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Theorem leibpi 26852

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 15651). (2) Using leibpilem2 26851 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 26848). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 26848, Abel's theorem (abelth2 26352) 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 26846) 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 12835	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12541	. . . . 5 ⊢ (⊤ → 0 ∈ ℤ)
3		eqidd 2730	. . . . 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 11167	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
5		ioran 985	. . . . . . . . . 10 ⊢ (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
6		neg1rr 12172	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℝ
7		leibpilem1 26850	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀))
8	7	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ₀)
9		reexpcl 14043	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
10	6, 8, 9	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
11	7	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
12	10, 11	nndivred 12240	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
13	12	recnd 11202	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
14	5, 13	sylan2b 594	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
15	4, 14	ifclda 4524	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ₀ → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
16	15	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
17	16	fmpttd 7087	. . . . . 6 ⊢ (⊤ → (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ₀⟶ℂ)
18	17	ffvelcdmda 7056	. . . . 5 ⊢ ((⊤ ∧ 𝑗 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
19		2nn0 12459	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 2 ∈ ℕ₀)
21		nn0mulcl 12478	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ₀ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
22	20, 21	sylan 580	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → (2 · 𝑛) ∈ ℕ₀)
23		nn0p1nn 12481	. . . . . . . . . . . 12 ⊢ ((2 · 𝑛) ∈ ℕ₀ → ((2 · 𝑛) + 1) ∈ ℕ)
24	22, 23	syl 17	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → ((2 · 𝑛) + 1) ∈ ℕ)
25	24	nnrecred 12237	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ₀) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ)
26	25	fmpttd 7087	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))):ℕ₀⟶ℝ)
27		nn0mulcl 12478	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℕ₀)
28	20, 27	sylan 580	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℕ₀)
29	28	nn0red 12504	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℝ)
30		peano2nn0 12482	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℕ₀)
31	30	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℕ₀)
32		nn0mulcl 12478	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ₀ ∧ (𝑘 + 1) ∈ ℕ₀) → (2 · (𝑘 + 1)) ∈ ℕ₀)
33	19, 31, 32	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · (𝑘 + 1)) ∈ ℕ₀)
34	33	nn0red 12504	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · (𝑘 + 1)) ∈ ℝ)
35		1red 11175	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 1 ∈ ℝ)
36		nn0re 12451	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
37	36	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℝ)
38	37	lep1d 12114	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ≤ (𝑘 + 1))
39		peano2re 11347	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
40	37, 39	syl 17	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 1) ∈ ℝ)
41		2re 12260	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
42	41	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 2 ∈ ℝ)
43		2pos 12289	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
44	43	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 0 < 2)
45		lemul2 12035	. . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
47	38, 46	mpbid 232	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ≤ (2 · (𝑘 + 1)))
48	29, 34, 35, 47	leadd1dd 11792	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1))
49		nn0p1nn 12481	. . . . . . . . . . . . . 14 ⊢ ((2 · 𝑘) ∈ ℕ₀ → ((2 · 𝑘) + 1) ∈ ℕ)
50	28, 49	syl 17	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ∈ ℕ)
51	50	nnred 12201	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ∈ ℝ)
52	50	nngt0d 12235	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 0 < ((2 · 𝑘) + 1))
53		nn0p1nn 12481	. . . . . . . . . . . . . 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 12201	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · (𝑘 + 1)) + 1) ∈ ℝ)
56	54	nngt0d 12235	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → 0 < ((2 · (𝑘 + 1)) + 1))
57		lerec 12066	. . . . . . . . . . . 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 838	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1))))
59	48, 58	mpbid 232	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1)))
60		oveq2 7395	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑘 + 1) → (2 · 𝑛) = (2 · (𝑘 + 1)))
61	60	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑘 + 1) → ((2 · 𝑛) + 1) = ((2 · (𝑘 + 1)) + 1))
62	61	oveq2d 7403	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑘 + 1) → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
63		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))
64		ovex 7420	. . . . . . . . . . . 12 ⊢ (1 / ((2 · (𝑘 + 1)) + 1)) ∈ V
65	62, 63, 64	fvmpt 6968	. . . . . . . . . . 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 7395	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
68	67	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((2 · 𝑛) + 1) = ((2 · 𝑘) + 1))
69	68	oveq2d 7403	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · 𝑘) + 1)))
