nconstwlpolemgt0 - Mathbox for Jim Kingdon

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Theorem nconstwlpolemgt0 14467

Description: Lemma for nconstwlpo 14469. 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 9269	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 1 ∈ ℤ)
3		simprl 529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℕ)
4	3	peano2nnd 8923	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝑥 + 1) ∈ ℕ)
5	4	nnzd 9363	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝑥 + 1) ∈ ℤ)
6	5, 2	zsubcld 9369	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((𝑥 + 1) − 1) ∈ ℤ)
7	2, 6	fzfigd 10417	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) ∈ Fin)
8		elfznn 10040	. . . . . . 7 ⊢ (𝑖 ∈ (1...((𝑥 + 1) − 1)) → 𝑖 ∈ ℕ)
9		2rp 9645	. . . . . . . . . . . 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 9363	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
13	10, 12	rpexpcld 10663	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈ ℝ⁺)
14	13	rpreccld 9694	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ⁺)
15	14	rpred 9683	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈ ℝ)
16		0re 7948	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17		1re 7947	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
18		prssi 3749	. . . . . . . . . 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 5648	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ {0, 1})
23	19, 22	sselid 3153	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → (𝐺‘𝑖) ∈ ℝ)
24	15, 23	remulcld 7978	. . . . . . 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 11393	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
27		eqid 2177	. . . . . 6 ⊢ (ℤ_≥‘(𝑥 + 1)) = (ℤ_≥‘(𝑥 + 1))
28		eqid 2177	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))
29		oveq2 5877	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖))
30	29	oveq2d 5885	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖)))
31		fveq2 5511	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (𝐺‘𝑛) = (𝐺‘𝑖))
32	30, 31	oveq12d 5887	. . . . . . 7 ⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐺‘𝑛)) = ((1 / (2↑𝑖)) · (𝐺‘𝑖)))
33		eluznn 9589	. . . . . . . 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 5605	. . . . . 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 14440	. . . . . . . 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 9552	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
40	28, 32, 11, 24	fvmptd3 5605	. . . . . . . . 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 7976	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℂ)
42	40, 41	eqeltrd 2254	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐺‘𝑛)))‘𝑖) ∈ ℂ)
43	39, 4, 42	iserex 11331	. . . . . . 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 11421	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)) ∈ ℝ)
46	3	nnzd 9363	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℤ)
47		fzofig 10418	. . . . . . . . . 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 10176	. . . . . . . . . . 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 11393	. . . . . . . 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 10663	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (2↑𝑥) ∈ ℝ⁺)
55	54	rpreccld 9694	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℝ⁺)
56	55	rpred 9683	. . . . . . . . 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 5648	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) ∈ {0, 1})
59	19, 58	sselid 3153	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (𝐺‘𝑥) ∈ ℝ)
60	56, 59	remulcld 7978	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) ∈ ℝ)
61	14	rpge0d 9687	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) → 0 ≤ (1 / (2↑𝑖)))
62		0le0 8997	. . . . . . . . . . . . 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 4038	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 0) → 0 ≤ (𝐺‘𝑖))
65		0le1 8428	. . . . . . . . . . . . 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 4038	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) ∧ 𝑖 ∈ ℕ) ∧ (𝐺‘𝑖) = 1) → 0 ≤ (𝐺‘𝑖))
68		elpri 3614	. . . . . . . . . . . . 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 8568	. . . . . . . . . 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 11451	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 ≤ Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)))
74	55	rpgt0d 9686	. . . . . . . . 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 5885	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) = ((1 / (2↑𝑥)) · 1))
77	56	recnd 7976	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1 / (2↑𝑥)) ∈ ℂ)
78	77	mulid1d 7965	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · 1) = (1 / (2↑𝑥)))
79	76, 78	eqtrd 2210	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) = (1 / (2↑𝑥)))
80	74, 79	breqtrrd 4028	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < ((1 / (2↑𝑥)) · (𝐺‘𝑥)))
81	52, 60, 73, 80	addgegt0d 8466	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐺‘𝑖)) + ((1 / (2↑𝑥)) · (𝐺‘𝑥))))
82		nfv 1528	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1))
83		nfcv 2319	. . . . . . . 8 ⊢ Ⅎ𝑖((1 / (2↑𝑥)) · (𝐺‘𝑥))
84		fzonel 10146	. . . . . . . . 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 5877	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (2↑𝑖) = (2↑𝑥))
88	87	oveq2d 5885	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (1 / (2↑𝑖)) = (1 / (2↑𝑥)))
89		fveq2 5511	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → (𝐺‘𝑖) = (𝐺‘𝑥))
90	88, 89	oveq12d 5887	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → ((1 / (2↑𝑖)) · (𝐺‘𝑖)) = ((1 / (2↑𝑥)) · (𝐺‘𝑥)))
91	60	recnd 7976	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((1 / (2↑𝑥)) · (𝐺‘𝑥)) ∈ ℂ)
92	82, 83, 48, 3, 85, 86, 90, 91	fsumsplitsn 11402	. . . . . . 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 4028	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐺‘𝑖)))
94	3	nncnd 8922	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ ℂ)
95		1cnd 7964	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 1 ∈ ℂ)
96	94, 95	pncand 8259	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → ((𝑥 + 1) − 1) = 𝑥)
97	96	oveq2d 5885	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) = (1...𝑥))
98	3, 39	eleqtrdi 2270	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 𝑥 ∈ (ℤ_≥‘1))
99		fzisfzounsn 10222	. . . . . . . . 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 2210	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → (1...((𝑥 + 1) − 1)) = ((1..^𝑥) ∪ {𝑥}))
102	101	sumeq1d 11358	. . . . . 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 4028	. . . . 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 9687	. . . . . . 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 8568	. . . . . 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 11422	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 ≤ Σ𝑖 ∈ (ℤ_≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐺‘𝑖)))
111	26, 45, 103, 110	addgtge0d 8467	. . . 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 11483	. . . 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 4028	. . 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 4042	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) = 1)) → 0 < 𝐴)
116	1, 115	rexlimddv 2599	1 ⊢ (𝜑 → 0 < 𝐴)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 ∪ cun 3127 ⊆ wss 3129 {csn 3591 {cpr 3592 class class class wbr 4000 ↦ cmpt 4061 dom cdm 4623 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 Fincfn 6734 ℂcc 7800 ℝcr 7801 0cc0 7802 1c1 7803 + caddc 7805 · cmul 7807 < clt 7982 ≤ cle 7983 − cmin 8118 / cdiv 8618 ℕcn 8908 2c2 8959 ℤcz 9242 ℤ_≥cuz 9517 ℝ⁺crp 9640 ...cfz 9995 ..^cfzo 10128 seqcseq 10431 ↑cexp 10505 ⇝ cli 11270 Σcsu 11345

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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

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 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-oadd 6415 df-er 6529 df-en 6735 df-dom 6736 df-fin 6737 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-ico 9881 df-fz 9996 df-fzo 10129 df-seqfrec 10432 df-exp 10506 df-ihash 10740 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-clim 11271 df-sumdc 11346

This theorem is referenced by: nconstwlpolem 14468

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