nconstwlpolemgt0 - Mathbox for Jim Kingdon

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Theorem nconstwlpolemgt0 14095

Description: Lemma for nconstwlpo 14097. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.)

**Hypotheses**
Ref	Expression
nconstwlpolem0.g	⊢ (𝜑 → 𝐺:ℕ⟶{0, 1})
nconstwlpolem0.a	⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖))
nconstwlpolemgt0.0	⊢ (𝜑 → ∃𝑥 ∈ ℕ (𝐺‘𝑥) = 1)

**Assertion**
Ref	Expression
nconstwlpolemgt0	⊢ (𝜑 → 0 < 𝐴)

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

Proof of Theorem nconstwlpolemgt0 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nconstwlpolemgt0.0	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℕ (𝐺‘𝑥) = 1)
2		1zzd 9239	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 1 ∈ ℤ)
3		simprl 526	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℕ)
4	3	peano2nnd 8893	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝑥 + 1) ∈ ℕ)
5	4	nnzd 9333	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝑥 + 1) ∈ ℤ)
6	5, 2	zsubcld 9339	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((𝑥 + 1) − 1) ∈ ℤ)
7	2, 6	fzfigd 10387	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) ∈ Fin)
8		elfznn 10010	. . . . . . 7 ⊢ (𝑖 ∈ (1...((𝑥 + 1) − 1)) → 𝑖 ∈ ℕ)
9		2rp 9615	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ⁺
10	9	a1i 9	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 2 ∈ ℝ⁺)
11		simpr 109	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
12	11	nnzd 9333	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
13	10, 12	rpexpcld 10633	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
14	13	rpreccld 9664	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
15	14	rpred 9653	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
16		0re 7920	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17		1re 7919	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
18		prssi 3738	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ)
19	16, 17, 18	mp2an 424	. . . . . . . . 9 ⊢ {0, 1} ⊆ ℝ
20		nconstwlpolem0.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶{0, 1})
21	20	ad2antrr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 𝐺:ℕ⟶{0, 1})
22	21, 11	ffvelrnd 5632	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ {0, 1})
23	19, 22	sselid 3145	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ ℝ)
24	15, 23	remulcld 7950	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
25	8, 24	sylan2 284	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (1...((𝑥 + 1) − 1))) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
26	7, 25	fsumrecl 11364	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
27		eqid 2170	. . . . . 6 ⊢ (ℤ_≥‘(𝑥 + 1)) = (ℤ_≥‘(𝑥 + 1))
28		eqid 2170	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))
29		oveq2 5861	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
30	29	oveq2d 5869	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
31		fveq2 5496	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (𝐺‘𝑛) = (𝐺‘𝑖))
32	30, 31	oveq12d 5871	. . . . . . 7 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐺‘𝑛)) = ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
33		eluznn 9559	. . . . . . . 8 ⊢ (((𝑥 + 1) ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 𝑖 ∈ ℕ)
34	4, 33	sylan 281	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 𝑖 ∈ ℕ)
35	34, 24	syldan 280	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
36	28, 32, 34, 35	fvmptd3 5589	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
37	20, 28	trilpolemclim 14068	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))) ∈ dom ⇝ )
38	37	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))) ∈ dom ⇝ )
39		nnuz 9522	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
40	28, 32, 11, 24	fvmptd3 5589	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
41	24	recnd 7948	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℂ)
42	40, 41	eqeltrd 2247	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))‘𝑖) ∈ ℂ)
43	39, 4, 42	iserex 11302	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))) ∈ dom ⇝ ↔ seq(𝑥 + 1)( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))) ∈ dom ⇝ ))
44	38, 43	mpbid 146	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → seq(𝑥 + 1)( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))) ∈ dom ⇝ )
45	27, 5, 36, 35, 44	isumrecl 11392	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
46	3	nnzd 9333	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℤ)
47		fzofig 10388	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (1..^𝑥) ∈ Fin)
48	2, 46, 47	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1..^𝑥) ∈ Fin)
49		elfzo1 10146	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1..^𝑥) ↔ (𝑖 ∈ ℕ ∧ 𝑥 ∈ ℕ ∧ 𝑖 < 𝑥))
50	49	simp1bi 1007	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1..^𝑥) → 𝑖 ∈ ℕ)
51	50, 24	sylan2 284	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (1..^𝑥)) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
52	48, 51	fsumrecl 11364	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
53	9	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 2 ∈ ℝ⁺)
54	53, 46	rpexpcld 10633	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (2↑𝑥) ∈ ℝ⁺)
55	54	rpreccld 9664	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℝ⁺)
56	55	rpred 9653	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℝ)
57	20	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝐺:ℕ⟶{0, 1})
58	57, 3	ffvelrnd 5632	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) ∈ {0, 1})
59	19, 58	sselid 3145	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) ∈ ℝ)
60	56, 59	remulcld 7950	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) ∈ ℝ)