70		ovex 7420	. . . . . . . . . . . 12 ⊢ (1 / ((2 · 𝑘) + 1)) ∈ V
71	69, 63, 70	fvmpt 6968	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
72	71	adantl 481	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
73	59, 66, 72	3brtr4d 5139	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) ≤ ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
74		nnuz 12836	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
75		1zzd 12564	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ)
76		ax-1cn 11126	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
77		divcnv 15819	. . . . . . . . . . 11 ⊢ (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
78	76, 77	mp1i 13	. . . . . . . . . 10 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
79		nn0ex 12448	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
80	79	mptex 7197	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) ∈ V
81	80	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) ∈ V)
82		oveq2 7395	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
83		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
84		ovex 7420	. . . . . . . . . . . . 13 ⊢ (1 / 𝑘) ∈ V
85	82, 83, 84	fvmpt 6968	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
86	85	adantl 481	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
87		nnrecre 12228	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
88	87	adantl 481	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
89	86, 88	eqeltrd 2828	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℝ)
90		nnnn0 12449	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
91	90	adantl 481	. . . . . . . . . . . 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 593	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℕ)
94	93	nnrecred 12237	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ)
95	92, 94	eqeltrd 2828	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ∈ ℝ)
96		nnre 12193	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
97	96	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
98	19, 91, 27	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℕ₀)
99	98	nn0red 12504	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℝ)
100		peano2re 11347	. . . . . . . . . . . . . 14 ⊢ ((2 · 𝑘) ∈ ℝ → ((2 · 𝑘) + 1) ∈ ℝ)
101	99, 100	syl 17	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
102		nn0addge1 12488	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ≤ (𝑘 + 𝑘))
103	97, 91, 102	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (𝑘 + 𝑘))
104	97	recnd 11202	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
105	104	2timesd 12425	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) = (𝑘 + 𝑘))
106	103, 105	breqtrrd 5135	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (2 · 𝑘))
107	99	lep1d 12114	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ≤ ((2 · 𝑘) + 1))
108	97, 99, 101, 106, 107	letrd 11331	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ ((2 · 𝑘) + 1))
109		nngt0 12217	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
110	109	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
111	93	nnred 12201	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
112	93	nngt0d 12235	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < ((2 · 𝑘) + 1))
113		lerec 12066	. . . . . . . . . . . . 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 838	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)))
115	108, 114	mpbid 232	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘))
116	115, 92, 86	3brtr4d 5139	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))
117	93	nnrpd 12993	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ⁺)
118	117	rpreccld 13005	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ⁺)
119	118	rpge0d 12999	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / ((2 · 𝑘) + 1)))
120	119, 92	breqtrrd 5135	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
121	74, 75, 78, 81, 89, 95, 116, 120	climsqz2 15608	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1))) ⇝ 0)
122		neg1cn 12171	. . . . . . . . . . . . 13 ⊢ -1 ∈ ℂ
123	122	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → -1 ∈ ℂ)
124		expcl 14044	. . . . . . . . . . . 12 ⊢ ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
125	123, 124	sylan 580	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
126	50	nncnd 12202	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ∈ ℂ)
127	50	nnne0d 12236	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((2 · 𝑘) + 1) ≠ 0)
128	125, 126, 127	divrecd 11961	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((-1↑𝑘) / ((2 · 𝑘) + 1)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
129		oveq2 7395	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (-1↑𝑛) = (-1↑𝑘))
130	129, 68	oveq12d 7405	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
131		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
132		ovex 7420	. . . . . . . . . . . 12 ⊢ ((-1↑𝑘) / ((2 · 𝑘) + 1)) ∈ V
133	130, 131, 132	fvmpt 6968	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
134	133	adantl 481	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
135	72	oveq2d 7403	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ₀) → ((-1↑𝑘) · ((𝑛 ∈ ℕ₀ ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
136	128, 134, 135	3eqtr4d 2774	. . . . . . . . 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 15651	. . . . . . . 8 ⊢ (⊤ → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝ )
138		climdm 15520	. . . . . . . 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 218	. . . . . . 7 ⊢ (⊤ → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
140		eqid 2729	. . . . . . . 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 6871	. . . . . . . 8 ⊢ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ₀ ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ∈ V
