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Theorem leibpilem2 27002

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 12368	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
3		nn0cn 12563	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
4		mulcl 11268	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (2 · 𝑛) ∈ ℂ)
5	2, 3, 4	sylancr 586	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℂ)
6		ax-1cn 11242	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
7		pncan 11542	. . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
8	5, 6, 7	sylancl 585	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
9	8	oveq1d 7463	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((((2 · 𝑛) + 1) − 1) / 2) = ((2 · 𝑛) / 2))
10		2ne0 12397	. . . . . . . . . . 11 ⊢ 2 ≠ 0
11		divcan3 11975	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑛) / 2) = 𝑛)
12	2, 10, 11	mp3an23 1453	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℂ → ((2 · 𝑛) / 2) = 𝑛)
13	3, 12	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) / 2) = 𝑛)
14	9, 13	eqtrd 2780	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((((2 · 𝑛) + 1) − 1) / 2) = 𝑛)
15	14	oveq2d 7464	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (-1↑((((2 · 𝑛) + 1) − 1) / 2)) = (-1↑𝑛))
16	15	oveq1d 7463	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) = ((-1↑𝑛) / ((2 · 𝑛) + 1)))
17	16	mpteq2ia 5269	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
18	1, 17	eqtr4i 2771	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))
19		seqeq3 14057	. . . 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 5173	. 2 ⊢ (seq0( + , 𝐹) ⇝ 𝐴 ↔ seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))) ⇝ 𝐴)
22		neg1rr 12408	. . . . . . . . 9 ⊢ -1 ∈ ℝ
23		reexpcl 14129	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 𝑛 ∈ ℕ₀) → (-1↑𝑛) ∈ ℝ)
24	22, 23	mpan 689	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (-1↑𝑛) ∈ ℝ)
25		2nn0 12570	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
26		nn0mulcl 12589	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
27	25, 26	mpan 689	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℕ₀)
28		nn0p1nn 12592	. . . . . . . . 9 ⊢ ((2 · 𝑛) ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
29	27, 28	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
30	24, 29	nndivred 12347	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℝ)
31	30	recnd 11318	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℂ)
32	16, 31	eqeltrd 2844	. . . . 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 7455	. . . . . . 7 ⊢ (𝑘 = ((2 · 𝑛) + 1) → (𝑘 − 1) = (((2 · 𝑛) + 1) − 1))
35	34	oveq1d 7463	. . . . . 6 ⊢ (𝑘 = ((2 · 𝑛) + 1) → ((𝑘 − 1) / 2) = ((((2 · 𝑛) + 1) − 1) / 2))
36	35	oveq2d 7464	. . . . 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 7466	. . . 4 ⊢ (𝑘 = ((2 · 𝑛) + 1) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) = ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))
39	33, 38	iserodd 16882	. . 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 11473	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
42	41	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
43		0cnd 11283	. . . . . . 7 ⊢ (⊤ → 0 ∈ ℂ)
44		1eluzge0 12957	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
45	44	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ∈ (ℤ_≥‘0))
46		1nn0 12569	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
47		leibpilem2.2	. . . . . . . . . 10 ⊢ 𝐺 = (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
48		0cnd 11283	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
49		ioran 984	. . . . . . . . . . . 12 ⊢ (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
50		leibpilem1 27001	. . . . . . . . . . . . . . . 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 14129	. . . . . . . . . . . . . . 15 ⊢ ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
53	22, 51, 52	sylancr 586	. . . . . . . . . . . . . 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 12347	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
56	55	recnd 11318	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
57	49, 56	sylan2b 593	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
58	48, 57	ifclda 4583	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
59	47, 58	fmpti 7146	. . . . . . . . 9 ⊢ 𝐺:ℕ₀⟶ℂ
60	59	ffvelcdmi 7117	. . . . . . . 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 12365	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
64	63	oveq2i 7459	. . . . . . . . . . 11 ⊢ (0...(1 − 1)) = (0...0)
65	62, 64	eleqtrdi 2854	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 ∈ (0...0))
66		elfz1eq 13595	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...0) → 𝑛 = 0)
67	65, 66	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 = 0)
68	67	fveq2d 6924	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → (𝐺‘𝑛) = (𝐺‘0))
69		0nn0 12568	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
70		iftrue 4554	. . . . . . . . . . 11 ⊢ ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = 0)
71	70	orcs 874	. . . . . . . . . 10 ⊢ (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = 0)
72		c0ex 11284	. . . . . . . . . 10 ⊢ 0 ∈ V
73	71, 47, 72	fvmpt 7029	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (𝐺‘0) = 0)
74	69, 73	ax-mp 5	. . . . . . . 8 ⊢ (𝐺‘0) = 0
75	68, 74	eqtrdi 2796	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → (𝐺‘𝑛) = 0)
76	42, 43, 45, 61, 75	seqid 14098	. . . . . 6 ⊢ (⊤ → (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , 𝐺))
77		1zzd 12674	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ)
78		simpr 484	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ (ℤ_≥‘1))
79		nnuz 12946	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
80	78, 79	eleqtrrdi 2855	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ ℕ)
81		nnne0 12327	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
82	81	neneqd 2951	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
83		biorf 935	. . . . . . . . . . 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 4571	. . . . . . . . 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 5170	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (2 ∥ 𝑘 ↔ 2 ∥ 𝑛))
87		oveq1 7455	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝑘 − 1) = (𝑛 − 1))
88	87	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝑘 − 1) / 2) = ((𝑛 − 1) / 2))
89	88	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (-1↑((𝑘 − 1) / 2)) = (-1↑((𝑛 − 1) / 2)))
90		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
91	89, 90	oveq12d 7466	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((-1↑((𝑘 − 1) / 2)) / 𝑘) = ((-1↑((𝑛 − 1) / 2)) / 𝑛))
92	86, 91	ifbieq2d 4574	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)))
93		eqid 2740	. . . . . . . . . 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 7481	. . . . . . . . . . 11 ⊢ ((-1↑((𝑛 − 1) / 2)) / 𝑛) ∈ V
95	72, 94	ifex 4598	. . . . . . . . . 10 ⊢ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) ∈ V
96	92, 93, 95	fvmpt 7029	. . . . . . . . 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 12560	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
98		eqeq1 2744	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
99	98, 86	orbi12d 917	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (𝑛 = 0 ∨ 2 ∥ 𝑛)))
100	99, 91	ifbieq2d 4574	. . . . . . . . . . 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 4598	. . . . . . . . . . 11 ⊢ if((𝑛 = 0 ∨ 2 ∥ 𝑛), 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) ∈ V
102	100, 47, 101	fvmpt 7029	. . . . . . . . . 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 2790	. . . . . . . 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 14078	. . . . . 6 ⊢ (⊤ → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) = seq1( + , 𝐺))
107	76, 106	eqtr4d 2783	. . . . 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 5173	. . 3 ⊢ ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ 𝐴)
110		1z 12673	. . . 4 ⊢ 1 ∈ ℤ
111		seqex 14054	. . . 4 ⊢ seq0( + , 𝐺) ∈ V
112		climres 15621	. . . 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 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 846 = wceq 1537 ⊤wtru 1538 ∈ wcel 2108 ≠ wne 2946 Vcvv 3488 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 ↾ cres 5702 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ...cfz 13567 seqcseq 14052 ↑cexp 14112 ⇝ cli 15530 ∥ cdvds 16302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-seq 14053 df-exp 14113 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-dvds 16303

This theorem is referenced by: leibpi 27003

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