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Theorem leibpi 26984

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 15695). (2) Using leibpilem2 26983 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 26980). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 26980, Abel's theorem (abelth2 26482) 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 26978) 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 12874	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12577	. . . . 5 ⊢ (⊤ → 0 ∈ ℤ)
3		eqidd 2762	. . . . 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 11169	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
5		ioran 996	. . . . . . . . . 10 ⊢ (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
6		neg1rr 12178	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℝ
7		leibpilem1 26982	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀))
8	7	simprd 499	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ₀)
9		reexpcl 14088	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
10	6, 8, 9	sylancr 596	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
11	7	simpld 498	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
12	10, 11	nndivred 12264	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
13	12	recnd 11207	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
14	5, 13	sylan2b 603	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
15	4, 14	ifclda 4515	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ₀ → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
16	15	adantl 485	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
17	16	fmpttd 7092	. . . . . 6 ⊢ (⊤ → (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ₀⟶ℂ)
18	17	ffvelcdmda 7061	. . . . 5 ⊢ ((⊤ ∧ 𝑗 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
19		2nn0 12495	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 2 ∈ ℕ₀)
21		nn0mulcl 12514	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
22	20, 21	sylan 589	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
23		nn0p1nn 12517	. . . . . . . . . . . 12 ⊢ ((2 · 𝑛) ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
24	22, 23	syl 17	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → ((2 · 𝑛) + 1) ∈ ℕ)
25	24	nnrecred 12261	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ)
26	25	fmpttd 7092	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))):ℕ₀⟶ℝ)
27		nn0mulcl 12514	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℕ₀)
28	20, 27	sylan 589	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℕ₀)
29	28	nn0red 12540	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℝ)
30		peano2nn0 12518	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℕ₀)
31	30	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℕ₀)
32		nn0mulcl 12514	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ₀ ∧ (𝑘 + 1) ∈ ℕ₀) → (2 · (𝑘 + 1)) ∈ ℕ₀)
33	19, 31, 32	sylancr 596	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · (𝑘 + 1)) ∈ ℕ₀)
34	33	nn0red 12540	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · (𝑘 + 1)) ∈ ℝ)
35		1red 11179	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 1 ∈ ℝ)
36		nn0re 12487	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
37	36	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℝ)
38	37	lep1d 12120	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ≤ (𝑘 + 1))
39		peano2re 11353	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
40	37, 39	syl 17	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℝ)
41		2re 12289	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
42	41	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 2 ∈ ℝ)
43		2pos 12319	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
44	43	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 0 < 2)
45		lemul2 12041	. . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
47	38, 46	mpbid 234	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ≤ (2 · (𝑘 + 1)))
48	29, 34, 35, 47	leadd1dd 11798	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1))
49		nn0p1nn 12517	. . . . . . . . . . . . . 14 ⊢ ((2 · 𝑘) ∈ ℕ₀ → ((2 · 𝑘) + 1) ∈ ℕ)
50	28, 49	syl 17	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ∈ ℕ)
51	50	nnred 12222	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ∈ ℝ)
52	50	nngt0d 12259	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 0 < ((2 · 𝑘) + 1))
53		nn0p1nn 12517	. . . . . . . . . . . . . 14 ⊢ ((2 · (𝑘 + 1)) ∈ ℕ₀ → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
54	33, 53	syl 17	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
55	54	nnred 12222	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · (𝑘 + 1)) + 1) ∈ ℝ)
56	54	nngt0d 12259	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 0 < ((2 · (𝑘 + 1)) + 1))
57		lerec 12072	. . . . . . . . . . . 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 849	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1))))
59	48, 58	mpbid 234	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1)))
60		oveq2 7400	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑘 + 1) → (2 · 𝑛) = (2 · (𝑘 + 1)))
61	60	oveq1d 7407	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑘 + 1) → ((2 · 𝑛) + 1) = ((2 · (𝑘 + 1)) + 1))
