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Theorem leibpi 24883
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 14616). (2) Using leibpilem2 24882 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 24879). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 24879, Abel's theorem (abelth2 24409) 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 24877) 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.
StepHypRef Expression
1 nn0uz 11922 . . . . 5 0 = (ℤ‘0)
2 0zd 11589 . . . . 5 (⊤ → 0 ∈ ℤ)
3 eqidd 2772 . . . . 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 10233 . . . . . . . . 9 ((𝑘 ∈ ℕ0 ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
5 ioran 968 . . . . . . . . . 10 (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
6 neg1rr 11325 . . . . . . . . . . . . 13 -1 ∈ ℝ
7 leibpilem1 24881 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ ((𝑘 − 1) / 2) ∈ ℕ0))
87simprd 483 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ0)
9 reexpcl 13077 . . . . . . . . . . . . 13 ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ0) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
106, 8, 9sylancr 575 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
117simpld 482 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
1210, 11nndivred 11269 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
1312recnd 10268 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
145, 13sylan2b 581 . . . . . . . . 9 ((𝑘 ∈ ℕ0 ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
154, 14ifclda 4259 . . . . . . . 8 (𝑘 ∈ ℕ0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
1615adantl 467 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
17 eqid 2771 . . . . . . 7 (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)))
1816, 17fmptd 6525 . . . . . 6 (⊤ → (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ0⟶ℂ)
1918ffvelrnda 6500 . . . . 5 ((⊤ ∧ 𝑗 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
20 2nn0 11509 . . . . . . . . . . . . . 14 2 ∈ ℕ0
2120a1i 11 . . . . . . . . . . . . 13 (⊤ → 2 ∈ ℕ0)
22 nn0mulcl 11529 . . . . . . . . . . . . 13 ((2 ∈ ℕ0𝑛 ∈ ℕ0) → (2 · 𝑛) ∈ ℕ0)
2321, 22sylan 569 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ0) → (2 · 𝑛) ∈ ℕ0)
24 nn0p1nn 11532 . . . . . . . . . . . 12 ((2 · 𝑛) ∈ ℕ0 → ((2 · 𝑛) + 1) ∈ ℕ)
2523, 24syl 17 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ0) → ((2 · 𝑛) + 1) ∈ ℕ)
2625nnrecred 11266 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ0) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ)
27 eqid 2771 . . . . . . . . . 10 (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))
2826, 27fmptd 6525 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))):ℕ0⟶ℝ)
29 nn0mulcl 11529 . . . . . . . . . . . . . 14 ((2 ∈ ℕ0𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
3021, 29sylan 569 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
3130nn0red 11552 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℝ)
32 peano2nn0 11533 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
3332adantl 467 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
34 nn0mulcl 11529 . . . . . . . . . . . . . 14 ((2 ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℕ0)
3520, 33, 34sylancr 575 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℕ0)
3635nn0red 11552 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℝ)
37 1red 10255 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 1 ∈ ℝ)
38 nn0re 11501 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
3938adantl 467 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
4039lep1d 11155 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → 𝑘 ≤ (𝑘 + 1))
41 peano2re 10409 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
4239, 41syl 17 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℝ)
43 2re 11290 . . . . . . . . . . . . . . 15 2 ∈ ℝ
4443a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 2 ∈ ℝ)
45 2pos 11312 . . . . . . . . . . . . . . 15 0 < 2
4645a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < 2)
47 lemul2 11076 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
4839, 42, 44, 46, 47syl112anc 1480 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
4940, 48mpbid 222 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ≤ (2 · (𝑘 + 1)))
5031, 36, 37, 49leadd1dd 10841 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1))
51 nn0p1nn 11532 . . . . . . . . . . . . . 14 ((2 · 𝑘) ∈ ℕ0 → ((2 · 𝑘) + 1) ∈ ℕ)
5230, 51syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℕ)
5352nnred 11235 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℝ)
5452nngt0d 11264 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < ((2 · 𝑘) + 1))
55 nn0p1nn 11532 . . . . . . . . . . . . . 14 ((2 · (𝑘 + 1)) ∈ ℕ0 → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
