nconstwlpolemgt0 - Mathbox for Jim Kingdon

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Theorem nconstwlpolemgt0 17321

Description: Lemma for nconstwlpo 17323. 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 9682	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 1 ∈ ℤ)
3		simprl 535	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℕ)
4	3	peano2nnd 9328	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝑥 + 1) ∈ ℕ)
5	4	nnzd 9778	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝑥 + 1) ∈ ℤ)
6	5, 2	zsubcld 9784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((𝑥 + 1) − 1) ∈ ℤ)
7	2, 6	fzfigd 10889	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) ∈ Fin)
8		elfznn 10477	. . . . . . 7 ⊢ (𝑖 ∈ (1...((𝑥 + 1) − 1)) → 𝑖 ∈ ℕ)
9		2rp 10076	. . . . . . . . . . . 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 9778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
13	10, 12	rpexpcld 11156	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
14	13	rpreccld 10125	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
15	14	rpred 10114	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
16		0re 8332	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17		1re 8331	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
18		prssi 3873	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆ ℝ)
19	16, 17, 18	mp2an 430	. . . . . . . . 9 ⊢ {0, 1} ⊆ ℝ
20		nconstwlpolem0.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ⟶{0, 1})
21	20	ad2antrr 492	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 𝐺:ℕ⟶{0, 1})
22	21, 11	ffvelcdmd 5847	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ {0, 1})
23	19, 22	sselid 3246	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ ℝ)
24	15, 23	remulcld 8362	. . . . . . 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 12193	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
27		eqid 2238	. . . . . 6 ⊢ (ℤ_≥‘(𝑥 + 1)) = (ℤ_≥‘(𝑥 + 1))
28		eqid 2238	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))
29		oveq2 6096	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
30	29	oveq2d 6104	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
31		fveq2 5698	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (𝐺‘𝑛) = (𝐺‘𝑖))
32	30, 31	oveq12d 6106	. . . . . . 7 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐺‘𝑛)) = ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
33		eluznn 10016	. . . . . . . 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 5802	. . . . . 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 17292	. . . . . . . 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 9974	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
40	28, 32, 11, 24	fvmptd3 5802	. . . . . . . . 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 8360	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℂ)
42	40, 41	eqeltrd 2315	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))‘𝑖) ∈ ℂ)
43	39, 4, 42	iserex 12130	. . . . . . 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 12221	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
46	3	nnzd 9778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℤ)
47		fzofig 10890	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (1..^𝑥) ∈ Fin)
48	2, 46, 47	syl2anc 415	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1..^𝑥) ∈ Fin)
49		elfzo1 10620	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1..^𝑥) ↔ (𝑖 ∈ ℕ ∧ 𝑥 ∈ ℕ ∧ 𝑖 < 𝑥))
50	49	simp1bi 1043	. . . . . . . . . 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 12193	. . . . . . . 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 11156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (2↑𝑥) ∈ ℝ⁺)
55	54	rpreccld 10125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℝ⁺)
56	55	rpred 10114	. . . . . . . . 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 5847	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) ∈ {0, 1})
59	19, 58	sselid 3246	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) ∈ ℝ)
60	56, 59	remulcld 8362	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) ∈ ℝ)
61	14	rpge0d 10118	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ (1 / (2↑𝑖)))
62		0le0 9402	. . . . . . . . . . . . 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 4169	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 0) → 0 ≤ (𝐺‘𝑖))
65		0le1 8816	. . . . . . . . . . . . 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 4169	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 1) → 0 ≤ (𝐺‘𝑖))
68		elpri 3732	. . . . . . . . . . . . 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 810	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ (𝐺‘𝑖))
71	15, 23, 61, 70	mulge0d 8957	. . . . . . . . . 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 12251	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 ≤ Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)))
74	55	rpgt0d 10117	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < (1 / (2↑𝑥)))
75		simprr 537	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) = 1)
76	75	oveq2d 6104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) = ((1 / (2↑𝑥)) · 1))
77	56	recnd 8360	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℂ)
78	77	mulridd 8349	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · 1) = (1 / (2↑𝑥)))
79	76, 78	eqtrd 2271	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) = (1 / (2↑𝑥)))
80	74, 79	breqtrrd 4159	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < ((1 / (2↑𝑥)) · (𝐺‘𝑥)))
81	52, 60, 73, 80	addgegt0d 8854	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)) + ((1 / (2↑𝑥)) · (𝐺‘𝑥))))
82		nfv 1581	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1))
83		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑖((1 / (2↑𝑥)) · (𝐺‘𝑥))
84		fzonel 10585	. . . . . . . . 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 6096	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (2↑𝑖) = (2↑𝑥))
88	87	oveq2d 6104	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (1 / (2↑𝑖)) = (1 / (2↑𝑥)))
89		fveq2 5698	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝐺‘𝑖) = (𝐺‘𝑥))
90	88, 89	oveq12d 6106	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) = ((1 / (2↑𝑥)) · (𝐺‘𝑥)))
91	60	recnd 8360	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) ∈ ℂ)
92	82, 83, 48, 3, 85, 86, 90, 91	fsumsplitsn 12202	. . . . . . 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 4159	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐺‘𝑖)))
94	3	nncnd 9327	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℂ)
95		1cnd 8348	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 1 ∈ ℂ)
96	94, 95	pncand 8645	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((𝑥 + 1) − 1) = 𝑥)
97	96	oveq2d 6104	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) = (1...𝑥))
98	3, 39	eleqtrdi 2331	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ (ℤ_≥‘1))
99		fzisfzounsn 10672	. . . . . . . . 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 2271	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) = ((1..^𝑥) ∪ {𝑥}))
102	101	sumeq1d 12157	. . . . . 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 4159	. . . . 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 10118	. . . . . . 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 8957	. . . . . 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 12222	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 ≤ Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)))
111	26, 45, 103, 110	addgtge0d 8855	. . . 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 12283	. . . 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 4159	. . 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 4173	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < 𝐴)
116	1, 115	rexlimddv 2673	1 ⊢ (𝜑 → 0 < 𝐴)

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 720 = wceq 1402 ∈ wcel 2209 ∃wrex 2529 ∪ cun 3218 ⊆ wss 3220 {csn 3709 {cpr 3710 class class class wbr 4131 ↦ cmpt 4193 dom cdm 4776 ⟶wf 5376 ‘cfv 5380 (class class class)co 6088 Fincfn 7026 ℂcc 8183 ℝcr 8184 0cc0 8185 1c1 8186 + caddc 8188 · cmul 8190 < clt 8366 ≤ cle 8367 − cmin 8504 / cdiv 9010 ℕcn 9313 2c2 9364 ℤcz 9655 ℤ_≥cuz 9937 ℝ⁺crp 10071 ...cfz 10428 ..^cfzo 10566 seqcseq 10905 ↑cexp 10996 ⇝ cli 12069 Σcsu 12144

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-en 7027 df-dom 7028 df-fin 7029 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-ico 10313 df-fz 10429 df-fzo 10567 df-seqfrec 10906 df-exp 10997 df-ihash 11237 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-clim 12070 df-sumdc 12145

This theorem is used by: nconstwlpolem 17322

W3C validator