62	61	oveq2d 7408	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑘 + 1) → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
63		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))
64		ovex 7425	. . . . . . . . . . . 12 ⊢ (1 / ((2 · (𝑘 + 1)) + 1)) ∈ V
65	62, 63, 64	fvmpt 6971	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
66	31, 65	syl 17	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
67		oveq2 7400	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
68	67	oveq1d 7407	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((2 · 𝑛) + 1) = ((2 · 𝑘) + 1))
69	68	oveq2d 7408	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · 𝑘) + 1)))
70		ovex 7425	. . . . . . . . . . . 12 ⊢ (1 / ((2 · 𝑘) + 1)) ∈ V
71	69, 63, 70	fvmpt 6971	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
72	71	adantl 485	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
73	59, 66, 72	3brtr4d 5131	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) ≤ ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
74		nnuz 12875	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
75		1zzd 12599	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ)
76		ax-1cn 11128	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
77		divcnv 15866	. . . . . . . . . . 11 ⊢ (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
78	76, 77	mp1i 13	. . . . . . . . . 10 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
79		nn0ex 12484	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
80	79	mptex 7203	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) ∈ V
81	80	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) ∈ V)
82		oveq2 7400	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
83		eqid 2761	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
84		ovex 7425	. . . . . . . . . . . . 13 ⊢ (1 / 𝑘) ∈ V
85	82, 83, 84	fvmpt 6971	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
86	85	adantl 485	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
87		nnrecre 12252	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
88	87	adantl 485	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
89	86, 88	eqeltrd 2861	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℝ)
90		nnnn0 12485	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
91	90	adantl 485	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ₀)
92	91, 71	syl 17	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
93	90, 50	sylan2 602	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℕ)
94	93	nnrecred 12261	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ)
95	92, 94	eqeltrd 2861	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ∈ ℝ)
96		nnre 12214	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
97	96	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
98	19, 91, 27	sylancr 596	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℕ₀)
99	98	nn0red 12540	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℝ)
100		peano2re 11353	. . . . . . . . . . . . . 14 ⊢ ((2 · 𝑘) ∈ ℝ → ((2 · 𝑘) + 1) ∈ ℝ)
101	99, 100	syl 17	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
102		nn0addge1 12524	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ≤ (𝑘 + 𝑘))
103	97, 91, 102	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (𝑘 + 𝑘))
104	97	recnd 11207	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
105	104	2timesd 12461	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) = (𝑘 + 𝑘))
106	103, 105	breqtrrd 5127	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (2 · 𝑘))
107	99	lep1d 12120	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ≤ ((2 · 𝑘) + 1))
108	97, 99, 101, 106, 107	letrd 11337	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ ((2 · 𝑘) + 1))
109		nngt0 12241	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
110	109	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
111	93	nnred 12222	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
112	93	nngt0d 12259	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < ((2 · 𝑘) + 1))
113		lerec 12072	. . . . . . . . . . . . 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 849	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)))
115	108, 114	mpbid 234	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘))
116	115, 92, 86	3brtr4d 5131	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))
117	93	nnrpd 13032	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ⁺)
118	117	rpreccld 13044	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ⁺)
119	118	rpge0d 13038	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / ((2 · 𝑘) + 1)))
120	119, 92	breqtrrd 5127	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
121	74, 75, 78, 81, 89, 95, 116, 120	climsqz2 15652	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) ⇝ 0)
122		neg1cn 12177	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
123	122	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → -1 ∈ ℂ)
124		expcl 14089	. . . . . . . . . . . 12 ⊢ ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
125	123, 124	sylan 589	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
126	50	nncnd 12223	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ∈ ℂ)
127	50	nnne0d 12260	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ≠ 0)
