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Theorem leibpilem2 25527

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 11700	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
3		nn0cn 11895	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
4		mulcl 10610	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (2 · 𝑛) ∈ ℂ)
5	2, 3, 4	sylancr 590	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℂ)
6		ax-1cn 10584	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
7		pncan 10881	. . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
8	5, 6, 7	sylancl 589	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
9	8	oveq1d 7150	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((((2 · 𝑛) + 1) − 1) / 2) = ((2 · 𝑛) / 2))
10		2ne0 11729	. . . . . . . . . . 11 ⊢ 2 ≠ 0
11		divcan3 11313	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑛) / 2) = 𝑛)
12	2, 10, 11	mp3an23 1450	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℂ → ((2 · 𝑛) / 2) = 𝑛)
13	3, 12	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) / 2) = 𝑛)
14	9, 13	eqtrd 2833	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((((2 · 𝑛) + 1) − 1) / 2) = 𝑛)
15	14	oveq2d 7151	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (-1↑((((2 · 𝑛) + 1) − 1) / 2)) = (-1↑𝑛))
16	15	oveq1d 7150	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) = ((-1↑𝑛) / ((2 · 𝑛) + 1)))
17	16	mpteq2ia 5121	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
18	1, 17	eqtr4i 2824	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))
19		seqeq3 13369	. . . 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 5037	. 2 ⊢ (seq0( + , 𝐹) ⇝ 𝐴 ↔ seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))) ⇝ 𝐴)
22		neg1rr 11740	. . . . . . . . 9 ⊢ -1 ∈ ℝ
23		reexpcl 13442	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 𝑛 ∈ ℕ₀) → (-1↑𝑛) ∈ ℝ)
24	22, 23	mpan 689	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (-1↑𝑛) ∈ ℝ)
25		2nn0 11902	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
26		nn0mulcl 11921	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
27	25, 26	mpan 689	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℕ₀)
28		nn0p1nn 11924	. . . . . . . . 9 ⊢ ((2 · 𝑛) ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
29	27, 28	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
30	24, 29	nndivred 11679	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℝ)
31	30	recnd 10658	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℂ)
32	16, 31	eqeltrd 2890	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) ∈ ℂ)
33	32	adantl 485	. . . 4 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) ∈ ℂ)
34		oveq1 7142	. . . . . . 7 ⊢ (𝑘 = ((2 · 𝑛) + 1) → (𝑘 − 1) = (((2 · 𝑛) + 1) − 1))
35	34	oveq1d 7150	. . . . . 6 ⊢ (𝑘 = ((2 · 𝑛) + 1) → ((𝑘 − 1) / 2) = ((((2 · 𝑛) + 1) − 1) / 2))
36	35	oveq2d 7151	. . . . 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 7153	. . . 4 ⊢ (𝑘 = ((2 · 𝑛) + 1) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) = ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))
39	33, 38	iserodd 16162	. . 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 1545	. 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		addid2 10812	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
42	41	adantl 485	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
43		0cnd 10623	. . . . . . 7 ⊢ (⊤ → 0 ∈ ℂ)
44		1eluzge0 12280	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
45	44	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ∈ (ℤ_≥‘0))
46		1nn0 11901	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
47		leibpilem2.2	. . . . . . . . . 10 ⊢ 𝐺 = (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
48		0cnd 10623	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
49		ioran 981	. . . . . . . . . . . 12 ⊢ (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
50		leibpilem1 25526	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀))
51	50	simprd 499	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ₀)
52		reexpcl 13442	. . . . . . . . . . . . . . 15 ⊢ ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
53	22, 51, 52	sylancr 590	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
54	50	simpld 498	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
55	53, 54	nndivred 11679	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
56	55	recnd 10658	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
57	49, 56	sylan2b 596	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
58	48, 57	ifclda 4459	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
59	47, 58	fmpti 6853	. . . . . . . . 9 ⊢ 𝐺:ℕ₀⟶ℂ
60	59	ffvelrni 6827	. . . . . . . 8 ⊢ (1 ∈ ℕ₀ → (𝐺‘1) ∈ ℂ)
