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Theorem leibpi 27065

Description: The Leibniz formula for π. This proof depends on three main facts: (1) the series 𝐹 is convergent, because it is an alternating series (iseralt 15726). (2) Using leibpilem2 27064 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 27061). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 27061, Abel's theorem (abelth2 26563) says that the limit along any sequence converging to 1, such as 1 − 1 / 𝑛, of the power series converges to the power series extended to 1, and then since arctan is continuous at 1 (atancn 27059) we get the desired result. This is Metamath 100 proof #26. (Contributed by Mario Carneiro, 7-Apr-2015.)

**Hypothesis**
Ref	Expression
leibpi.1	⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))

**Assertion**
Ref	Expression
leibpi	⊢ seq0( + , 𝐹) ⇝ (π / 4)

Proof of Theorem leibpi Dummy variables 𝑗 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 12891	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12594	. . . . 5 ⊢ (⊤ → 0 ∈ ℤ)
3		eqidd 2766	. . . . 5 ⊢ ((⊤ ∧ 𝑗 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
4		0cnd 11187	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
5		ioran 999	. . . . . . . . . 10 ⊢ (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
6		neg1rr 12195	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℝ
7		leibpilem1 27063	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀))
8	7	simprd 500	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ₀)
9		reexpcl 14105	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
10	6, 8, 9	sylancr 598	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
11	7	simpld 499	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
12	10, 11	nndivred 12281	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
13	12	recnd 11225	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
14	5, 13	sylan2b 605	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
15	4, 14	ifclda 4519	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ₀ → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
16	15	adantl 486	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
17	16	fmpttd 7100	. . . . . 6 ⊢ (⊤ → (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ₀⟶ℂ)
18	17	ffvelcdmda 7069	. . . . 5 ⊢ ((⊤ ∧ 𝑗 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
19		2nn0 12512	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 2 ∈ ℕ₀)
21		nn0mulcl 12531	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
22	20, 21	sylan 591	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
23		nn0p1nn 12534	. . . . . . . . . . . 12 ⊢ ((2 · 𝑛) ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
24	22, 23	syl 18	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → ((2 · 𝑛) + 1) ∈ ℕ)
25	24	nnrecred 12278	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ)
26	25	fmpttd 7100	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))):ℕ₀⟶ℝ)
27		nn0mulcl 12531	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℕ₀)
28	20, 27	sylan 591	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℕ₀)
29	28	nn0red 12557	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℝ)
30		peano2nn0 12535	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℕ₀)
31	30	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℕ₀)
32		nn0mulcl 12531	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ₀ ∧ (𝑘 + 1) ∈ ℕ₀) → (2 · (𝑘 + 1)) ∈ ℕ₀)
33	19, 31, 32	sylancr 598	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · (𝑘 + 1)) ∈ ℕ₀)
34	33	nn0red 12557	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · (𝑘 + 1)) ∈ ℝ)
35		1red 11197	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 1 ∈ ℝ)
36		nn0re 12504	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
37	36	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℝ)
38	37	lep1d 12137	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ≤ (𝑘 + 1))
39		peano2re 11371	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
40	37, 39	syl 18	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℝ)
41		2re 12306	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
42	41	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 2 ∈ ℝ)
43		2pos 12336	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
44	43	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 0 < 2)
45		lemul2 12059	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
46	37, 40, 42, 44, 45	syl112anc 1397	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
47	38, 46	mpbid 235	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ≤ (2 · (𝑘 + 1)))
48	29, 34, 35, 47	leadd1dd 11816	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1))
49		nn0p1nn 12534	. . . . . . . . . . . . . 14 ⊢ ((2 · 𝑘) ∈ ℕ₀ → ((2 · 𝑘) + 1) ∈ ℕ)
50	28, 49	syl 18	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ∈ ℕ)
51	50	nnred 12239	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ∈ ℝ)
52	50	nngt0d 12276	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 0 < ((2 · 𝑘) + 1))
53		nn0p1nn 12534	. . . . . . . . . . . . . 14 ⊢ ((2 · (𝑘 + 1)) ∈ ℕ₀ → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
54	33, 53	syl 18	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
55	54	nnred 12239	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · (𝑘 + 1)) + 1) ∈ ℝ)
56	54	nngt0d 12276	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 0 < ((2 · (𝑘 + 1)) + 1))