128	125, 126, 127	divrecd 11967	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((-1↑𝑘) / ((2 · 𝑘) + 1)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
129		oveq2 7400	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (-1↑𝑛) = (-1↑𝑘))
130	129, 68	oveq12d 7410	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
131		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
132		ovex 7425	. . . . . . . . . . . 12 ⊢ ((-1↑𝑘) / ((2 · 𝑘) + 1)) ∈ V
133	130, 131, 132	fvmpt 6971	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
134	133	adantl 485	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
135	72	oveq2d 7408	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((-1↑𝑘) · ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
136	128, 134, 135	3eqtr4d 2806	. . . . . . . . 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 15695	. . . . . . . 8 ⊢ (⊤ → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝ )
138		climdm 15564	. . . . . . . 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 220	. . . . . . 7 ⊢ (⊤ → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
140		eqid 2761	. . . . . . . 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 6876	. . . . . . . 8 ⊢ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ∈ V
142	131, 140, 141	leibpilem2 26983	. . . . . . 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 220	. . . . . 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 14013	. . . . . . 7 ⊢ seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ V
145	144, 141	breldm 5882	. . . . . 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 17	. . . . 5 ⊢ (⊤ → seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ dom ⇝ )
147	1, 2, 3, 18, 146	isumclim2 15768	. . . 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 2761	. . . . . . . 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 26482	. . . . . . 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 13002	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
151	150	adantl 485	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
152	151	rpreccld 13044	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
153	152	rpred 13034	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
154	152	rpge0d 13038	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ≤ (1 / 𝑛))
155		nnge1 12238	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 1 ≤ 𝑛)
156	155	adantl 485	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
157		nnre 12214	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
158	157	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
159	158	recnd 11207	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
160	159	mulridd 11196	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑛 · 1) = 𝑛)
161	156, 160	breqtrrd 5127	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ (𝑛 · 1))
162		1red 11179	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
163		nngt0 12241	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
164	163	adantl 485	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
165		ledivmul 12065	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
166	162, 162, 158, 164, 165	syl112anc 1392	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
167	161, 166	mpbird 259	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ≤ 1)
168		elicc01 13467	. . . . . . . . . 10 ⊢ ((1 / 𝑛) ∈ (0[,]1) ↔ ((1 / 𝑛) ∈ ℝ ∧ 0 ≤ (1 / 𝑛) ∧ (1 / 𝑛) ≤ 1))
169	153, 154, 167, 168	syl3anbrc 1356	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ (0[,]1))
170		iirev 24971	. . . . . . . . 9 ⊢ ((1 / 𝑛) ∈ (0[,]1) → (1 − (1 / 𝑛)) ∈ (0[,]1))
171	169, 170	syl 17	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ (0[,]1))
172	171	fmpttd 7092	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶(0[,]1))
173		1cnd 11172	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
174		nnex 12213	. . . . . . . . . . 11 ⊢ ℕ ∈ V
175	174	mptex 7203	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V
176	175	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V)
177	89	recnd 11207	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℂ)
178	82	oveq2d 7408	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (1 − (1 / 𝑛)) = (1 − (1 / 𝑘)))
179		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))
180		ovex 7425	. . . . . . . . . . . 12 ⊢ (1 − (1 / 𝑘)) ∈ V
181	178, 179, 180	fvmpt 6971	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − (1 / 𝑘)))
182	85	oveq2d 7408	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)) = (1 − (1 / 𝑘)))
183	181, 182	eqtr4d 2799	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
184	183	adantl 485	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
185	74, 75, 78, 173, 176, 177, 184	climsubc2 15649	. . . . . . . 8 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ (1 − 0))
186		1m0e1 12334	. . . . . . . 8 ⊢ (1 − 0) = 1
187	185, 186	breqtrdi 5140	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ 1)
188		1elunit 13471	. . . . . . . 8 ⊢ 1 ∈ (0[,]1)
189	188	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ∈ (0[,]1))
190	74, 75, 149, 172, 187, 189	climcncf 24942	. . . . . 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 2762	. . . . . . . 8 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))))
192		eqidd 2762	. . . . . . . 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 7399	. . . . . . . . . 10 ⊢ (𝑥 = (1 − (1 / 𝑛)) → (𝑥↑𝑗) = ((1 − (1 / 𝑛))↑𝑗))