142	131, 140, 141	leibpilem2 26851	. . . . . . 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 218	. . . . . 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 13968	. . . . . . 7 ⊢ seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ V
145	144, 141	breldm 5872	. . . . . 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 15724	. . . 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 2729	. . . . . . . 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 26352	. . . . . . 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 12963	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
151	150	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
152	151	rpreccld 13005	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
153	152	rpred 12995	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
154	152	rpge0d 12999	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ≤ (1 / 𝑛))
155		nnge1 12214	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 1 ≤ 𝑛)
156	155	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
157		nnre 12193	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
158	157	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
159	158	recnd 11202	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
160	159	mulridd 11191	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑛 · 1) = 𝑛)
161	156, 160	breqtrrd 5135	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ (𝑛 · 1))
162		1red 11175	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
163		nngt0 12217	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
164	163	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
165		ledivmul 12059	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
166	162, 162, 158, 164, 165	syl112anc 1376	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
167	161, 166	mpbird 257	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ≤ 1)
168		elicc01 13427	. . . . . . . . . 10 ⊢ ((1 / 𝑛) ∈ (0[,]1) ↔ ((1 / 𝑛) ∈ ℝ ∧ 0 ≤ (1 / 𝑛) ∧ (1 / 𝑛) ≤ 1))
169	153, 154, 167, 168	syl3anbrc 1344	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ (0[,]1))
170		iirev 24823	. . . . . . . . 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 7087	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶(0[,]1))
173		1cnd 11169	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
174		nnex 12192	. . . . . . . . . . 11 ⊢ ℕ ∈ V
175	174	mptex 7197	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V
176	175	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V)
177	89	recnd 11202	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℂ)
178	82	oveq2d 7403	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (1 − (1 / 𝑛)) = (1 − (1 / 𝑘)))
179		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))
180		ovex 7420	. . . . . . . . . . . 12 ⊢ (1 − (1 / 𝑘)) ∈ V
181	178, 179, 180	fvmpt 6968	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − (1 / 𝑘)))
182	85	oveq2d 7403	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)) = (1 − (1 / 𝑘)))
183	181, 182	eqtr4d 2767	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
184	183	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
185	74, 75, 78, 173, 176, 177, 184	climsubc2 15605	. . . . . . . 8 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ (1 − 0))
186		1m0e1 12302	. . . . . . . 8 ⊢ (1 − 0) = 1
187	185, 186	breqtrdi 5148	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ 1)
188		1elunit 13431	. . . . . . . 8 ⊢ 1 ∈ (0[,]1)
189	188	a1i 11	. . . . . . 7 ⊢ (⊤ → 1 ∈ (0[,]1))
190	74, 75, 149, 172, 187, 189	climcncf 24793	. . . . . 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 2730	. . . . . . . 8 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))))
192		eqidd 2730	. . . . . . . 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 7394	. . . . . . . . . 10 ⊢ (𝑥 = (1 − (1 / 𝑛)) → (𝑥↑𝑗) = ((1 − (1 / 𝑛))↑𝑗))
194	193	oveq2d 7403	. . . . . . . . 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 15669	. . . . . . . 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 7101	. . . . . . 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 12541	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℤ)
198	8	adantll 714	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ₀)
199	6, 198, 9	sylancr 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
200	199	recnd 11202	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
201	200	adantllr 719	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
202		1re 11174	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℝ
203		resubcl 11486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ) → (1 − (1 / 𝑛)) ∈ ℝ)
204	202, 153, 203	sylancr 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℝ)
205	204	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (1 − (1 / 𝑛)) ∈ ℝ)
206		simplr 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ₀)
207	205, 206	reexpcld 14128	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℝ)
208	207	recnd 11202	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
209		nn0cn 12452	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
210	209	ad2antlr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℂ)
211	11	adantll 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
212	211	nnne0d 12236	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ≠ 0)
213	201, 208, 210, 212	div12d 11994	. . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
215	208, 214	mulcomd 11195	. . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . 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 4521	. . . . . . . . . . . . . . 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 11202	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℂ)