5635, 55syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
5756nnred 11235 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℝ)
5856nngt0d 11264 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < ((2 · (𝑘 + 1)) + 1))
59 lerec 11106 . . . . . . . . . . . 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))))
6053, 54, 57, 58, 59syl22anc 1477 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1))))
6150, 60mpbid 222 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1)))
62 oveq2 6799 . . . . . . . . . . . . . 14 (𝑛 = (𝑘 + 1) → (2 · 𝑛) = (2 · (𝑘 + 1)))
6362oveq1d 6806 . . . . . . . . . . . . 13 (𝑛 = (𝑘 + 1) → ((2 · 𝑛) + 1) = ((2 · (𝑘 + 1)) + 1))
6463oveq2d 6807 . . . . . . . . . . . 12 (𝑛 = (𝑘 + 1) → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
65 ovex 6821 . . . . . . . . . . . 12 (1 / ((2 · (𝑘 + 1)) + 1)) ∈ V
6664, 27, 65fvmpt 6422 . . . . . . . . . . 11 ((𝑘 + 1) ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
6733, 66syl 17 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
68 oveq2 6799 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
6968oveq1d 6806 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((2 · 𝑛) + 1) = ((2 · 𝑘) + 1))
7069oveq2d 6807 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · 𝑘) + 1)))
71 ovex 6821 . . . . . . . . . . . 12 (1 / ((2 · 𝑘) + 1)) ∈ V
7270, 27, 71fvmpt 6422 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
7372adantl 467 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
7461, 67, 733brtr4d 4818 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) ≤ ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
75 nnuz 11923 . . . . . . . . . 10 ℕ = (ℤ‘1)
76 1zzd 11608 . . . . . . . . . 10 (⊤ → 1 ∈ ℤ)
77 ax-1cn 10194 . . . . . . . . . . 11 1 ∈ ℂ
78 divcnv 14785 . . . . . . . . . . 11 (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
7977, 78mp1i 13 . . . . . . . . . 10 (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
80 nn0ex 11498 . . . . . . . . . . . 12 0 ∈ V
8180mptex 6628 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ∈ V
8281a1i 11 . . . . . . . . . 10 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ∈ V)
83 oveq2 6799 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
84 eqid 2771 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
85 ovex 6821 . . . . . . . . . . . . 13 (1 / 𝑘) ∈ V
8683, 84, 85fvmpt 6422 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
8786adantl 467 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
88 nnrecre 11257 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
8988adantl 467 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
9087, 89eqeltrd 2850 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℝ)
91 nnnn0 11499 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
9291adantl 467 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
9392, 72syl 17 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
9491, 52sylan2 580 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℕ)
9594nnrecred 11266 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ)
9693, 95eqeltrd 2850 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ∈ ℝ)
97 nnre 11227 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
9897adantl 467 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
9920, 92, 29sylancr 575 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℕ0)
10099nn0red 11552 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℝ)
101 peano2re 10409 . . . . . . . . . . . . . 14 ((2 · 𝑘) ∈ ℝ → ((2 · 𝑘) + 1) ∈ ℝ)
102100, 101syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
103 nn0addge1 11539 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → 𝑘 ≤ (𝑘 + 𝑘))
10498, 92, 103syl2anc 573 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (𝑘 + 𝑘))
10598recnd 10268 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
1061052timesd 11475 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) = (𝑘 + 𝑘))
107104, 106breqtrrd 4814 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (2 · 𝑘))
108100lep1d 11155 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ≤ ((2 · 𝑘) + 1))
10998, 100, 102, 107, 108letrd 10394 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ ((2 · 𝑘) + 1))
110 nngt0 11249 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 0 < 𝑘)
111110adantl 467 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
11294nnred 11235 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
11394nngt0d 11264 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < ((2 · 𝑘) + 1))
114 lerec 11106 . . . . . . . . . . . . 13 (((𝑘 ∈ ℝ ∧ 0 < 𝑘) ∧ (((2 · 𝑘) + 1) ∈ ℝ ∧ 0 < ((2 · 𝑘) + 1))) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)))
11598, 111, 112, 113, 114syl22anc 1477 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)))