194	193	oveq2d 7408	. . . . . . . . 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 15713	. . . . . . . 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 7107	. . . . . . 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 12577	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℤ)
198	8	adantll 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ₀)
199	6, 198, 9	sylancr 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
200	199	recnd 11207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
201	200	adantllr 729	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
202		1re 11178	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℝ
203		resubcl 11492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ) → (1 − (1 / 𝑛)) ∈ ℝ)
204	202, 153, 203	sylancr 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℝ)
205	204	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (1 − (1 / 𝑛)) ∈ ℝ)
206		simplr 778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ₀)
207	205, 206	reexpcld 14173	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℝ)
208	207	recnd 11207	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
209		nn0cn 12488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
210	209	ad2antlr 737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℂ)
211	11	adantll 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
212	211	nnne0d 12260	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ≠ 0)
213	201, 208, 210, 212	div12d 12000	. . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
215	208, 214	mulcomd 11200	. . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . . . 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 4512	. . . . . . . . . . . . . . 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 11207	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℂ)
220		expcl 14089	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 − (1 / 𝑛)) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
221	219, 220	sylan 589	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
222	221	mul02d 11378	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (0 · ((1 − (1 / 𝑛))↑𝑘)) = 0)
223	222	ifeq1d 4499	. . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . 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 7490	. . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
228		c0ex 11170	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
229		ovex 7425	. . . . . . . . . . . . . . 15 ⊢ ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ V
230	228, 229	ifex 4530	. . . . . . . . . . . . . 14 ⊢ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
231		eqid 2761	. . . . . . . . . . . . . . 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 6983	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . . . 16 ⊢ ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ V
235	228, 234	ifex 4530	. . . . . . . . . . . . . . 15 ⊢ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ V
236	140	fvmpt2 6983	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 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 7407	. . . . . . . . . . . . 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 2806	. . . . . . . . . . . 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 3153	. . . . . . . . . . 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 1933	. . . . . . . . . . . 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 6874	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
243		nffvmpt1 6874	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)
244		nfcv 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 ·
245		nfcv 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((1 − (1 / 𝑛))↑𝑗)
246	243, 244, 245	nfov 7422	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))
247	242, 246	nfeq 2936	. . . . . . . . . . . 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 6863	. . . . . . . . . . . . 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 6863	. . . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((1 − (1 / 𝑛))↑𝑘) = ((1 − (1 / 𝑛))↑𝑗))
251	249, 250	oveq12d 7410	. . . . . . . . . . . . 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 2777	. . . . . . . . . . . 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 3303	. . . . . . . . . . 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 220	. . . . . . . . . 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 3253	. . . . . . . . 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 11169	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
257	207, 211	nndivred 12264	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℝ)
258	257	recnd 11207	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℂ)
259	201, 258	mulcld 11199	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
260	5, 259	sylan2b 603	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
261	256, 260	ifclda 4515	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ ℂ)
262	261	fmpttd 7092	. . . . . . . . . . 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 7061	. . . . . . . . . 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 2862	. . . . . . . . 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 12493	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
