trilpolemeq1 - Mathbox for Jim Kingdon

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Theorem trilpolemeq1 14072

Description: Lemma for trilpo 14075. The 𝐴 = 1 case. This is proved by noting that if any (𝐹‘𝑥) is zero, then the infinite sum 𝐴 is less than one based on the term which is zero. We are using the fact that the 𝐹 sequence is decidable (in the sense that each element is either zero or one). (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))
trilpolemeq1.a	⊢ (𝜑 → 𝐴 = 1)

**Assertion**
Ref	Expression
trilpolemeq1	⊢ (𝜑 → ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)

Distinct variable groups: 𝑖,𝐹 𝜑,𝑖,𝑥 Allowed substitution hints: 𝐴(𝑥,𝑖) 𝐹(𝑥)

Proof of Theorem trilpolemeq1 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trilpolemeq1.a	. . . . 5 ⊢ (𝜑 → 𝐴 = 1)
2	1	ad2antrr 485	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 = 1)
3		trilpolemgt1.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1})
4		trilpolemgt1.a	. . . . . . . 8 ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖))
5	3, 4	trilpolemcl 14069	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6	5	ad2antrr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 ∈ ℝ)
7		nnuz 9522	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
8		eqid 2170	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘(𝑥 + 1)) = (ℤ_≥‘(𝑥 + 1))
9		simplr 525	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝑥 ∈ ℕ)
10	9	peano2nnd 8893	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (𝑥 + 1) ∈ ℕ)
11		eqid 2170	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))
12		oveq2 5861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
13	12	oveq2d 5869	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
14		fveq2 5496	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑖 → (𝐹‘𝑛) = (𝐹‘𝑖))
15	13, 14	oveq12d 5871	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐹‘𝑛)) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
16		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
17		2rp 9615	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ⁺
18	17	a1i 9	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → 2 ∈ ℝ⁺)
19	16	nnzd 9333	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
20	18, 19	rpexpcld 10633	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
21	20	rpreccld 9664	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
22	21	rpred 9653	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
23		0re 7920	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
24		1re 7919	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
25		prssi 3738	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ)
26	23, 24, 25	mp2an 424	. . . . . . . . . . . . . . . . 17 ⊢ {0, 1} ⊆ ℝ
27	3	ad3antrrr 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → 𝐹:ℕ⟶{0, 1})
28	27, 16	ffvelrnd 5632	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ {0, 1})
29	26, 28	sselid 3145	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ)
30	22, 29	remulcld 7950	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
31	11, 15, 16, 30	fvmptd3 5589	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
32	30	recnd 7948	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℂ)
33	3, 11	trilpolemclim 14068	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )
34	33	ad2antrr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )
35	7, 8, 10, 31, 32, 34	isumsplit 11454	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))))
36	9	nncnd 8892	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝑥 ∈ ℂ)
37		1cnd 7936	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 1 ∈ ℂ)
38	36, 37	pncand 8231	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((𝑥 + 1) − 1) = 𝑥)
39	38	oveq2d 5869	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1...((𝑥 + 1) − 1)) = (1...𝑥))
40	9, 7	eleqtrdi 2263	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝑥 ∈ (ℤ_≥‘1))
41		fzisfzounsn 10192	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (ℤ_≥‘1) → (1...𝑥) = ((1..^𝑥) ∪ {𝑥}))
42	40, 41	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1...𝑥) = ((1..^𝑥) ∪ {𝑥}))
43	39, 42	eqtrd 2203	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1...((𝑥 + 1) − 1)) = ((1..^𝑥) ∪ {𝑥}))
44	43	sumeq1d 11329	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) = Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐹‘𝑖)))
45		nfv 1521	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0)
46		nfcv 2312	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖((1 / (2↑𝑥)) · (𝐹‘𝑥))
47		1zzd 9239	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 1 ∈ ℤ)
48	9	nnzd 9333	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝑥 ∈ ℤ)
49		fzofig 10388	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (1..^𝑥) ∈ Fin)
50	47, 48, 49	syl2anc 409	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1..^𝑥) ∈ Fin)
51		fzonel 10116	. . . . . . . . . . . . . . . . 17 ⊢ ¬ 𝑥 ∈ (1..^𝑥)
52	51	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ¬ 𝑥 ∈ (1..^𝑥))
53	17	a1i 9	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → 2 ∈ ℝ⁺)
54		elfzoelz 10103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (1..^𝑥) → 𝑖 ∈ ℤ)
55	54	adantl 275	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → 𝑖 ∈ ℤ)
56	53, 55	rpexpcld 10633	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (2↑𝑖) ∈ ℝ⁺)
57	56	rpreccld 9664	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (1 / (2↑𝑖)) ∈ ℝ⁺)
58	57	rpred 9653	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (1 / (2↑𝑖)) ∈ ℝ)
59		elfzouz 10107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1..^𝑥) → 𝑖 ∈ (ℤ_≥‘1))
60	59, 7	eleqtrrdi 2264	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1..^𝑥) → 𝑖 ∈ ℕ)
61	60, 29	sylan2 284	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (𝐹‘𝑖) ∈ ℝ)
62	58, 61	remulcld 7950	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
63	62	recnd 7948	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℂ)