57		lerec 12089	. . . . . . . . . . . 12 ⊢ (((((2 · 𝑘) + 1) ∈ ℝ ∧ 0 < ((2 · 𝑘) + 1)) ∧ (((2 · (𝑘 + 1)) + 1) ∈ ℝ ∧ 0 < ((2 · (𝑘 + 1)) + 1))) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1))))
58	51, 52, 55, 56, 57	syl22anc 851	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1))))
59	48, 58	mpbid 235	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1)))
60		oveq2 7408	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑘 + 1) → (2 · 𝑛) = (2 · (𝑘 + 1)))
61	60	oveq1d 7415	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑘 + 1) → ((2 · 𝑛) + 1) = ((2 · (𝑘 + 1)) + 1))
62	61	oveq2d 7416	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑘 + 1) → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
63		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))
64		ovex 7433	. . . . . . . . . . . 12 ⊢ (1 / ((2 · (𝑘 + 1)) + 1)) ∈ V
65	62, 63, 64	fvmpt 6979	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
66	31, 65	syl 18	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
67		oveq2 7408	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
68	67	oveq1d 7415	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((2 · 𝑛) + 1) = ((2 · 𝑘) + 1))
69	68	oveq2d 7416	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · 𝑘) + 1)))
70		ovex 7433	. . . . . . . . . . . 12 ⊢ (1 / ((2 · 𝑘) + 1)) ∈ V
71	69, 63, 70	fvmpt 6979	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
72	71	adantl 486	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
73	59, 66, 72	3brtr4d 5137	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) ≤ ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
74		nnuz 12892	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
75		1zzd 12616	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ)
76		ax-1cn 11146	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
77		divcnv 15897	. . . . . . . . . . 11 ⊢ (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
78	76, 77	mp1i 14	. . . . . . . . . 10 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
79		nn0ex 12501	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
80	79	mptex 7211	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) ∈ V
81	80	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) ∈ V)
82		oveq2 7408	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
83		eqid 2765	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
84		ovex 7433	. . . . . . . . . . . . 13 ⊢ (1 / 𝑘) ∈ V
85	82, 83, 84	fvmpt 6979	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
86	85	adantl 486	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
87		nnrecre 12269	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
88	87	adantl 486	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
89	86, 88	eqeltrd 2865	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℝ)
90		nnnn0 12502	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
91	90	adantl 486	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ₀)
92	91, 71	syl 18	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
93	90, 50	sylan2 604	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℕ)
94	93	nnrecred 12278	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ)
95	92, 94	eqeltrd 2865	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ∈ ℝ)
96		nnre 12231	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
97	96	adantl 486	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
98	19, 91, 27	sylancr 598	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℕ₀)
99	98	nn0red 12557	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℝ)
100		peano2re 11371	. . . . . . . . . . . . . 14 ⊢ ((2 · 𝑘) ∈ ℝ → ((2 · 𝑘) + 1) ∈ ℝ)
101	99, 100	syl 18	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
102		nn0addge1 12541	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ≤ (𝑘 + 𝑘))
103	97, 91, 102	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (𝑘 + 𝑘))
104	97	recnd 11225	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
105	104	2timesd 12478	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) = (𝑘 + 𝑘))
106	103, 105	breqtrrd 5133	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (2 · 𝑘))
107	99	lep1d 12137	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ≤ ((2 · 𝑘) + 1))
108	97, 99, 101, 106, 107	letrd 11355	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ ((2 · 𝑘) + 1))
109		nngt0 12258	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
110	109	adantl 486	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
111	93	nnred 12239	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
112	93	nngt0d 12276	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < ((2 · 𝑘) + 1))
113		lerec 12089	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℝ ∧ 0 < 𝑘) ∧ (((2 · 𝑘) + 1) ∈ ℝ ∧ 0 < ((2 · 𝑘) + 1))) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)))
114	97, 110, 111, 112, 113	syl22anc 851	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)))
115	108, 114	mpbid 235	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘))
116	115, 92, 86	3brtr4d 5137	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))
117	93	nnrpd 13049	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ⁺)
118	117	rpreccld 13061	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ⁺)
119	118	rpge0d 13055	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / ((2 · 𝑘) + 1)))
120	119, 92	breqtrrd 5133	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
121	74, 75, 78, 81, 89, 95, 116, 120	climsqz2 15683	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) ⇝ 0)
122		neg1cn 12194	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