220		expcl 14044	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 − (1 / 𝑛)) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
221	219, 220	sylan 580	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
222	221	mul02d 11372	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (0 · ((1 − (1 / 𝑛))↑𝑘)) = 0)
223	222	ifeq1d 4508	. . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . 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 7487	. . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
228		c0ex 11168	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
229		ovex 7420	. . . . . . . . . . . . . . 15 ⊢ ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ V
230	228, 229	ifex 4539	. . . . . . . . . . . . . 14 ⊢ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
231		eqid 2729	. . . . . . . . . . . . . . 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 6979	. . . . . . . . . . . . . 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 586	. . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . . . 16 ⊢ ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ V
235	228, 234	ifex 4539	. . . . . . . . . . . . . . 15 ⊢ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ V
236	140	fvmpt2 6979	. . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 3125	. . . . . . . . . . 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 1914	. . . . . . . . . . . 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 6869	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
243		nffvmpt1 6869	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)
244		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 ·
245		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((1 − (1 / 𝑛))↑𝑗)
246	243, 244, 245	nfov 7417	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))
247	242, 246	nfeq 2905	. . . . . . . . . . . 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 6858	. . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . 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 7395	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((1 − (1 / 𝑛))↑𝑘) = ((1 − (1 / 𝑛))↑𝑗))
251	249, 250	oveq12d 7405	. . . . . . . . . . . . 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 2745	. . . . . . . . . . . 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 3280	. . . . . . . . . . 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 218	. . . . . . . . . 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 3229	. . . . . . . . 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 11167	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
257	207, 211	nndivred 12240	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℝ)
258	257	recnd 11202	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℂ)
259	201, 258	mulcld 11194	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
260	5, 259	sylan2b 594	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
261	256, 260	ifclda 4524	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ ℂ)
262	261	fmpttd 7087	. . . . . . . . . . 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 7056	. . . . . . . . . 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 2829	. . . . . . . . 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 12457	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
266	265	a1i 11	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℕ₀)
267		0p1e1 12303	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
268		seqeq1 13969	. . . . . . . . . . . . 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 12564	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
271		elnnuz 12837	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ_≥‘1))
272		nnne0 12220	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
273	272	neneqd 2930	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ ℕ → ¬ 𝑘 = 0)
274		biorf 936	. . . . . . . . . . . . . . . . . . . . . . 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 223	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ 2 ∥ 𝑘))
277	276	ifbid 4512	. . . . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . . . . 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 4539	. . . . . . . . . . . . . . . . . . . . 21 ⊢ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
280		eqid 2729	. . . . . . . . . . . . . . . . . . . . . 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 6979	. . . . . . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . 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 3046	. . . . . . . . . . . . . . . . . 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 1914	. . . . . . . . . . . . . . . . . 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 6869	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
288	242, 287	nfeq 2905	. . . . . . . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . . . . 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 3280	. . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . 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 3229	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 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 13992	. . . . . . . . . . . . 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 15400	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) = (1 − (1 / 𝑛)))
297		ltsubrp 12989	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ⁺) → (1 − (1 / 𝑛)) < 1)
298	202, 152, 297	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) < 1)
299	296, 298	eqbrtrd 5129	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) < 1)
