leibpilem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > leibpilem2		Structured version Visualization version GIF version

Theorem leibpilem2 25519

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 11713	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
3		nn0cn 11908	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
4		mulcl 10621	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (2 · 𝑛) ∈ ℂ)
5	2, 3, 4	sylancr 589	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℂ)
6		ax-1cn 10595	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
7		pncan 10892	. . . . . . . . . . 11 ⊢ (((2 · 𝑛) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
8	5, 6, 7	sylancl 588	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (((2 · 𝑛) + 1) − 1) = (2 · 𝑛))
9	8	oveq1d 7171	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((((2 · 𝑛) + 1) − 1) / 2) = ((2 · 𝑛) / 2))
10		2ne0 11742	. . . . . . . . . . 11 ⊢ 2 ≠ 0
11		divcan3 11324	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑛) / 2) = 𝑛)
12	2, 10, 11	mp3an23 1449	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℂ → ((2 · 𝑛) / 2) = 𝑛)
13	3, 12	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) / 2) = 𝑛)
14	9, 13	eqtrd 2856	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((((2 · 𝑛) + 1) − 1) / 2) = 𝑛)
15	14	oveq2d 7172	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → (-1↑((((2 · 𝑛) + 1) − 1) / 2)) = (-1↑𝑛))
16	15	oveq1d 7171	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) = ((-1↑𝑛) / ((2 · 𝑛) + 1)))
17	16	mpteq2ia 5157	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
18	1, 17	eqtr4i 2847	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))
19		seqeq3 13375	. . . 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 5073	. 2 ⊢ (seq0( + , 𝐹) ⇝ 𝐴 ↔ seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))) ⇝ 𝐴)
22		neg1rr 11753	. . . . . . . . 9 ⊢ -1 ∈ ℝ
23		reexpcl 13447	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 𝑛 ∈ ℕ₀) → (-1↑𝑛) ∈ ℝ)
24	22, 23	mpan 688	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (-1↑𝑛) ∈ ℝ)
25		2nn0 11915	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
26		nn0mulcl 11934	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
27	25, 26	mpan 688	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℕ₀)
28		nn0p1nn 11937	. . . . . . . . 9 ⊢ ((2 · 𝑛) ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
29	27, 28	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
30	24, 29	nndivred 11692	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℝ)
31	30	recnd 10669	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑𝑛) / ((2 · 𝑛) + 1)) ∈ ℂ)
32	16, 31	eqeltrd 2913	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) ∈ ℂ)
33	32	adantl 484	. . . 4 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)) ∈ ℂ)
34		oveq1 7163	. . . . . . 7 ⊢ (𝑘 = ((2 · 𝑛) + 1) → (𝑘 − 1) = (((2 · 𝑛) + 1) − 1))
35	34	oveq1d 7171	. . . . . 6 ⊢ (𝑘 = ((2 · 𝑛) + 1) → ((𝑘 − 1) / 2) = ((((2 · 𝑛) + 1) − 1) / 2))
36	35	oveq2d 7172	. . . . 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 7174	. . . 4 ⊢ (𝑘 = ((2 · 𝑛) + 1) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) = ((-1↑((((2 · 𝑛) + 1) − 1) / 2)) / ((2 · 𝑛) + 1)))
39	33, 38	iserodd 16172	. . 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		addid2 10823	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
42	41	adantl 484	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
43		0cnd 10634	. . . . . . 7 ⊢ (⊤ → 0 ∈ ℂ)
44		1eluzge0 12293	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
45	44	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ∈ (ℤ_≥‘0))
46		1nn0 11914	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
47		leibpilem2.2	. . . . . . . . . 10 ⊢ 𝐺 = (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
48		0cnd 10634	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
49		ioran 980	. . . . . . . . . . . 12 ⊢ (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
50		leibpilem1 25518	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀))
51	50	simprd 498	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ₀)
52		reexpcl 13447	. . . . . . . . . . . . . . 15 ⊢ ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
53	22, 51, 52	sylancr 589	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
54	50	simpld 497	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
55	53, 54	nndivred 11692	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
56	55	recnd 10669	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
57	49, 56	sylan2b 595	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
58	48, 57	ifclda 4501	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
59	47, 58	fmpti 6876	. . . . . . . . 9 ⊢ 𝐺:ℕ₀⟶ℂ