64		oveq2 5861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑥 → (2↑𝑖) = (2↑𝑥))
65	64	oveq2d 5869	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑥 → (1 / (2↑𝑖)) = (1 / (2↑𝑥)))
66		fveq2 5496	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑥 → (𝐹‘𝑖) = (𝐹‘𝑥))
67	65, 66	oveq12d 5871	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑥 → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑥)) · (𝐹‘𝑥)))
68	17	a1i 9	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 2 ∈ ℝ⁺)
69	68, 48	rpexpcld 10633	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (2↑𝑥) ∈ ℝ⁺)
70	69	rpreccld 9664	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1 / (2↑𝑥)) ∈ ℝ⁺)
71	70	rpred 9653	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1 / (2↑𝑥)) ∈ ℝ)
72	3	ad2antrr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐹:ℕ⟶{0, 1})
73	72, 9	ffvelrnd 5632	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (𝐹‘𝑥) ∈ {0, 1})
74	26, 73	sselid 3145	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (𝐹‘𝑥) ∈ ℝ)
75	71, 74	remulcld 7950	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((1 / (2↑𝑥)) · (𝐹‘𝑥)) ∈ ℝ)
76	75	recnd 7948	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((1 / (2↑𝑥)) · (𝐹‘𝑥)) ∈ ℂ)
77	45, 46, 50, 9, 52, 63, 67, 76	fsumsplitsn 11373	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + ((1 / (2↑𝑥)) · (𝐹‘𝑥))))
78		simpr 109	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (𝐹‘𝑥) = 0)
79	78	oveq2d 5869	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((1 / (2↑𝑥)) · (𝐹‘𝑥)) = ((1 / (2↑𝑥)) · 0))
80	70	rpcnd 9655	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1 / (2↑𝑥)) ∈ ℂ)
81	80	mul01d 8312	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((1 / (2↑𝑥)) · 0) = 0)
82	79, 81	eqtrd 2203	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((1 / (2↑𝑥)) · (𝐹‘𝑥)) = 0)
83	82	oveq2d 5869	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + ((1 / (2↑𝑥)) · (𝐹‘𝑥))) = (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0))
84	44, 77, 83	3eqtrd 2207	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0))
85	84	oveq1d 5868	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))) = ((Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))))
86	35, 85	eqtrd 2203	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))))
87	50, 62	fsumrecl 11364	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
88		0red 7921	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 0 ∈ ℝ)
89	87, 88	readdcld 7949	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0) ∈ ℝ)
90	10	nnzd 9333	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (𝑥 + 1) ∈ ℤ)
91		eluznn 9559	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 + 1) ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 𝑖 ∈ ℕ)
92	10, 91	sylan 281	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 𝑖 ∈ ℕ)
93	92, 30	syldan 280	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
94	11, 15, 92, 93	fvmptd3 5589	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐹‘𝑖)))
95	31, 32	eqeltrd 2247	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℂ)
96	7, 10, 95	iserex 11302	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ ↔ seq(𝑥 + 1)( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ ))
97	34, 96	mpbid 146	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → seq(𝑥 + 1)( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )
98	8, 90, 94, 93, 97	isumrecl 11392	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ)
99	50, 58	fsumrecl 11364	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) ∈ ℝ)
100	99, 71	readdcld 7949	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))) ∈ ℝ)
101		eqid 2170	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))
102	92, 21	syldan 280	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → (1 / (2↑𝑖)) ∈ ℝ⁺)
103	101, 13, 92, 102	fvmptd3 5589	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) = (1 / (2↑𝑖)))
104	92, 22	syldan 280	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → (1 / (2↑𝑖)) ∈ ℝ)
105		seqex 10403	. . . . . . . . . . . . . . . . 17 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ V
106		ax-1cn 7867	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
107	101	geo2lim 11479	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℂ → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ 1)
108	106, 107	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ 1
109		breldmg 4817	. . . . . . . . . . . . . . . . 17 ⊢ ((seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ V ∧ 1 ∈ ℂ ∧ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ 1) → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
110	105, 106, 108, 109	mp3an 1332	. . . . . . . . . . . . . . . 16 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝
111	110	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
112	101, 13, 16, 21	fvmptd3 5589	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) = (1 / (2↑𝑖)))
113	21	rpcnd 9655	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℂ)
114	112, 113	eqeltrd 2247	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) ∈ ℂ)
115	7, 10, 114	iserex 11302	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ ↔ seq(𝑥 + 1)( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ ))
116	111, 115	mpbid 146	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → seq(𝑥 + 1)( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ )
117	8, 90, 103, 104, 116	isumrecl 11392	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖)) ∈ ℝ)
118		simpr 109	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → (𝐹‘𝑖) = 0)
119	118	oveq2d 5869	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 0))
120	57	adantr 274	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → (1 / (2↑𝑖)) ∈ ℝ⁺)
121	120	rpcnd 9655	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → (1 / (2↑𝑖)) ∈ ℂ)
122	121	mul01d 8312	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · 0) = 0)