123	122	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → -1 ∈ ℂ)
124		expcl 14106	. . . . . . . . . . . 12 ⊢ ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
125	123, 124	sylan 591	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
126	50	nncnd 12240	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ∈ ℂ)
127	50	nnne0d 12277	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ≠ 0)
128	125, 126, 127	divrecd 11985	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((-1↑𝑘) / ((2 · 𝑘) + 1)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
129		oveq2 7408	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (-1↑𝑛) = (-1↑𝑘))
130	129, 68	oveq12d 7418	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
131		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
132		ovex 7433	. . . . . . . . . . . 12 ⊢ ((-1↑𝑘) / ((2 · 𝑘) + 1)) ∈ V
133	130, 131, 132	fvmpt 6979	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
134	133	adantl 486	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
135	72	oveq2d 7416	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((-1↑𝑘) · ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
136	128, 134, 135	3eqtr4d 2810	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) · ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)))
137	1, 2, 26, 73, 121, 136	iseralt 15726	. . . . . . . 8 ⊢ (⊤ → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝ )
138		climdm 15595	. . . . . . . 8 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝ ↔ seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
139	137, 138	sylib 221	. . . . . . 7 ⊢ (⊤ → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
140		eqid 2765	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))) = (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
141		fvex 6884	. . . . . . . 8 ⊢ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ∈ V
142	131, 140, 141	leibpilem2 27064	. . . . . . 7 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ↔ seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
143	139, 142	sylib 221	. . . . . 6 ⊢ (⊤ → seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
144		seqex 14030	. . . . . . 7 ⊢ seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ V
145	144, 141	breldm 5889	. . . . . 6 ⊢ (seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) → seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ dom ⇝ )
146	143, 145	syl 18	. . . . 5 ⊢ (⊤ → seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ dom ⇝ )
147	1, 2, 3, 18, 146	isumclim2 15799	. . . 4 ⊢ (⊤ → seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
148		eqid 2765	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) = (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)))
149	17, 146, 148	abelth2 26563	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) ∈ ((0[,]1)–cn→ℂ))
150		nnrp 13019	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
151	150	adantl 486	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
152	151	rpreccld 13061	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
153	152	rpred 13051	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
154	152	rpge0d 13055	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ≤ (1 / 𝑛))
155		nnge1 12255	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 1 ≤ 𝑛)
156	155	adantl 486	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
157		nnre 12231	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
158	157	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
159	158	recnd 11225	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
160	159	mulridd 11214	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑛 · 1) = 𝑛)
161	156, 160	breqtrrd 5133	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ (𝑛 · 1))
162		1red 11197	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
163		nngt0 12258	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
164	163	adantl 486	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
165		ledivmul 12082	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
166	162, 162, 158, 164, 165	syl112anc 1397	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
167	161, 166	mpbird 260	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ≤ 1)
168		elicc01 13484	. . . . . . . . . 10 ⊢ ((1 / 𝑛) ∈ (0[,]1) ↔ ((1 / 𝑛) ∈ ℝ ∧ 0 ≤ (1 / 𝑛) ∧ (1 / 𝑛) ≤ 1))
169	153, 154, 167, 168	syl3anbrc 1360	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ (0[,]1))
170		iirev 25049	. . . . . . . . 9 ⊢ ((1 / 𝑛) ∈ (0[,]1) → (1 − (1 / 𝑛)) ∈ (0[,]1))
171	169, 170	syl 18	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ (0[,]1))
172	171	fmpttd 7100	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶(0[,]1))
173		1cnd 11190	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
174		nnex 12230	. . . . . . . . . . 11 ⊢ ℕ ∈ V
175	174	mptex 7211	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V
176	175	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V)
177	89	recnd 11225	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℂ)
178	82	oveq2d 7416	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (1 − (1 / 𝑛)) = (1 − (1 / 𝑘)))
179		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))
180		ovex 7433	. . . . . . . . . . . 12 ⊢ (1 − (1 / 𝑘)) ∈ V
181	178, 179, 180	fvmpt 6979	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − (1 / 𝑘)))
182	85	oveq2d 7416	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)) = (1 − (1 / 𝑘)))
183	181, 182	eqtr4d 2803	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
