 146	. . . 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 9867	. . . . . . 7 ⊢ (1...𝑛) ⊆ ℕ
153	152	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (1...𝑛) ⊆ ℕ)
154	65, 153	fssresd 5307	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶{0, 1})
155		0red 7791	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 0 ∈ ℝ)
156		1red 7805	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → 1 ∈ ℝ)
157		prexg 4141	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ∈ V)
158	155, 156, 157	syl2anc 409	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → {0, 1} ∈ V)
159	158, 82	elmapd 6564	. . . . 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 166	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → (𝐹 ↾ (1...𝑛)) ∈ ({0, 1} ↑_𝑚 (1...𝑛)))
161	146, 151, 160	rspcdva 2798	. . 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 708	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (1 / 𝑛) < (1 − 𝐴))) → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)
163	10, 162	rexlimddv 2557	1 ⊢ (𝜑 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 = wceq 1332 ∈ wcel 1481 ∀wral 2417 ∃wrex 2418 Vcvv 2689 ⊆ wss 3076 {cpr 3533 class class class wbr 3937 ↦ cmpt 3997 dom cdm 4547 ↾ cres 4549 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 ↑_𝑚 cmap 6550 Fincfn 6642 Omnicomni 7012 ℂcc 7642 ℝcr 7643 0cc0 7644 1c1 7645 + caddc 7647 · cmul 7649 < clt 7824 ≤ cle 7825 − cmin 7957 / cdiv 8456 ℕcn 8744 2c2 8795 ℕ₀cn0 9001 ℤcz 9078 ℤ_≥cuz 9350 ℝ⁺crp 9470 ...cfz 9821 seqcseq 10249 ↑cexp 10323 ⇝ cli 11079 Σcsu 11154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-2o 6322 df-oadd 6325 df-er 6437 df-map 6552 df-en 6643 df-dom 6644 df-fin 6645 df-omni 7014 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-ico 9707 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-ihash 10554 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155

This theorem is referenced by: trilpolemres 13410 neapmkvlem 13424

W3C validator