123	119, 122	eqtrd 2203	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = 0)
124	120	rpge0d 9657	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → 0 ≤ (1 / (2↑𝑖)))
125	123, 124	eqbrtrd 4011	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
126		simpr 109	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → (𝐹‘𝑖) = 1)
127	126	oveq2d 5869	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 1))
128	58	adantr 274	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ∈ ℝ)
129	128	recnd 7948	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ∈ ℂ)
130	129	mulid1d 7937	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · 1) = (1 / (2↑𝑖)))
131	127, 130	eqtrd 2203	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (1 / (2↑𝑖)))
132	128	leidd 8433	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ≤ (1 / (2↑𝑖)))
133	131, 132	eqbrtrd 4011	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
134	72	adantr 274	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → 𝐹:ℕ⟶{0, 1})
135	60	adantl 275	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → 𝑖 ∈ ℕ)
136	134, 135	ffvelrnd 5632	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (𝐹‘𝑖) ∈ {0, 1})
137		elpri 3606	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑖) ∈ {0, 1} → ((𝐹‘𝑖) = 0 ∨ (𝐹‘𝑖) = 1))
138	136, 137	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → ((𝐹‘𝑖) = 0 ∨ (𝐹‘𝑖) = 1))
139	125, 133, 138	mpjaodan 793	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖)))
140	50, 62, 58, 139	fsumle 11426	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)))
141	70	rpgt0d 9656	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 0 < (1 / (2↑𝑥)))
142	87, 88, 99, 71, 140, 141	leltaddd 8485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0) < (Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))))
143	72, 4, 8, 10	trilpolemisumle 14070	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖)))
144	89, 98, 100, 117, 142, 143	ltleaddd 8484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))) < ((Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))))
145	86, 144	eqbrtrd 4011	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) < ((Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))))
146	4, 145	eqbrtrid 4024	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 < ((Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))))
147		nfcv 2312	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(1 / (2↑𝑥))
148	57	rpcnd 9655	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (1 / (2↑𝑖)) ∈ ℂ)
149	45, 147, 50, 9, 52, 148, 65, 80	fsumsplitsn 11373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})(1 / (2↑𝑖)) = (Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))))
150	149	oveq1d 5868	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))) = ((Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))))
151	146, 150	breqtrrd 4017	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 < (Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))))
152	42	sumeq1d 11329	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)) = Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})(1 / (2↑𝑖)))
153	152	oveq1d 5868	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))) = (Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))))
154	151, 153	breqtrrd 4017	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 < (Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))))
155	7, 8, 10, 112, 113, 111	isumsplit 11454	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ℕ (1 / (2↑𝑖)) = (Σ𝑖 ∈ (1...((𝑥 + 1) − 1))(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))))
156	39	sumeq1d 11329	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))(1 / (2↑𝑖)) = Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)))
157	156	oveq1d 5868	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1...((𝑥 + 1) − 1))(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))) = (Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))))
158	155, 157	eqtrd 2203	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ℕ (1 / (2↑𝑖)) = (Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))(1 / (2↑𝑖))))
159	154, 158	breqtrrd 4017	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 < Σ𝑖 ∈ ℕ (1 / (2↑𝑖)))
160		geoihalfsum 11485	. . . . . . 7 ⊢ Σ𝑖 ∈ ℕ (1 / (2↑𝑖)) = 1
161	159, 160	breqtrdi 4030	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 < 1)
162	6, 161	ltned 8033	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 ≠ 1)
163	162	neneqd 2361	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ¬ 𝐴 = 1)
164	2, 163	pm2.65da 656	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ¬ (𝐹‘𝑥) = 0)
165	3	ffvelrnda 5631	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ {0, 1})
166		elpri 3606	. . . . 5 ⊢ ((𝐹‘𝑥) ∈ {0, 1} → ((𝐹‘𝑥) = 0 ∨ (𝐹‘𝑥) = 1))
167	165, 166	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝐹‘𝑥) = 0 ∨ (𝐹‘𝑥) = 1))
168	167	orcomd 724	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝐹‘𝑥) = 1 ∨ (𝐹‘𝑥) = 0))
169	164, 168	ecased 1344	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) = 1)
170	169	ralrimiva 2543	1 ⊢ (𝜑 → ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 703 = wceq 1348 ∈ wcel 2141 ∀wral 2448 Vcvv 2730 ∪ cun 3119 ⊆ wss 3121 {csn 3583 {cpr 3584 class class class wbr 3989 ↦ cmpt 4050 dom cdm 4611 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 Fincfn 6718 ℂcc 7772 ℝcr 7773 0cc0 7774 1c1 7775 + caddc 7777 · cmul 7779 < clt 7954 ≤ cle 7955 − cmin 8090 / cdiv 8589 ℕcn 8878 2c2 8929 ℤcz 9212 ℤ_≥cuz 9487 ℝ⁺crp 9610 ...cfz 9965 ..^cfzo 10098 seqcseq 10401 ↑cexp 10475 ⇝ cli 11241 Σcsu 11316

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-ico 9851 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317

This theorem is referenced by: trilpolemres 14074 redcwlpolemeq1 14086

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