116109, 115mpbid 222 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘))
117116, 93, 873brtr4d 4818 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))
11894nnrpd 12066 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ+)
119118rpreccld 12078 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ+)
120119rpge0d 12072 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / ((2 · 𝑘) + 1)))
121120, 93breqtrrd 4814 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
12275, 76, 79, 82, 90, 96, 117, 121climsqz2 14573 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ⇝ 0)
123 neg1cn 11324 . . . . . . . . . . . . 13 -1 ∈ ℂ
124123a1i 11 . . . . . . . . . . . 12 (⊤ → -1 ∈ ℂ)
125 expcl 13078 . . . . . . . . . . . 12 ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℂ)
126124, 125sylan 569 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℂ)
12752nncnd 11236 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℂ)
12852nnne0d 11265 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ≠ 0)
129126, 127, 128divrecd 11004 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) / ((2 · 𝑘) + 1)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
130 oveq2 6799 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (-1↑𝑛) = (-1↑𝑘))
131130, 69oveq12d 6809 . . . . . . . . . . . 12 (𝑛 = 𝑘 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
132 eqid 2771 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
133 ovex 6821 . . . . . . . . . . . 12 ((-1↑𝑘) / ((2 · 𝑘) + 1)) ∈ V
134131, 132, 133fvmpt 6422 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
135134adantl 467 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
13673oveq2d 6807 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) · ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
137129, 135, 1363eqtr4d 2815 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) · ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)))
1381, 2, 28, 74, 122, 137iseralt 14616 . . . . . . . 8 (⊤ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝ )
139 climdm 14486 . . . . . . . 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))))))
140138, 139sylib 208 . . . . . . 7 (⊤ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
141 fvex 6340 . . . . . . . 8 ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ∈ V
142132, 17, 141leibpilem2 24882 . . . . . . 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))))))
143140, 142sylib 208 . . . . . 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 13003 . . . . . . 7 seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ V
145144, 141breldm 5465 . . . . . 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 ⇝ )
146143, 145syl 17 . . . . 5 (⊤ → seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ dom ⇝ )
1471, 2, 3, 19, 146isumclim2 14690 . . . 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 2771 . . . . . . . 8 (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗))) = (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗)))
14918, 146, 148abelth2 24409 . . . . . . 7 (⊤ → (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗))) ∈ ((0[,]1)–cn→ℂ))
150 nnrp 12038 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
151150adantl 467 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+)
152151rpreccld 12078 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ+)
153152rpred 12068 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
154152rpge0d 12072 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ≤ (1 / 𝑛))
155 nnge1 11246 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 1 ≤ 𝑛)
156155adantl 467 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
157 nnre 11227 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
158157adantl 467 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
159158recnd 10268 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
160159mulid1d 10257 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑛 · 1) = 𝑛)
161156, 160breqtrrd 4814 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ (𝑛 · 1))
162 1red 10255 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
163 nngt0 11249 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 0 < 𝑛)
164163adantl 467 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
165 ledivmul 11099 . . . . . . . . . . . 12 ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
166162, 162, 158, 164, 165syl112anc 1480 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
167161, 166mpbird 247 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ≤ 1)
168 0re 10240 . . . . . . . . . . 11 0 ∈ ℝ
169 1re 10239 . . . . . . . . . . 11 1 ∈ ℝ
170168, 169elicc2i 12437 . . . . . . . . . 10 ((1 / 𝑛) ∈ (0[,]1) ↔ ((1 / 𝑛) ∈ ℝ ∧ 0 ≤ (1 / 𝑛) ∧ (1 / 𝑛) ≤ 1))