184	183	adantl 486	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
185	74, 75, 78, 173, 176, 177, 184	climsubc2 15680	. . . . . . . 8 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ (1 − 0))
186		1m0e1 12351	. . . . . . . 8 ⊢ (1 − 0) = 1
187	185, 186	breqtrdi 5146	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ 1)
188		1elunit 13488	. . . . . . . 8 ⊢ 1 ∈ (0[,]1)
189	188	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ∈ (0[,]1))
190	74, 75, 149, 172, 187, 189	climcncf 25020	. . . . . 6 ⊢ (⊤ → ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) ⇝ ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)))‘1))
191		eqidd 2766	. . . . . . . 8 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))))
192		eqidd 2766	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) = (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))))
193		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑥 = (1 − (1 / 𝑛)) → (𝑥↑𝑗) = ((1 − (1 / 𝑛))↑𝑗))
194	193	oveq2d 7416	. . . . . . . . 9 ⊢ (𝑥 = (1 − (1 / 𝑛)) → (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
195	194	sumeq2sdv 15744	. . . . . . . 8 ⊢ (𝑥 = (1 − (1 / 𝑛)) → Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)) = Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
196	171, 191, 192, 195	fmptco 7115	. . . . . . 7 ⊢ (⊤ → ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))))
197		0zd 12594	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℤ)
198	8	adantll 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ₀)
199	6, 198, 9	sylancr 598	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
200	199	recnd 11225	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
201	200	adantllr 731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
202		1re 11196	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℝ
203		resubcl 11510	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ) → (1 − (1 / 𝑛)) ∈ ℝ)
204	202, 153, 203	sylancr 598	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℝ)
205	204	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (1 − (1 / 𝑛)) ∈ ℝ)
206		simplr 780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ₀)
207	205, 206	reexpcld 14190	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℝ)
208	207	recnd 11225	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
209		nn0cn 12505	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
210	209	ad2antlr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℂ)
211	11	adantll 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
212	211	nnne0d 12277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ≠ 0)
213	201, 208, 210, 212	div12d 12018	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) = (((1 − (1 / 𝑛))↑𝑘) · ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
214	13	adantll 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
215	208, 214	mulcomd 11218	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) · ((-1↑((𝑘 − 1) / 2)) / 𝑘)) = (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘)))
216	213, 215	eqtrd 2800	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) = (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘)))
217	5, 216	sylan2b 605	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) = (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘)))
218	217	ifeq2da 4516	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘))))
219	204	recnd 11225	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℂ)
220		expcl 14106	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 − (1 / 𝑛)) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
221	219, 220	sylan 591	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
222	221	mul02d 11396	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (0 · ((1 − (1 / 𝑛))↑𝑘)) = 0)
223	222	ifeq1d 4503	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), (0 · ((1 − (1 / 𝑛))↑𝑘)), (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘))) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘))))
224	218, 223	eqtr4d 2803	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), (0 · ((1 − (1 / 𝑛))↑𝑘)), (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘))))
225		ovif 7498	. . . . . . . . . . . . . 14 ⊢ (if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) · ((1 − (1 / 𝑛))↑𝑘)) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), (0 · ((1 − (1 / 𝑛))↑𝑘)), (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘)))
226	224, 225	eqtr4di 2818	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = (if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) · ((1 − (1 / 𝑛))↑𝑘)))
227		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
228		c0ex 11188	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
229		ovex 7433	. . . . . . . . . . . . . . 15 ⊢ ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ V
230	228, 229	ifex 4534	. . . . . . . . . . . . . 14 ⊢ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
231		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))) = (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
232	231	fvmpt2 6991	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
233	227, 230, 232	sylancl 597	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
234		ovex 7433	. . . . . . . . . . . . . . . 16 ⊢ ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ V
235	228, 234	ifex 4534	. . . . . . . . . . . . . . 15 ⊢ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ V