266	265	a1i 11	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℕ₀)
267		0p1e1 12335	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
268		seqeq1 14014	. . . . . . . . . . . . 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 12599	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
271		elnnuz 12876	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ_≥‘1))
272		nnne0 12244	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
273	272	neneqd 2961	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ ℕ → ¬ 𝑘 = 0)
274		biorf 947	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑘 = 0 → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
275	273, 274	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
276	275	bicomd 225	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ 2 ∥ 𝑘))
277	276	ifbid 4503	. . . . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . . . . 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 4530	. . . . . . . . . . . . . . . . . . . . 21 ⊢ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
280		eqid 2761	. . . . . . . . . . . . . . . . . . . . . 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 6983	. . . . . . . . . . . . . . . . . . . . 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 701	. . . . . . . . . . . . . . . . . . . 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 2806	. . . . . . . . . . . . . . . . . . 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 3077	. . . . . . . . . . . . . . . . . 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 1933	. . . . . . . . . . . . . . . . . 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 6874	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
288	242, 287	nfeq 2936	. . . . . . . . . . . . . . . . . 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 6863	. . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . 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 3303	. . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . 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 3253	. . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . 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 14037	. . . . . . . . . . . . 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 15444	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) = (1 − (1 / 𝑛)))
297		ltsubrp 13028	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ⁺) → (1 − (1 / 𝑛)) < 1)
298	202, 152, 297	sylancr 596	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) < 1)
299	296, 298	eqbrtrd 5121	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) < 1)
300	280	atantayl2 26980	. . . . . . . . . . . . . 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 593	. . . . . . . . . . . . 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 5121	. . . . . . . . . . . 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 5134	. . . . . . . . . . 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 15666	. . . . . . . . . 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 12576	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℤ
306		seq1 14024	. . . . . . . . . . . . . 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 4485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
309	308	orcs 886	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
310	309, 231, 228	fvmpt 6971	. . . . . . . . . . . . . 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 2784	. . . . . . . . . . . 12 ⊢ (seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = 0
313	312	oveq2i 7403	. . . . . . . . . . 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 26953	. . . . . . . . . . . . . 14 ⊢ ((1 − (1 / 𝑛)) ∈ ℝ → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
315	204, 314	syl 17	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
316	315	recnd 11207	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℂ)
317	316	addridd 11380	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + 0) = (arctan‘(1 − (1 / 𝑛))))
318	313, 317	eqtrid 2808	. . . . . . . . . 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 5125	. . . . . . . . 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 15767	. . . . . . . 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 5192	. . . . . . 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 2796	. . . . . 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 7399	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥↑𝑗) = (1↑𝑗))
324		nn0z 12589	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → 𝑗 ∈ ℤ)
325		1exp 14101	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℤ → (1↑𝑗) = 1)
326	324, 325	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ₀ → (1↑𝑗) = 1)
327	323, 326	sylan9eq 2816	. . . . . . . . . . 11 ⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ₀) → (𝑥↑𝑗) = 1)
328	327	oveq2d 7408	. . . . . . . . . 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 1566	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ₀⟶ℂ
330	329	ffvelcdmi 7060	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