60	59	ffvelrni 6850	. . . . . . . 8 ⊢ (1 ∈ ℕ₀ → (𝐺‘1) ∈ ℂ)
61	46, 60	mp1i 13	. . . . . . 7 ⊢ (⊤ → (𝐺‘1) ∈ ℂ)
62		simpr 487	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 ∈ (0...(1 − 1)))
63		1m1e0 11710	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
64	63	oveq2i 7167	. . . . . . . . . . 11 ⊢ (0...(1 − 1)) = (0...0)
65	62, 64	eleqtrdi 2923	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 ∈ (0...0))
66		elfz1eq 12919	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...0) → 𝑛 = 0)
67	65, 66	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → 𝑛 = 0)
68	67	fveq2d 6674	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → (𝐺‘𝑛) = (𝐺‘0))
69		0nn0 11913	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
70		iftrue 4473	. . . . . . . . . . 11 ⊢ ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = 0)
71	70	orcs 871	. . . . . . . . . 10 ⊢ (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = 0)
72		c0ex 10635	. . . . . . . . . 10 ⊢ 0 ∈ V
73	71, 47, 72	fvmpt 6768	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (𝐺‘0) = 0)
74	69, 73	ax-mp 5	. . . . . . . 8 ⊢ (𝐺‘0) = 0
75	68, 74	syl6eq 2872	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ (0...(1 − 1))) → (𝐺‘𝑛) = 0)
76	42, 43, 45, 61, 75	seqid 13416	. . . . . 6 ⊢ (⊤ → (seq0( + , 𝐺) ↾ (ℤ_≥‘1)) = seq1( + , 𝐺))
77		1zzd 12014	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ)
78		simpr 487	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ (ℤ_≥‘1))
79		nnuz 12282	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
80	78, 79	eleqtrrdi 2924	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ ℕ)
81		nnne0 11672	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
82	81	neneqd 3021	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
83		biorf 933	. . . . . . . . . . 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 4489	. . . . . . . . 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 5070	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (2 ∥ 𝑘 ↔ 2 ∥ 𝑛))
87		oveq1 7163	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝑘 − 1) = (𝑛 − 1))
88	87	oveq1d 7171	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝑘 − 1) / 2) = ((𝑛 − 1) / 2))
89	88	oveq2d 7172	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (-1↑((𝑘 − 1) / 2)) = (-1↑((𝑛 − 1) / 2)))
90		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
91	89, 90	oveq12d 7174	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((-1↑((𝑘 − 1) / 2)) / 𝑘) = ((-1↑((𝑛 − 1) / 2)) / 𝑛))
92	86, 91	ifbieq2d 4492	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)))
93		eqid 2821	. . . . . . . . . 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 7189	. . . . . . . . . . 11 ⊢ ((-1↑((𝑛 − 1) / 2)) / 𝑛) ∈ V
95	72, 94	ifex 4515	. . . . . . . . . 10 ⊢ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) ∈ V
96	92, 93, 95	fvmpt 6768	. . . . . . . . 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 11905	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
98		eqeq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
99	98, 86	orbi12d 915	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (𝑛 = 0 ∨ 2 ∥ 𝑛)))
100	99, 91	ifbieq2d 4492	. . . . . . . . . . 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 4515	. . . . . . . . . . 11 ⊢ if((𝑛 = 0 ∨ 2 ∥ 𝑛), 0, ((-1↑((𝑛 − 1) / 2)) / 𝑛)) ∈ V
102	100, 47, 101	fvmpt 6768	. . . . . . . . . 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 2866	. . . . . . . 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 13396	. . . . . 6 ⊢ (⊤ → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) = seq1( + , 𝐺))
107	76, 106	eqtr4d 2859	. . . . 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 5073	. . 3 ⊢ ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ 𝐴)
110		1z 12013	. . . 4 ⊢ 1 ∈ ℤ
111		seqex 13372	. . . 4 ⊢ seq0( + , 𝐺) ∈ V
112		climres 14932	. . . 4 ⊢ ((1 ∈ ℤ ∧ seq0( + , 𝐺) ∈ V) → ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴))
113	110, 111, 112	mp2an 690	. . 3 ⊢ ((seq0( + , 𝐺) ↾ (ℤ_≥‘1)) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴)
114	109, 113	bitr3i 279	. 2 ⊢ (seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴)
115	21, 40, 114	3bitri 299	1 ⊢ (seq0( + , 𝐹) ⇝ 𝐴 ↔ seq0( + , 𝐺) ⇝ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ifcif 4467 class class class wbr 5066 ↦ cmpt 5146 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 − cmin 10870 -cneg 10871 / cdiv 11297 ℕcn 11638 2c2 11693 ℕ₀cn0 11898 ℤcz 11982 ℤ_≥cuz 12244 ...cfz 12893 seqcseq 13370 ↑cexp 13430 ⇝ cli 14841 ∥ cdvds 15607

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-seq 13371 df-exp 13431 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-dvds 15608

This theorem is referenced by: leibpi 25520

W3C validator