236	140	fvmpt2 6991	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ₀ ∧ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ V) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
237	227, 235, 236	sylancl 597	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
238	237	oveq1d 7415	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)) = (if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) · ((1 − (1 / 𝑛))↑𝑘)))
239	226, 233, 238	3eqtr4d 2810	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)))
240	239	ralrimiva 3157	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)))
241		nfv 1937	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘))
242		nffvmpt1 6882	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
243		nffvmpt1 6882	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)
244		nfcv 2927	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 ·
245		nfcv 2927	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((1 − (1 / 𝑛))↑𝑗)
246	243, 244, 245	nfov 7430	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))
247	242, 246	nfeq 2940	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))
248		fveq2 6871	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
249		fveq2 6871	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) = ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
250		oveq2 7408	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((1 − (1 / 𝑛))↑𝑘) = ((1 − (1 / 𝑛))↑𝑗))
251	249, 250	oveq12d 7418	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
252	248, 251	eqeq12d 2781	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)) ↔ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))))
253	241, 247, 252	cbvralw 3307	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)) ↔ ∀𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
254	240, 253	sylib 221	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
255	254	r19.21bi 3257	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
256		0cnd 11187	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
257	207, 211	nndivred 12281	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℝ)
258	257	recnd 11225	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℂ)
259	201, 258	mulcld 11217	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
260	5, 259	sylan2b 605	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
261	256, 260	ifclda 4519	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ ℂ)
262	261	fmpttd 7100	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))):ℕ₀⟶ℂ)
263	262	ffvelcdmda 7069	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) ∈ ℂ)
264	255, 263	eqeltrrd 2866	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ₀) → (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)) ∈ ℂ)
265		0nn0 12510	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
266	265	a1i 11	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℕ₀)
267		0p1e1 12352	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
268		seqeq1 14031	. . . . . . . . . . . . 13 ⊢ ((0 + 1) = 1 → seq(0 + 1)( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) = seq1( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))))
269	267, 268	ax-mp 5	. . . . . . . . . . . 12 ⊢ seq(0 + 1)( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) = seq1( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))
270		1zzd 12616	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
271		elnnuz 12893	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ_≥‘1))
272		nnne0 12261	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
273	272	neneqd 2965	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ ℕ → ¬ 𝑘 = 0)
274		biorf 949	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑘 = 0 → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
275	273, 274	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
276	275	bicomd 226	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ 2 ∥ 𝑘))
277	276	ifbid 4507	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
278	90, 230, 232	sylancl 597	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
279	228, 229	ifex 4534	. . . . . . . . . . . . . . . . . . . . 21 ⊢ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
280		eqid 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))) = (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
281	280	fvmpt2 6991	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ ∧ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V) → ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
282	279, 281	mpan2 703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
283	277, 278, 282	3eqtr4d 2810	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘))
284	283	rgen 3081	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑘 ∈ ℕ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘)
285	284	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘))
286		nfv 1937	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘)
287		nffvmpt1 6882	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
288	242, 287	nfeq 2940	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
289		fveq2 6871	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑗 → ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
290	248, 289	eqeq12d 2781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑗 → (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) ↔ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)))
291	286, 288, 290	cbvralw 3307	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ ℕ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) ↔ ∀𝑗 ∈ ℕ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
