nconstwlpolemgt0 - Mathbox for Jim Kingdon

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Theorem nconstwlpolemgt0 14352

Description: Lemma for nconstwlpo 14354. 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 9251	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 1 ∈ ℤ)
3		simprl 529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℕ)
4	3	peano2nnd 8905	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝑥 + 1) ∈ ℕ)
5	4	nnzd 9345	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝑥 + 1) ∈ ℤ)
6	5, 2	zsubcld 9351	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((𝑥 + 1) − 1) ∈ ℤ)
7	2, 6	fzfigd 10399	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) ∈ Fin)
8		elfznn 10022	. . . . . . 7 ⊢ (𝑖 ∈ (1...((𝑥 + 1) − 1)) → 𝑖 ∈ ℕ)
9		2rp 9627	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ⁺
10	9	a1i 9	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 2 ∈ ℝ⁺)
11		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
12	11	nnzd 9345	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
13	10, 12	rpexpcld 10645	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
14	13	rpreccld 9676	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
15	14	rpred 9665	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
16		0re 7932	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17		1re 7931	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
18		prssi 3747	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ)
19	16, 17, 18	mp2an 426	. . . . . . . . 9 ⊢ {0, 1} ⊆ ℝ
20		nconstwlpolem0.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶{0, 1})
21	20	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 𝐺:ℕ⟶{0, 1})
22	21, 11	ffvelcdmd 5644	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ {0, 1})
23	19, 22	sselid 3151	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ ℝ)
24	15, 23	remulcld 7962	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
25	8, 24	sylan2 286	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (1...((𝑥 + 1) − 1))) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
26	7, 25	fsumrecl 11375	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
27		eqid 2175	. . . . . 6 ⊢ (ℤ_≥‘(𝑥 + 1)) = (ℤ_≥‘(𝑥 + 1))
28		eqid 2175	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))
29		oveq2 5873	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
30	29	oveq2d 5881	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
31		fveq2 5507	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (𝐺‘𝑛) = (𝐺‘𝑖))
32	30, 31	oveq12d 5883	. . . . . . 7 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐺‘𝑛)) = ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
33		eluznn 9571	. . . . . . . 8 ⊢ (((𝑥 + 1) ∈ ℕ ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 𝑖 ∈ ℕ)
34	4, 33	sylan 283	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 𝑖 ∈ ℕ)
35	34, 24	syldan 282	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
36	28, 32, 34, 35	fvmptd3 5601	. . . . . 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 14325	. . . . . . . 8 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))) ∈ dom ⇝ )
38	37	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))) ∈ dom ⇝ )
39		nnuz 9534	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
40	28, 32, 11, 24	fvmptd3 5601	. . . . . . . . 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 7960	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℂ)
42	40, 41	eqeltrd 2252	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))‘𝑖) ∈ ℂ)
43	39, 4, 42	iserex 11313	. . . . . . 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 147	. . . . . 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 11403	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
46	3	nnzd 9345	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℤ)
47		fzofig 10400	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (1..^𝑥) ∈ Fin)
48	2, 46, 47	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1..^𝑥) ∈ Fin)
49		elfzo1 10158	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1..^𝑥) ↔ (𝑖 ∈ ℕ ∧ 𝑥 ∈ ℕ ∧ 𝑖 < 𝑥))
50	49	simp1bi 1012	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1..^𝑥) → 𝑖 ∈ ℕ)
51	50, 24	sylan2 286	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (1..^𝑥)) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
52	48, 51	fsumrecl 11375	. . . . . . . 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 10645	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (2↑𝑥) ∈ ℝ⁺)
55	54	rpreccld 9676	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℝ⁺)
56	55	rpred 9665	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℝ)
57	20	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝐺:ℕ⟶{0, 1})
58	57, 3	ffvelcdmd 5644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) ∈ {0, 1})
59	19, 58	sselid 3151	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) ∈ ℝ)
60	56, 59	remulcld 7962	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) ∈ ℝ)
61	14	rpge0d 9669	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ (1 / (2↑𝑖)))
62		0le0 8979	. . . . . . . . . . . . 13 ⊢ 0 ≤ 0
63		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 0) → (𝐺‘𝑖) = 0)
64	62, 63	breqtrrid 4036	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 0) → 0 ≤ (𝐺‘𝑖))
65		0le1 8412	. . . . . . . . . . . . 13 ⊢ 0 ≤ 1
66		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 1) → (𝐺‘𝑖) = 1)
