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Theorem leibpilem2 26999

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 12339	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
3		nn0cn 12534	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
4		mulcl 11237	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (2 · 𝑛) ∈ ℂ)
5	2, 3, 4	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℂ)
6		ax-1cn 11211	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
7		pncan 11512	. . . . . . . . . . 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 7446	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((((2 · 𝑛) + 1) − 1) / 2) = ((2 · 𝑛) / 2))
10		2ne0 12368	. . . . . . . . . . 11 ⊢ 2 ≠ 0
11		divcan3 11946	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑛) / 2) = 𝑛)
12	2, 10, 11	mp3an23 1452	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℂ → ((2 · 𝑛) / 2) = 𝑛)
13	3, 12	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) / 2) = 𝑛)
14	9, 13	eqtrd 2775	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((((2 · 𝑛) + 1) − 1) / 2) = 𝑛)
15	14	oveq2d 7447	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (-1↑((((2 · 𝑛) + 1) − 1) / 2)) = (-1↑𝑛))
16	15	oveq1d 7446	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) = ((-1↑𝑛) / ((2 · 𝑛) + 1)))
17	16	mpteq2ia 5251	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
18	1, 17	eqtr4i 2766	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))
19		seqeq3 14044	. . . 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 5155	. 2 ⊢ (seq0( + , 𝐹) ⇝ 𝐴 ↔ seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))) ⇝ 𝐴)
22		neg1rr 12379	. . . . . . . . 9 ⊢ -1 ∈ ℝ
23		reexpcl 14116	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 𝑛 ∈ ℕ₀) → (-1↑𝑛) ∈ ℝ)
24	22, 23	mpan 690	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (-1↑𝑛) ∈ ℝ)
25		2nn0 12541	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
26		nn0mulcl 12560	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
27	25, 26	mpan 690	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℕ₀)
28		nn0p1nn 12563	. . . . . . . . 9 ⊢ ((2 · 𝑛) ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
29	27, 28	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
30	24, 29	nndivred 12318	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℝ)
31	30	recnd 11287	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℂ)
32	16, 31	eqeltrd 2839	. . . . 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 7438	. . . . . . 7 ⊢ (𝑘 = ((2 · 𝑛) + 1) → (𝑘 − 1) = (((2 · 𝑛) + 1) − 1))
35	34	oveq1d 7446	. . . . . 6 ⊢ (𝑘 = ((2 · 𝑛) + 1) → ((𝑘 − 1) / 2) = ((((2 · 𝑛) + 1) − 1) / 2))
36	35	oveq2d 7447	. . . . 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 7449	. . . 4 ⊢ (𝑘 = ((2 · 𝑛) + 1) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) = ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))
39	33, 38	iserodd 16869	. . 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 1544	. 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 11442	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
42	41	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
43		0cnd 11252	. . . . . . 7 ⊢ (⊤ → 0 ∈ ℂ)
44		1eluzge0 12932	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
45	44	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ∈ (ℤ_≥‘0))
46		1nn0 12540	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
47		leibpilem2.2	. . . . . . . . . 10 ⊢ 𝐺 = (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
48		0cnd 11252	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
49		ioran 985	. . . . . . . . . . . 12 ⊢ (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
50		leibpilem1 26998	. . . . . . . . . . . . . . . 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 14116	. . . . . . . . . . . . . . 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 12318	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
56	55	recnd 11287	. . . . . . . . . . . 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 4566	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
59	47, 58	fmpti 7132	. . . . . . . . 9 ⊢ 𝐺:ℕ₀⟶ℂ
60	59	ffvelcdmi 7103	. . . . . . . 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 12336	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
64	63	oveq2i 7442	. . . . . . . . . . 11 ⊢ (0...(1 − 1)) = (0...0)
65	62, 64	eleqtrdi 2849	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 ∈ (0...0))
66		elfz1eq 13572	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...0) → 𝑛 = 0)
67	65, 66	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 = 0)
68	67	fveq2d 6911	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → (𝐺‘𝑛) = (𝐺‘0))
69		0nn0 12539	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
70		iftrue 4537	. . . . . . . . . . 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 11253	. . . . . . . . . 10 ⊢ 0 ∈ V
73	71, 47, 72	fvmpt 7016	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (𝐺‘0) = 0)
74	69, 73	ax-mp 5	. . . . . . . 8 ⊢ (𝐺‘0) = 0
75	68, 74	eqtrdi 2791	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → (𝐺‘𝑛) = 0)
76	42, 43, 45, 61, 75	seqid 14085	. . . . . 6 ⊢ (⊤ → (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , 𝐺))
77		1zzd 12646	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ)
78		simpr 484	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ (ℤ_≥‘1))
79		nnuz 12919	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
80	78, 79	eleqtrrdi 2850	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ ℕ)
81		nnne0 12298	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
82	81	neneqd 2943	. . . . . . . . . . 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 4554	. . . . . . . . 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 5152	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (2 ∥ 𝑘 ↔ 2 ∥ 𝑛))
87		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝑘 − 1) = (𝑛 − 1))
88	87	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝑘 − 1) / 2) = ((𝑛 − 1) / 2))
89	88	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (-1↑((𝑘 − 1) / 2)) = (-1↑((𝑛 − 1) / 2)))
90		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
91	89, 90	oveq12d 7449	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((-1↑((𝑘 − 1) / 2)) / 𝑘) = ((-1↑((𝑛 − 1) / 2)) / 𝑛))
92	86, 91	ifbieq2d 4557	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)))
93		eqid 2735	. . . . . . . . . 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 7464	. . . . . . . . . . 11 ⊢ ((-1↑((𝑛 − 1) / 2)) / 𝑛) ∈ V
95	72, 94	ifex 4581	. . . . . . . . . 10 ⊢ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) ∈ V
96	92, 93, 95	fvmpt 7016	. . . . . . . . 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 12531	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
98		eqeq1 2739	. . . . . . . . . . . . 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 4557	. . . . . . . . . . 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 4581	. . . . . . . . . . 11 ⊢ if((𝑛 = 0 ∨ 2 ∥ 𝑛), 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) ∈ V
102	100, 47, 101	fvmpt 7016	. . . . . . . . . 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 2785	. . . . . . . 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 14065	. . . . . 6 ⊢ (⊤ → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) = seq1( + , 𝐺))
107	76, 106	eqtr4d 2778	. . . . 5 ⊢ (⊤ → (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))))
108	107	mptru 1544	. . . 4 ⊢ (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))))
109	108	breq1i 5155	. . 3 ⊢ ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ 𝐴)
110		1z 12645	. . . 4 ⊢ 1 ∈ ℤ
111		seqex 14041	. . . 4 ⊢ seq0( + , 𝐺) ∈ V
112		climres 15608	. . . 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 1537 ⊤wtru 1538 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 -cneg 11491 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 seqcseq 14039 ↑cexp 14099 ⇝ cli 15517 ∥ cdvds 16287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-seq 14040 df-exp 14100 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-dvds 16288

This theorem is referenced by: leibpi 27000

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