292	285, 291	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
293	292	r19.21bi 3257	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
294	271, 293	sylan2br 606	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ (ℤ_≥‘1)) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
295	270, 294	seqfeq 14054	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → seq1( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) = seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))))
296	153, 162, 167	abssubge0d 15475	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) = (1 − (1 / 𝑛)))
297		ltsubrp 13045	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ⁺) → (1 − (1 / 𝑛)) < 1)
298	202, 152, 297	sylancr 598	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) < 1)
299	296, 298	eqbrtrd 5127	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) < 1)
300	280	atantayl2 27061	. . . . . . . . . . . . . 14 ⊢ (((1 − (1 / 𝑛)) ∈ ℂ ∧ (abs‘(1 − (1 / 𝑛))) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 / 𝑛))))
301	219, 299, 300	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 / 𝑛))))
302	295, 301	eqbrtrd 5127	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → seq1( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 / 𝑛))))
303	269, 302	eqbrtrid 5140	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → seq(0 + 1)( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 / 𝑛))))
304	1, 266, 263, 303	clim2ser2 15697	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ ((arctan‘(1 − (1 / 𝑛))) + (seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0)))
305		0z 12593	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℤ
306		seq1 14041	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℤ → (seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0))
307	305, 306	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0)
308		iftrue 4489	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
309	308	orcs 888	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
310	309, 231, 228	fvmpt 6979	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0) = 0)
311	265, 310	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0) = 0
312	307, 311	eqtri 2788	. . . . . . . . . . . 12 ⊢ (seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = 0
313	312	oveq2i 7411	. . . . . . . . . . 11 ⊢ ((arctan‘(1 − (1 / 𝑛))) + (seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0)) = ((arctan‘(1 − (1 / 𝑛))) + 0)
314		atanrecl 27034	. . . . . . . . . . . . . 14 ⊢ ((1 − (1 / 𝑛)) ∈ ℝ → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
315	204, 314	syl 18	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
316	315	recnd 11225	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℂ)
317	316	addridd 11398	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + 0) = (arctan‘(1 − (1 / 𝑛))))
318	313, 317	eqtrid 2812	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + (seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0)) = (arctan‘(1 − (1 / 𝑛))))
319	304, 318	breqtrd 5131	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 / 𝑛))))
320	1, 197, 255, 264, 319	isumclim 15798	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)) = (arctan‘(1 − (1 / 𝑛))))
321	320	mpteq2dva 5198	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))) = (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))))
322	196, 321	eqtrd 2800	. . . . . 6 ⊢ (⊤ → ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗))) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))))
323		oveq1 7407	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥↑𝑗) = (1↑𝑗))
324		nn0z 12606	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → 𝑗 ∈ ℤ)
325		1exp 14118	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℤ → (1↑𝑗) = 1)
326	324, 325	syl 18	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ₀ → (1↑𝑗) = 1)
327	323, 326	sylan9eq 2820	. . . . . . . . . . 11 ⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ₀) → (𝑥↑𝑗) = 1)
328	327	oveq2d 7416	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ₀) → (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)) = (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · 1))
329	17	mptru 1570	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ₀⟶ℂ
330	329	ffvelcdmi 7068	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
331	330	mulridd 11214	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ₀ → (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · 1) = ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
332	331	adantl 486	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ₀) → (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · 1) = ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
333	328, 332	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ₀) → (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)) = ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
334	333	sumeq2dv 15743	. . . . . . . 8 ⊢ (𝑥 = 1 → Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)) = Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
335		sumex 15729	. . . . . . . 8 ⊢ Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ V
336	334, 148, 335	fvmpt 6979	. . . . . . 7 ⊢ (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)))‘1) = Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
337	188, 336	mp1i 14	. . . . . 6 ⊢ (⊤ → ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ₀ (((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥↑𝑗)))‘1) = Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
