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Theorem leibpilem2 26849

Description: The Leibniz formula for π. (Contributed by Mario Carneiro, 7-Apr-2015.)

**Hypotheses**
Ref	Expression
leibpi.1	⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
leibpilem2.2	⊢ 𝐺 = (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
leibpilem2.3	⊢ 𝐴 ∈ V

**Assertion**
Ref	Expression
leibpilem2	⊢ (seq0( + , 𝐹) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴)

Distinct variable groups: 𝑘,𝑛 𝑛,𝐺 Allowed substitution hints: 𝐴(𝑘,𝑛) 𝐹(𝑘,𝑛) 𝐺(𝑘)

**Proof of Theorem leibpilem2**
Step	Hyp	Ref	Expression
1		leibpi.1	. . . . 5 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
2		2cn 12203	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
3		nn0cn 12394	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
4		mulcl 11093	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (2 · 𝑛) ∈ ℂ)
5	2, 3, 4	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℂ)
6		ax-1cn 11067	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
7		pncan 11369	. . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
8	5, 6, 7	sylancl 586	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
9	8	oveq1d 7364	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((((2 · 𝑛) + 1) − 1) / 2) = ((2 · 𝑛) / 2))
10		2ne0 12232	. . . . . . . . . . 11 ⊢ 2 ≠ 0
11		divcan3 11805	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑛) / 2) = 𝑛)
12	2, 10, 11	mp3an23 1455	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℂ → ((2 · 𝑛) / 2) = 𝑛)
13	3, 12	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) / 2) = 𝑛)
14	9, 13	eqtrd 2764	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((((2 · 𝑛) + 1) − 1) / 2) = 𝑛)
15	14	oveq2d 7365	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (-1↑((((2 · 𝑛) + 1) − 1) / 2)) = (-1↑𝑛))
16	15	oveq1d 7364	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) = ((-1↑𝑛) / ((2 · 𝑛) + 1)))
17	16	mpteq2ia 5187	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
18	1, 17	eqtr4i 2755	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))
19		seqeq3 13913	. . . 4 ⊢ (𝐹 = (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1))) → seq0( + , 𝐹) = seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))))
20	18, 19	ax-mp 5	. . 3 ⊢ seq0( + , 𝐹) = seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1))))
21	20	breq1i 5099	. 2 ⊢ (seq0( + , 𝐹) ⇝ 𝐴 ↔ seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))) ⇝ 𝐴)
22		neg1rr 12114	. . . . . . . . 9 ⊢ -1 ∈ ℝ
23		reexpcl 13985	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 𝑛 ∈ ℕ₀) → (-1↑𝑛) ∈ ℝ)
24	22, 23	mpan 690	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (-1↑𝑛) ∈ ℝ)
25		2nn0 12401	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
26		nn0mulcl 12420	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
27	25, 26	mpan 690	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℕ₀)
28		nn0p1nn 12423	. . . . . . . . 9 ⊢ ((2 · 𝑛) ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
29	27, 28	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
30	24, 29	nndivred 12182	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℝ)
31	30	recnd 11143	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℂ)
32	16, 31	eqeltrd 2828	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) ∈ ℂ)
33	32	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) ∈ ℂ)
34		oveq1 7356	. . . . . . 7 ⊢ (𝑘 = ((2 · 𝑛) + 1) → (𝑘 − 1) = (((2 · 𝑛) + 1) − 1))
35	34	oveq1d 7364	. . . . . 6 ⊢ (𝑘 = ((2 · 𝑛) + 1) → ((𝑘 − 1) / 2) = ((((2 · 𝑛) + 1) − 1) / 2))
36	35	oveq2d 7365	. . . . 5 ⊢ (𝑘 = ((2 · 𝑛) + 1) → (-1↑((𝑘 − 1) / 2)) = (-1↑((((2 · 𝑛) + 1) − 1) / 2)))
37		id 22	. . . . 5 ⊢ (𝑘 = ((2 · 𝑛) + 1) → 𝑘 = ((2 · 𝑛) + 1))
38	36, 37	oveq12d 7367	. . . 4 ⊢ (𝑘 = ((2 · 𝑛) + 1) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) = ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))
39	33, 38	iserodd 16747	. . 3 ⊢ (⊤ → (seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))) ⇝ 𝐴 ↔ seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ 𝐴))
40	39	mptru 1547	. 2 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))) ⇝ 𝐴 ↔ seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ 𝐴)
41		addlid 11299	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
42	41	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
43		0cnd 11108	. . . . . . 7 ⊢ (⊤ → 0 ∈ ℂ)
44		1eluzge0 12781	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
45	44	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ∈ (ℤ_≥‘0))
46		1nn0 12400	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
47		leibpilem2.2	. . . . . . . . . 10 ⊢ 𝐺 = (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
48		0cnd 11108	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
49		ioran 985	. . . . . . . . . . . 12 ⊢ (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
50		leibpilem1 26848	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀))
51	50	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ₀)
52		reexpcl 13985	. . . . . . . . . . . . . . 15 ⊢ ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
53	22, 51, 52	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
54	50	simpld 494	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
55	53, 54	nndivred 12182	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
56	55	recnd 11143	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
57	49, 56	sylan2b 594	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
58	48, 57	ifclda 4512	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
59	47, 58	fmpti 7046	. . . . . . . . 9 ⊢ 𝐺:ℕ₀⟶ℂ
60	59	ffvelcdmi 7017	. . . . . . . 8 ⊢ (1 ∈ ℕ₀ → (𝐺‘1) ∈ ℂ)
