nconstwlpolemgt0 - Mathbox for Jim Kingdon

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Theorem nconstwlpolemgt0 16850

Description: Lemma for nconstwlpo 16852. 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 9604	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 1 ∈ ℤ)
3		simprl 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℕ)
4	3	peano2nnd 9252	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝑥 + 1) ∈ ℕ)
5	4	nnzd 9699	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝑥 + 1) ∈ ℤ)
6	5, 2	zsubcld 9705	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((𝑥 + 1) − 1) ∈ ℤ)
7	2, 6	fzfigd 10793	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) ∈ Fin)
8		elfznn 10388	. . . . . . 7 ⊢ (𝑖 ∈ (1...((𝑥 + 1) − 1)) → 𝑖 ∈ ℕ)
9		2rp 9991	. . . . . . . . . . . 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 9699	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
13	10, 12	rpexpcld 11059	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
14	13	rpreccld 10040	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
15	14	rpred 10029	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
16		0re 8274	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17		1re 8273	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
18		prssi 3852	. . . . . . . . . 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 5813	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ {0, 1})
23	19, 22	sselid 3236	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ ℝ)
24	15, 23	remulcld 8304	. . . . . . 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 12087	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
27		eqid 2232	. . . . . 6 ⊢ (ℤ_≥‘(𝑥 + 1)) = (ℤ_≥‘(𝑥 + 1))
28		eqid 2232	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))
29		oveq2 6058	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
30	29	oveq2d 6066	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
31		fveq2 5670	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (𝐺‘𝑛) = (𝐺‘𝑖))
32	30, 31	oveq12d 6068	. . . . . . 7 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐺‘𝑛)) = ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
33		eluznn 9932	. . . . . . . 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 5771	. . . . . 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 16820	. . . . . . . 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 9890	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
40	28, 32, 11, 24	fvmptd3 5771	. . . . . . . . 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 8302	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℂ)
42	40, 41	eqeltrd 2309	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))‘𝑖) ∈ ℂ)
43	39, 4, 42	iserex 12024	. . . . . . 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 12115	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
46	3	nnzd 9699	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℤ)
47		fzofig 10794	. . . . . . . . . 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 10530	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1..^𝑥) ↔ (𝑖 ∈ ℕ ∧ 𝑥 ∈ ℕ ∧ 𝑖 < 𝑥))
50	49	simp1bi 1039	. . . . . . . . . 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 12087	. . . . . . . 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 11059	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (2↑𝑥) ∈ ℝ⁺)
55	54	rpreccld 10040	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℝ⁺)
56	55	rpred 10029	. . . . . . . . 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 5813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) ∈ {0, 1})
59	19, 58	sselid 3236	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) ∈ ℝ)
60	56, 59	remulcld 8304	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) ∈ ℝ)
61	14	rpge0d 10033	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ (1 / (2↑𝑖)))
62		0le0 9326	. . . . . . . . . . . . 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 4147	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 0) → 0 ≤ (𝐺‘𝑖))
65		0le1 8755	. . . . . . . . . . . . 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 4147	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 1) → 0 ≤ (𝐺‘𝑖))
68		elpri 3712	. . . . . . . . . . . . 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 806	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ (𝐺‘𝑖))
71	15, 23, 61, 70	mulge0d 8895	. . . . . . . . . 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 12145	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 ≤ Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)))
74	55	rpgt0d 10032	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < (1 / (2↑𝑥)))
75		simprr 533	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) = 1)
76	75	oveq2d 6066	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) = ((1 / (2↑𝑥)) · 1))
77	56	recnd 8302	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℂ)
78	77	mulridd 8291	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · 1) = (1 / (2↑𝑥)))
79	76, 78	eqtrd 2265	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) = (1 / (2↑𝑥)))
80	74, 79	breqtrrd 4137	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < ((1 / (2↑𝑥)) · (𝐺‘𝑥)))
81	52, 60, 73, 80	addgegt0d 8793	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)) + ((1 / (2↑𝑥)) · (𝐺‘𝑥))))
82		nfv 1577	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1))
83		nfcv 2384	. . . . . . . 8 ⊢ Ⅎ𝑖((1 / (2↑𝑥)) · (𝐺‘𝑥))
84		fzonel 10495	. . . . . . . . 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 6058	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (2↑𝑖) = (2↑𝑥))
88	87	oveq2d 6066	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (1 / (2↑𝑖)) = (1 / (2↑𝑥)))
89		fveq2 5670	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝐺‘𝑖) = (𝐺‘𝑥))
90	88, 89	oveq12d 6068	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) = ((1 / (2↑𝑥)) · (𝐺‘𝑥)))
91	60	recnd 8302	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) ∈ ℂ)
92	82, 83, 48, 3, 85, 86, 90, 91	fsumsplitsn 12096	. . . . . . 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 4137	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐺‘𝑖)))
94	3	nncnd 9251	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℂ)
95		1cnd 8290	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 1 ∈ ℂ)
96	94, 95	pncand 8585	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((𝑥 + 1) − 1) = 𝑥)
97	96	oveq2d 6066	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) = (1...𝑥))
98	3, 39	eleqtrdi 2325	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ (ℤ_≥‘1))
99		fzisfzounsn 10582	. . . . . . . . 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 2265	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) = ((1..^𝑥) ∪ {𝑥}))
102	101	sumeq1d 12051	. . . . . 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 4137	. . . . 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 10033	. . . . . . 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 8895	. . . . . 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 12116	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 ≤ Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)))
111	26, 45, 103, 110	addgtge0d 8794	. . . 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 12177	. . . 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 4137	. . 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 4151	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < 𝐴)
116	1, 115	rexlimddv 2665	1 ⊢ (𝜑 → 0 < 𝐴)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 716 = wceq 1398 ∈ wcel 2203 ∃wrex 2521 ∪ cun 3209 ⊆ wss 3211 {csn 3689 {cpr 3690 class class class wbr 4109 ↦ cmpt 4171 dom cdm 4749 ⟶wf 5348 ‘cfv 5352 (class class class)co 6050 Fincfn 6975 ℂcc 8125 ℝcr 8126 0cc0 8127 1c1 8128 + caddc 8130 · cmul 8132 < clt 8308 ≤ cle 8309 − cmin 8444 / cdiv 8946 ℕcn 9237 2c2 9288 ℤcz 9577 ℤ_≥cuz 9853 ℝ⁺crp 9986 ...cfz 10342 ..^cfzo 10476 seqcseq 10809 ↑cexp 10900 ⇝ cli 11963 Σcsu 12038

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-frec 6622 df-1o 6647 df-oadd 6651 df-er 6767 df-en 6976 df-dom 6977 df-fin 6978 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-ico 10227 df-fz 10343 df-fzo 10477 df-seqfrec 10810 df-exp 10901 df-ihash 11139 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-clim 11964 df-sumdc 12039

This theorem is referenced by: nconstwlpolem 16851

W3C validator