338	190, 322, 337	3brtr3d 5136	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
339		eqid 2765	. . . . . . . . 9 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
340		eqid 2765	. . . . . . . . 9 ⊢ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} = {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
341	339, 340	atancn 27059	. . . . . . . 8 ⊢ (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}–cn→ℂ)
342	341	a1i 11	. . . . . . 7 ⊢ (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}–cn→ℂ))
343		unitssre 13517	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℝ
344	339, 340	ressatans 27057	. . . . . . . . 9 ⊢ ℝ ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
345	343, 344	sstri 3948	. . . . . . . 8 ⊢ (0[,]1) ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
346		fss 6712	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶(0[,]1) ∧ (0[,]1) ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
347	172, 345, 346	sylancl 597	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
348	344, 202	sselii 3936	. . . . . . . 8 ⊢ 1 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
349	348	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
350	74, 75, 342, 347, 187, 349	climcncf 25020	. . . . . 6 ⊢ (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) ⇝ ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1))
351	345, 171	sselid 3937	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
352		cncff 25013	. . . . . . . . . 10 ⊢ ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}–cn→ℂ) → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}):{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}⟶ℂ)
353	341, 352	mp1i 14	. . . . . . . . 9 ⊢ (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}):{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}⟶ℂ)
354	353	feqmptd 6939	. . . . . . . 8 ⊢ (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘𝑘)))
355		fvres 6890	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘𝑘) = (arctan‘𝑘))
356	355	mpteq2ia 5200	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘𝑘)) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ (arctan‘𝑘))
357	354, 356	eqtrdi 2816	. . . . . . 7 ⊢ (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ (arctan‘𝑘)))
358		fveq2 6871	. . . . . . 7 ⊢ (𝑘 = (1 − (1 / 𝑛)) → (arctan‘𝑘) = (arctan‘(1 − (1 / 𝑛))))
359	351, 191, 357, 358	fmptco 7115	. . . . . 6 ⊢ (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))))
360		fvres 6890	. . . . . . . 8 ⊢ (1 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (arctan‘1))
361	348, 360	mp1i 14	. . . . . . 7 ⊢ (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (arctan‘1))
362		atan1 27051	. . . . . . 7 ⊢ (arctan‘1) = (π / 4)
363	361, 362	eqtrdi 2816	. . . . . 6 ⊢ (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (π / 4))
364	350, 359, 363	3brtr3d 5136	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ (π / 4))
365		climuni 15593	. . . . 5 ⊢ (((𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∧ (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ (π / 4)) → Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4))
366	338, 364, 365	syl2anc 595	. . . 4 ⊢ (⊤ → Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4))
367	147, 366	breqtrd 5131	. . 3 ⊢ (⊤ → seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
368	367	mptru 1570	. 2 ⊢ seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4)
369		leibpi.1	. . 3 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
370		ovex 7433	. . 3 ⊢ (π / 4) ∈ V
371	369, 140, 370	leibpilem2 27064	. 2 ⊢ (seq0( + , 𝐹) ⇝ (π / 4) ↔ seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
372	368, 371	mpbir 234	1 ⊢ seq0( + , 𝐹) ⇝ (π / 4)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1563 ⊤wtru 1564 ∈ wcel 2145 ∀wral 3079 {crab 3417 Vcvv 3457 ∖ cdif 3904 ⊆ wss 3907 ifcif 4483 class class class wbr 5105 ↦ cmpt 5186 dom cdm 5652 ↾ cres 5654 ∘ ccom 5656 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 ℝcr 11087 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 -∞cmnf 11229 < clt 11231 ≤ cle 11232 − cmin 11429 -cneg 11430 / cdiv 11859 ℕcn 12224 2c2 12286 4c4 12288 ℕ₀cn0 12495 ℤcz 12582 ℤ_≥cuz 12853 ℝ⁺crp 13007 (,]cioc 13364 [,]cicc 13366 seqcseq 14028 ↑cexp 14088 abscabs 15275 ⇝ cli 15525 Σcsu 15727 πcpi 16110 ∥ cdvds 16300 –cn→ccncf 24996 arctancatan 26987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-oadd 8445 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-xnn0 12569 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-fac 14301 df-bc 14330 df-hash 14358 df-shft 15094 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-limsup 15512 df-clim 15529 df-rlim 15530 df-sum 15728 df-ef 16111 df-sin 16113 df-cos 16114 df-tan 16115 df-pi 16116 df-dvds 16301 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-mulg 19125 df-cntz 19378 df-cmn 19843 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-fbas 21479 df-fg 21480 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-nei 23216 df-lp 23254 df-perf 23255 df-cn 23345 df-cnp 23346 df-t1 23432 df-haus 23433 df-cmp 23505 df-tx 23680 df-hmeo 23873 df-fil 23964 df-fm 24056 df-flim 24057 df-flf 24058 df-xms 24438 df-ms 24439 df-tms 24440 df-cncf 24998 df-limc 25986 df-dv 25987 df-ulm 26498 df-log 26679 df-atan 26990

This theorem is referenced by: leibpisum 27066

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