331	330	mulridd 11196	. . . . . . . . . . 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 485	. . . . . . . . . 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 2796	. . . . . . . . 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 15712	. . . . . . . 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 15698	. . . . . . . 8 ⊢ Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ V
336	334, 148, 335	fvmpt 6971	. . . . . . 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 13	. . . . . 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 5130	. . . . 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 2761	. . . . . . . . 9 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
340		eqid 2761	. . . . . . . . 9 ⊢ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} = {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
341	339, 340	atancn 26978	. . . . . . . 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 13500	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℝ
344	339, 340	ressatans 26976	. . . . . . . . 9 ⊢ ℝ ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
345	343, 344	sstri 3945	. . . . . . . 8 ⊢ (0[,]1) ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
346		fss 6704	. . . . . . . 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 595	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
348	344, 202	sselii 3933	. . . . . . . 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 24942	. . . . . 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 3934	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
352		cncff 24935	. . . . . . . . . 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 13	. . . . . . . . 9 ⊢ (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}):{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}⟶ℂ)
354	353	feqmptd 6931	. . . . . . . 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 6882	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘𝑘) = (arctan‘𝑘))
356	355	mpteq2ia 5194	. . . . . . . 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 2812	. . . . . . 7 ⊢ (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ (arctan‘𝑘)))
358		fveq2 6863	. . . . . . 7 ⊢ (𝑘 = (1 − (1 / 𝑛)) → (arctan‘𝑘) = (arctan‘(1 − (1 / 𝑛))))
359	351, 191, 357, 358	fmptco 7107	. . . . . 6 ⊢ (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))))
360		fvres 6882	. . . . . . . 8 ⊢ (1 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (arctan‘1))
361	348, 360	mp1i 13	. . . . . . 7 ⊢ (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (arctan‘1))
362		atan1 26970	. . . . . . 7 ⊢ (arctan‘1) = (π / 4)
363	361, 362	eqtrdi 2812	. . . . . 6 ⊢ (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (π / 4))
364	350, 359, 363	3brtr3d 5130	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ (π / 4))
365		climuni 15562	. . . . 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 593	. . . 4 ⊢ (⊤ → Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4))
367	147, 366	breqtrd 5125	. . 3 ⊢ (⊤ → seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
368	367	mptru 1566	. 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 7425	. . 3 ⊢ (π / 4) ∈ V
371	369, 140, 370	leibpilem2 26983	. 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 233	1 ⊢ seq0( + , 𝐹) ⇝ (π / 4)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 399 ∨ wo 858 = wceq 1559 ⊤wtru 1560 ∈ wcel 2141 ∀wral 3075 {crab 3413 Vcvv 3453 ∖ cdif 3901 ⊆ wss 3904 ifcif 4479 class class class wbr 5099 ↦ cmpt 5180 dom cdm 5645 ↾ cres 5647 ∘ ccom 5649 ⟶wf 6513 ‘cfv 6517 (class class class)co 7392 ℂcc 11068 ℝcr 11069 0cc0 11070 1c1 11071 + caddc 11073 · cmul 11075 -∞cmnf 11211 < clt 11213 ≤ cle 11214 − cmin 11411 -cneg 11412 / cdiv 11841 ℕcn 12207 2c2 12269 4c4 12271 ℕ₀cn0 12478 ℤcz 12565 ℤ_≥cuz 12836 ℝ⁺crp 12990 (,]cioc 13347 [,]cicc 13349 seqcseq 14011 ↑cexp 14071 abscabs 15244 ⇝ cli 15494 Σcsu 15696 πcpi 16079 ∥ cdvds 16269 –cn→ccncf 24918 arctancatan 26906

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148 ax-addf 11149

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-oadd 8436 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-xnn0 12552 df-z 12566 df-dec 12686 df-uz 12837 df-q 12947 df-rp 12991 df-xneg 13111 df-xadd 13112 df-xmul 13113 df-ioo 13350 df-ioc 13351 df-ico 13352 df-icc 13353 df-fz 13510 df-fzo 13657 df-fl 13799 df-mod 13877 df-seq 14012 df-exp 14072 df-fac 14284 df-bc 14313 df-hash 14341 df-shft 15077 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-limsup 15481 df-clim 15498 df-rlim 15499 df-sum 15697 df-ef 16080 df-sin 16082 df-cos 16083 df-tan 16084 df-pi 16085 df-dvds 16270 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17515 df-qtop 17520 df-imas 17521 df-xps 17523 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-mulg 19093 df-cntz 19340 df-cmn 19805 df-psmet 21396 df-xmet 21397 df-met 21398 df-bl 21399 df-mopn 21400 df-fbas 21401 df-fg 21402 df-cnfld 21405 df-top 22934 df-topon 22951 df-topsp 22973 df-bases 22986 df-cld 23059 df-ntr 23060 df-cls 23061 df-nei 23138 df-lp 23176 df-perf 23177 df-cn 23267 df-cnp 23268 df-t1 23354 df-haus 23355 df-cmp 23427 df-tx 23602 df-hmeo 23795 df-fil 23886 df-fm 23978 df-flim 23979 df-flf 23980 df-xms 24360 df-ms 24361 df-tms 24362 df-cncf 24920 df-limc 25908 df-dv 25909 df-ulm 26417 df-log 26598 df-atan 26909

This theorem is referenced by: leibpisum 26985

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