300	280	atantayl2 26848	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 5129	. . . . . . . . . . . 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 5142	. . . . . . . . . . 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 15622	. . . . . . . . . 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 12540	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℤ
306		seq1 13979	. . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
309	308	orcs 875	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
310	309, 231, 228	fvmpt 6968	. . . . . . . . . . . . . 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 2752	. . . . . . . . . . . 12 ⊢ (seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = 0
313	312	oveq2i 7398	. . . . . . . . . . 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 26821	. . . . . . . . . . . . . 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 11202	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℂ)
317	316	addridd 11374	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + 0) = (arctan‘(1 − (1 / 𝑛))))
318	313, 317	eqtrid 2776	. . . . . . . . . 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 5133	. . . . . . . . 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 15723	. . . . . . . 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 5200	. . . . . . 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 2764	. . . . . 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 7394	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥↑𝑗) = (1↑𝑗))
324		nn0z 12554	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → 𝑗 ∈ ℤ)
325		1exp 14056	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℤ → (1↑𝑗) = 1)
326	324, 325	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ₀ → (1↑𝑗) = 1)
327	323, 326	sylan9eq 2784	. . . . . . . . . . 11 ⊢ ((𝑥 = 1 ∧ 𝑗 ∈ ℕ₀) → (𝑥↑𝑗) = 1)
328	327	oveq2d 7403	. . . . . . . . . 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 1547	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ₀⟶ℂ
330	329	ffvelcdmi 7055	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
331	330	mulridd 11191	. . . . . . . . . . 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 481	. . . . . . . . . 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 2764	. . . . . . . . 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 15668	. . . . . . . 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 15654	. . . . . . . 8 ⊢ Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ V
336	334, 148, 335	fvmpt 6968	. . . . . . 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 5138	. . . . 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 2729	. . . . . . . . 9 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
340		eqid 2729	. . . . . . . . 9 ⊢ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} = {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
341	339, 340	atancn 26846	. . . . . . . 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 13460	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℝ
344	339, 340	ressatans 26844	. . . . . . . . 9 ⊢ ℝ ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
345	343, 344	sstri 3956	. . . . . . . 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 586	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
348	344, 202	sselii 3943	. . . . . . . 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 24793	. . . . . 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 3944	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
352		cncff 24786	. . . . . . . . . 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 6929	. . . . . . . 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 6877	. . . . . . . . 9 ⊢ (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘𝑘) = (arctan‘𝑘))
356	355	mpteq2ia 5202	. . . . . . . 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 2780	. . . . . . 7 ⊢ (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ (arctan‘𝑘)))
358		fveq2 6858	. . . . . . 7 ⊢ (𝑘 = (1 − (1 / 𝑛)) → (arctan‘𝑘) = (arctan‘(1 − (1 / 𝑛))))
359	351, 191, 357, 358	fmptco 7101	. . . . . 6 ⊢ (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))))
360		fvres 6877	. . . . . . . 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 26838	. . . . . . 7 ⊢ (arctan‘1) = (π / 4)
363	361, 362	eqtrdi 2780	. . . . . 6 ⊢ (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (π / 4))
364	350, 359, 363	3brtr3d 5138	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ (π / 4))
365		climuni 15518	. . . . 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 584	. . . 4 ⊢ (⊤ → Σ𝑗 ∈ ℕ₀ ((𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4))
367	147, 366	breqtrd 5133	. . 3 ⊢ (⊤ → seq0( + , (𝑘 ∈ ℕ₀ ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
368	367	mptru 1547	. 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 7420	. . 3 ⊢ (π / 4) ∈ V
371	369, 140, 370	leibpilem2 26851	. 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 231	1 ⊢ seq0( + , 𝐹) ⇝ (π / 4)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ∀wral 3044 {crab 3405 Vcvv 3447 ∖ cdif 3911 ⊆ wss 3914 ifcif 4488 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 ↾ cres 5640 ∘ ccom 5642 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 -∞cmnf 11206 < clt 11208 ≤ cle 11209 − cmin 11405 -cneg 11406 / cdiv 11835 ℕcn 12186 2c2 12241 4c4 12243 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 (,]cioc 13307 [,]cicc 13309 seqcseq 13966 ↑cexp 14026 abscabs 15200 ⇝ cli 15450 Σcsu 15652 πcpi 16032 ∥ cdvds 16222 –cn→ccncf 24769 arctancatan 26774

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-tan 16037 df-pi 16038 df-dvds 16223 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-t1 23201 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-ulm 26286 df-log 26465 df-atan 26777

This theorem is referenced by: leibpisum 26853

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