61	46, 60	mp1i 13	. . . . . . 7 ⊢ (⊤ → (𝐺‘1) ∈ ℂ)
62		simpr 484	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 ∈ (0...(1 − 1)))
63		1m1e0 12200	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
64	63	oveq2i 7360	. . . . . . . . . . 11 ⊢ (0...(1 − 1)) = (0...0)
65	62, 64	eleqtrdi 2838	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 ∈ (0...0))
66		elfz1eq 13438	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...0) → 𝑛 = 0)
67	65, 66	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 = 0)
68	67	fveq2d 6826	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → (𝐺‘𝑛) = (𝐺‘0))
69		0nn0 12399	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
70		iftrue 4482	. . . . . . . . . . 11 ⊢ ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = 0)
71	70	orcs 875	. . . . . . . . . 10 ⊢ (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = 0)
72		c0ex 11109	. . . . . . . . . 10 ⊢ 0 ∈ V
73	71, 47, 72	fvmpt 6930	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (𝐺‘0) = 0)
74	69, 73	ax-mp 5	. . . . . . . 8 ⊢ (𝐺‘0) = 0
75	68, 74	eqtrdi 2780	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → (𝐺‘𝑛) = 0)
76	42, 43, 45, 61, 75	seqid 13954	. . . . . 6 ⊢ (⊤ → (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , 𝐺))
77		1zzd 12506	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ)
78		simpr 484	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ (ℤ_≥‘1))
79		nnuz 12778	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
80	78, 79	eleqtrrdi 2839	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ ℕ)
81		nnne0 12162	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
82	81	neneqd 2930	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
83		biorf 936	. . . . . . . . . . 11 ⊢ (¬ 𝑛 = 0 → (2 ∥ 𝑛 ↔ (𝑛 = 0 ∨ 2 ∥ 𝑛)))
84	82, 83	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (2 ∥ 𝑛 ↔ (𝑛 = 0 ∨ 2 ∥ 𝑛)))
85	84	ifbid 4500	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) = if((𝑛 = 0 ∨ 2 ∥ 𝑛), 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)))
86		breq2 5096	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (2 ∥ 𝑘 ↔ 2 ∥ 𝑛))
87		oveq1 7356	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝑘 − 1) = (𝑛 − 1))
88	87	oveq1d 7364	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝑘 − 1) / 2) = ((𝑛 − 1) / 2))
89	88	oveq2d 7365	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (-1↑((𝑘 − 1) / 2)) = (-1↑((𝑛 − 1) / 2)))
90		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
91	89, 90	oveq12d 7367	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((-1↑((𝑘 − 1) / 2)) / 𝑘) = ((-1↑((𝑛 − 1) / 2)) / 𝑛))
92	86, 91	ifbieq2d 4503	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)))
93		eqid 2729	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))) = (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
94		ovex 7382	. . . . . . . . . . 11 ⊢ ((-1↑((𝑛 − 1) / 2)) / 𝑛) ∈ V
95	72, 94	ifex 4527	. . . . . . . . . 10 ⊢ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) ∈ V
96	92, 93, 95	fvmpt 6930	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑛) = if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)))
97		nnnn0 12391	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
98		eqeq1 2733	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
99	98, 86	orbi12d 918	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (𝑛 = 0 ∨ 2 ∥ 𝑛)))
100	99, 91	ifbieq2d 4503	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = if((𝑛 = 0 ∨ 2 ∥ 𝑛), 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)))
101	72, 94	ifex 4527	. . . . . . . . . . 11 ⊢ if((𝑛 = 0 ∨ 2 ∥ 𝑛), 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) ∈ V
102	100, 47, 101	fvmpt 6930	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (𝐺‘𝑛) = if((𝑛 = 0 ∨ 2 ∥ 𝑛), 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)))
103	97, 102	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = if((𝑛 = 0 ∨ 2 ∥ 𝑛), 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)))
104	85, 96, 103	3eqtr4d 2774	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑛) = (𝐺‘𝑛))
105	80, 104	syl 17	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑛) = (𝐺‘𝑛))
106	77, 105	seqfeq 13934	. . . . . 6 ⊢ (⊤ → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) = seq1( + , 𝐺))
107	76, 106	eqtr4d 2767	. . . . 5 ⊢ (⊤ → (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))))
108	107	mptru 1547	. . . 4 ⊢ (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))))
109	108	breq1i 5099	. . 3 ⊢ ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ 𝐴)
110		1z 12505	. . . 4 ⊢ 1 ∈ ℤ
111		seqex 13910	. . . 4 ⊢ seq0( + , 𝐺) ∈ V
112		climres 15482	. . . 4 ⊢ ((1 ∈ ℤ ∧ seq0( + , 𝐺) ∈ V) → ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴))
113	110, 111, 112	mp2an 692	. . 3 ⊢ ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴)
114	109, 113	bitr3i 277	. 2 ⊢ (seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴)
115	21, 40, 114	3bitri 297	1 ⊢ (seq0( + , 𝐹) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 Vcvv 3436 ifcif 4476 class class class wbr 5092 ↦ cmpt 5173 ↾ cres 5621 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 − cmin 11347 -cneg 11348 / cdiv 11777 ℕcn 12128 2c2 12183 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ...cfz 13410 seqcseq 13908 ↑cexp 13968 ⇝ cli 15391 ∥ cdvds 16163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-oadd 8392 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-n0 12385 df-xnn0 12458 df-z 12472 df-uz 12736 df-rp 12894 df-fz 13411 df-seq 13909 df-exp 13969 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-dvds 16164

This theorem is referenced by: leibpi 26850

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