61	14	rpge0d 9657	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ (1 / (2↑𝑖)))
62		0le0 8967	. . . . . . . . . . . . 13 ⊢ 0 ≤ 0
63		simpr 109	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 0) → (𝐺‘𝑖) = 0)
64	62, 63	breqtrrid 4027	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 0) → 0 ≤ (𝐺‘𝑖))
65		0le1 8400	. . . . . . . . . . . . 13 ⊢ 0 ≤ 1
66		simpr 109	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 1) → (𝐺‘𝑖) = 1)
67	65, 66	breqtrrid 4027	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 1) → 0 ≤ (𝐺‘𝑖))
68		elpri 3606	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑖) ∈ {0, 1} → ((𝐺‘𝑖) = 0 ∨ (𝐺‘𝑖) = 1))
69	22, 68	syl 14	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((𝐺‘𝑖) = 0 ∨ (𝐺‘𝑖) = 1))
70	64, 67, 69	mpjaodan 793	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ (𝐺‘𝑖))
71	15, 23, 61, 70	mulge0d 8540	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
72	50, 71	sylan2 284	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (1..^𝑥)) → 0 ≤ ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
73	48, 51, 72	fsumge0 11422	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 ≤ Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)))
74	55	rpgt0d 9656	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < (1 / (2↑𝑥)))
75		simprr 527	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) = 1)
76	75	oveq2d 5869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) = ((1 / (2↑𝑥)) · 1))
77	56	recnd 7948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℂ)
78	77	mulid1d 7937	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · 1) = (1 / (2↑𝑥)))
79	76, 78	eqtrd 2203	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) = (1 / (2↑𝑥)))
80	74, 79	breqtrrd 4017	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < ((1 / (2↑𝑥)) · (𝐺‘𝑥)))
81	52, 60, 73, 80	addgegt0d 8438	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)) + ((1 / (2↑𝑥)) · (𝐺‘𝑥))))
82		nfv 1521	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1))
83		nfcv 2312	. . . . . . . 8 ⊢ Ⅎ𝑖((1 / (2↑𝑥)) · (𝐺‘𝑥))
84		fzonel 10116	. . . . . . . . 9 ⊢ ¬ 𝑥 ∈ (1..^𝑥)
85	84	a1i 9	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ¬ 𝑥 ∈ (1..^𝑥))
86	50, 41	sylan2 284	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (1..^𝑥)) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℂ)
87		oveq2 5861	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (2↑𝑖) = (2↑𝑥))
88	87	oveq2d 5869	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (1 / (2↑𝑖)) = (1 / (2↑𝑥)))
89		fveq2 5496	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝐺‘𝑖) = (𝐺‘𝑥))
90	88, 89	oveq12d 5871	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) = ((1 / (2↑𝑥)) · (𝐺‘𝑥)))
91	60	recnd 7948	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) ∈ ℂ)
92	82, 83, 48, 3, 85, 86, 90, 91	fsumsplitsn 11373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐺‘𝑖)) = (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)) + ((1 / (2↑𝑥)) · (𝐺‘𝑥))))
93	81, 92	breqtrrd 4017	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐺‘𝑖)))
94	3	nncnd 8892	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℂ)
95		1cnd 7936	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 1 ∈ ℂ)
96	94, 95	pncand 8231	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((𝑥 + 1) − 1) = 𝑥)
97	96	oveq2d 5869	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) = (1...𝑥))
98	3, 39	eleqtrdi 2263	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ (ℤ_≥‘1))
99		fzisfzounsn 10192	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℤ_≥‘1) → (1...𝑥) = ((1..^𝑥) ∪ {𝑥}))
100	98, 99	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...𝑥) = ((1..^𝑥) ∪ {𝑥}))
101	97, 100	eqtrd 2203	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) = ((1..^𝑥) ∪ {𝑥}))
102	101	sumeq1d 11329	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) = Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐺‘𝑖)))
103	93, 102	breqtrrd 4017	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)))
104	34, 15	syldan 280	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → (1 / (2↑𝑖)) ∈ ℝ)
105	34, 23	syldan 280	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → (𝐺‘𝑖) ∈ ℝ)
106	34, 14	syldan 280	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → (1 / (2↑𝑖)) ∈ ℝ⁺)
107	106	rpge0d 9657	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 0 ≤ (1 / (2↑𝑖)))
108	34, 70	syldan 280	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 0 ≤ (𝐺‘𝑖))
109	104, 105, 107, 108	mulge0d 8540	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 0 ≤ ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
110	27, 5, 36, 35, 44, 109	isumge0 11393	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 ≤ Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)))
111	26, 45, 103, 110	addgtge0d 8439	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < (Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖))))
112	39, 27, 4, 40, 41, 38	isumsplit 11454	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖)) = (Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) + Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖))))
113	111, 112	breqtrrd 4017	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
114		nconstwlpolem0.a	. . 3 ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖))
115	113, 114	breqtrrdi 4031	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < 𝐴)
116	1, 115	rexlimddv 2592	1 ⊢ (𝜑 → 0 < 𝐴)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 703 = wceq 1348 ∈ wcel 2141 ∃wrex 2449 ∪ 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: nconstwlpolem 14096

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