61	46, 60	mp1i 13	. . . . . . 7 ⊢ (⊤ → (𝐺‘1) ∈ ℂ)
62		simpr 488	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 ∈ (0...(1 − 1)))
63		1m1e0 11697	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
64	63	oveq2i 7146	. . . . . . . . . . 11 ⊢ (0...(1 − 1)) = (0...0)
65	62, 64	eleqtrdi 2900	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 ∈ (0...0))
66		elfz1eq 12913	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...0) → 𝑛 = 0)
67	65, 66	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 = 0)
68	67	fveq2d 6649	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → (𝐺‘𝑛) = (𝐺‘0))
69		0nn0 11900	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
70		iftrue 4431	. . . . . . . . . . 11 ⊢ ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = 0)
71	70	orcs 872	. . . . . . . . . 10 ⊢ (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = 0)
72		c0ex 10624	. . . . . . . . . 10 ⊢ 0 ∈ V
73	71, 47, 72	fvmpt 6745	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (𝐺‘0) = 0)
74	69, 73	ax-mp 5	. . . . . . . 8 ⊢ (𝐺‘0) = 0
75	68, 74	eqtrdi 2849	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → (𝐺‘𝑛) = 0)
76	42, 43, 45, 61, 75	seqid 13411	. . . . . 6 ⊢ (⊤ → (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , 𝐺))
77		1zzd 12001	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ)
78		simpr 488	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ (ℤ_≥‘1))
79		nnuz 12269	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
80	78, 79	eleqtrrdi 2901	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ ℕ)
81		nnne0 11659	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
82	81	neneqd 2992	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
83		biorf 934	. . . . . . . . . . 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 4447	. . . . . . . . 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 5034	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (2 ∥ 𝑘 ↔ 2 ∥ 𝑛))
87		oveq1 7142	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝑘 − 1) = (𝑛 − 1))
88	87	oveq1d 7150	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝑘 − 1) / 2) = ((𝑛 − 1) / 2))
89	88	oveq2d 7151	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (-1↑((𝑘 − 1) / 2)) = (-1↑((𝑛 − 1) / 2)))
90		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
91	89, 90	oveq12d 7153	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((-1↑((𝑘 − 1) / 2)) / 𝑘) = ((-1↑((𝑛 − 1) / 2)) / 𝑛))
92	86, 91	ifbieq2d 4450	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)))
93		eqid 2798	. . . . . . . . . 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 7168	. . . . . . . . . . 11 ⊢ ((-1↑((𝑛 − 1) / 2)) / 𝑛) ∈ V
95	72, 94	ifex 4473	. . . . . . . . . 10 ⊢ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) ∈ V
96	92, 93, 95	fvmpt 6745	. . . . . . . . 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 11892	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
98		eqeq1 2802	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
99	98, 86	orbi12d 916	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (𝑛 = 0 ∨ 2 ∥ 𝑛)))
100	99, 91	ifbieq2d 4450	. . . . . . . . . . 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 4473	. . . . . . . . . . 11 ⊢ if((𝑛 = 0 ∨ 2 ∥ 𝑛), 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) ∈ V
102	100, 47, 101	fvmpt 6745	. . . . . . . . . 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 2843	. . . . . . . 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 13391	. . . . . 6 ⊢ (⊤ → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) = seq1( + , 𝐺))
107	76, 106	eqtr4d 2836	. . . . 5 ⊢ (⊤ → (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))))
108	107	mptru 1545	. . . 4 ⊢ (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))))
109	108	breq1i 5037	. . 3 ⊢ ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ 𝐴)
110		1z 12000	. . . 4 ⊢ 1 ∈ ℤ
111		seqex 13366	. . . 4 ⊢ seq0( + , 𝐺) ∈ V
112		climres 14924	. . . 4 ⊢ ((1 ∈ ℤ ∧ seq0( + , 𝐺) ∈ V) → ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴))
113	110, 111, 112	mp2an 691	. . 3 ⊢ ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴)
114	109, 113	bitr3i 280	. 2 ⊢ (seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴)
115	21, 40, 114	3bitri 300	1 ⊢ (seq0( + , 𝐹) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ifcif 4425 class class class wbr 5030 ↦ cmpt 5110 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 ...cfz 12885 seqcseq 13364 ↑cexp 13425 ⇝ cli 14833 ∥ cdvds 15599

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-dvds 15600

This theorem is referenced by: leibpi 25528

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