67	65, 66	breqtrrid 4036	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 1) → 0 ≤ (𝐺‘𝑖))
68		elpri 3612	. . . . . . . . . . . . 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 798	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ (𝐺‘𝑖))
71	15, 23, 61, 70	mulge0d 8552	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
72	50, 71	sylan2 286	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (1..^𝑥)) → 0 ≤ ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
73	48, 51, 72	fsumge0 11433	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 ≤ Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)))
74	55	rpgt0d 9668	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < (1 / (2↑𝑥)))
75		simprr 531	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) = 1)
76	75	oveq2d 5881	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) = ((1 / (2↑𝑥)) · 1))
77	56	recnd 7960	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℂ)
78	77	mulid1d 7949	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · 1) = (1 / (2↑𝑥)))
79	76, 78	eqtrd 2208	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) = (1 / (2↑𝑥)))
80	74, 79	breqtrrd 4026	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < ((1 / (2↑𝑥)) · (𝐺‘𝑥)))
81	52, 60, 73, 80	addgegt0d 8450	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)) + ((1 / (2↑𝑥)) · (𝐺‘𝑥))))
82		nfv 1526	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1))
83		nfcv 2317	. . . . . . . 8 ⊢ Ⅎ𝑖((1 / (2↑𝑥)) · (𝐺‘𝑥))
84		fzonel 10128	. . . . . . . . 9 ⊢ ¬ 𝑥 ∈ (1..^𝑥)
85	84	a1i 9	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ¬ 𝑥 ∈ (1..^𝑥))
86	50, 41	sylan2 286	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (1..^𝑥)) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℂ)
87		oveq2 5873	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (2↑𝑖) = (2↑𝑥))
88	87	oveq2d 5881	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (1 / (2↑𝑖)) = (1 / (2↑𝑥)))
89		fveq2 5507	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝐺‘𝑖) = (𝐺‘𝑥))
90	88, 89	oveq12d 5883	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) = ((1 / (2↑𝑥)) · (𝐺‘𝑥)))
91	60	recnd 7960	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) ∈ ℂ)
92	82, 83, 48, 3, 85, 86, 90, 91	fsumsplitsn 11384	. . . . . . 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 4026	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐺‘𝑖)))
94	3	nncnd 8904	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℂ)
95		1cnd 7948	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 1 ∈ ℂ)
96	94, 95	pncand 8243	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((𝑥 + 1) − 1) = 𝑥)
97	96	oveq2d 5881	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) = (1...𝑥))
98	3, 39	eleqtrdi 2268	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ (ℤ_≥‘1))
99		fzisfzounsn 10204	. . . . . . . . 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 2208	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) = ((1..^𝑥) ∪ {𝑥}))
102	101	sumeq1d 11340	. . . . . 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 4026	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)))
104	34, 15	syldan 282	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → (1 / (2↑𝑖)) ∈ ℝ)
105	34, 23	syldan 282	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → (𝐺‘𝑖) ∈ ℝ)
106	34, 14	syldan 282	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → (1 / (2↑𝑖)) ∈ ℝ⁺)
107	106	rpge0d 9669	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 0 ≤ (1 / (2↑𝑖)))
108	34, 70	syldan 282	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ (ℤ_≥‘(𝑥 + 1))) → 0 ≤ (𝐺‘𝑖))
109	104, 105, 107, 108	mulge0d 8552	. . . . . 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 11404	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 ≤ Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)))
111	26, 45, 103, 110	addgtge0d 8451	. . . 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 11465	. . . 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 4026	. . 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 4040	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < 𝐴)
116	1, 115	rexlimddv 2597	1 ⊢ (𝜑 → 0 < 𝐴)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2146 ∃wrex 2454 ∪ cun 3125 ⊆ wss 3127 {csn 3589 {cpr 3590 class class class wbr 3998 ↦ cmpt 4059 dom cdm 4620 ⟶wf 5204 ‘cfv 5208 (class class class)co 5865 Fincfn 6730 ℂcc 7784 ℝcr 7785 0cc0 7786 1c1 7787 + caddc 7789 · cmul 7791 < clt 7966 ≤ cle 7967 − cmin 8102 / cdiv 8601 ℕcn 8890 2c2 8941 ℤcz 9224 ℤ_≥cuz 9499 ℝ⁺crp 9622 ...cfz 9977 ..^cfzo 10110 seqcseq 10413 ↑cexp 10487 ⇝ cli 11252 Σcsu 11327

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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906

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 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-isom 5217 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-frec 6382 df-1o 6407 df-oadd 6411 df-er 6525 df-en 6731 df-dom 6732 df-fin 6733 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-inn 8891 df-2 8949 df-3 8950 df-4 8951 df-n0 9148 df-z 9225 df-uz 9500 df-q 9591 df-rp 9623 df-ico 9863 df-fz 9978 df-fzo 10111 df-seqfrec 10414 df-exp 10488 df-ihash 10722 df-cj 10817 df-re 10818 df-im 10819 df-rsqrt 10973 df-abs 10974 df-clim 11253 df-sumdc 11328

This theorem is referenced by: nconstwlpolem 14353

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