171153, 154, 167, 170syl3anbrc 1428 . . . . . . . . 9 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ (0[,]1))
172 iirev 22941 . . . . . . . . 9 ((1 / 𝑛) ∈ (0[,]1) → (1 − (1 / 𝑛)) ∈ (0[,]1))
173171, 172syl 17 . . . . . . . 8 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ (0[,]1))
174 eqid 2771 . . . . . . . 8 (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))
175173, 174fmptd 6525 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶(0[,]1))
176 1cnd 10256 . . . . . . . . 9 (⊤ → 1 ∈ ℂ)
177 nnex 11226 . . . . . . . . . . 11 ℕ ∈ V
178177mptex 6628 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V
179178a1i 11 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V)
18090recnd 10268 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℂ)
18183oveq2d 6807 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (1 − (1 / 𝑛)) = (1 − (1 / 𝑘)))
182 ovex 6821 . . . . . . . . . . . 12 (1 − (1 / 𝑘)) ∈ V
183181, 174, 182fvmpt 6422 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − (1 / 𝑘)))
18486oveq2d 6807 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)) = (1 − (1 / 𝑘)))
185183, 184eqtr4d 2808 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
186185adantl 467 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
18775, 76, 79, 176, 179, 180, 186climsubc2 14570 . . . . . . . 8 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ (1 − 0))
188 1m0e1 11331 . . . . . . . 8 (1 − 0) = 1
189187, 188syl6breq 4827 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ 1)
190 1elunit 12491 . . . . . . . 8 1 ∈ (0[,]1)
191190a1i 11 . . . . . . 7 (⊤ → 1 ∈ (0[,]1))
19275, 76, 149, 175, 189, 191climcncf 22916 . . . . . 6 (⊤ → ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗))) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) ⇝ ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗)))‘1))
193 eqidd 2772 . . . . . . . 8 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))))
194 eqidd 2772 . . . . . . . 8 (⊤ → (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗))) = (𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗))))
195 oveq1 6798 . . . . . . . . . 10 (𝑥 = (1 − (1 / 𝑛)) → (𝑥𝑗) = ((1 − (1 / 𝑛))↑𝑗))
196195oveq2d 6807 . . . . . . . . 9 (𝑥 = (1 − (1 / 𝑛)) → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗)) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
197196sumeq2sdv 14636 . . . . . . . 8 (𝑥 = (1 − (1 / 𝑛)) → Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗)) = Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
198173, 193, 194, 197fmptco 6537 . . . . . . 7 (⊤ → ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗))) ∘ (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))))
199 0zd 11589 . . . . . . . . 9 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℤ)
2008adantll 693 . . . . . . . . . . . . . . . . . . . . . 22 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ0)
2016, 200, 9sylancr 575 . . . . . . . . . . . . . . . . . . . . 21 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
202201recnd 10268 . . . . . . . . . . . . . . . . . . . 20 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
203202adantllr 698 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
204 resubcl 10545 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ) → (1 − (1 / 𝑛)) ∈ ℝ)
205169, 153, 204sylancr 575 . . . . . . . . . . . . . . . . . . . . . 22 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℝ)
206205ad2antrr 705 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (1 − (1 / 𝑛)) ∈ ℝ)
207 simplr 752 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ0)
208206, 207reexpcld 13225 . . . . . . . . . . . . . . . . . . . 20 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℝ)
209208recnd 10268 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
210 nn0cn 11502 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
211210ad2antlr 706 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℂ)
21211adantll 693 . . . . . . . . . . . . . . . . . . . 20 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
213212nnne0d 11265 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ≠ 0)
214203, 209, 211, 213div12d 11037 . . . . . . . . . . . . . . . . . 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)))
21513adantll 693 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
216209, 215mulcomd 10261 . . . . . . . . . . . . . . . . . 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)))
217214, 216eqtrd 2805 . . . . . . . . . . . . . . . . 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)))
2185, 217sylan2b 581 . . . . . . . . . . . . . . . 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)))
219218ifeq2da 4256 . . . . . . . . . . . . . . 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))))
220205recnd 10268 . . . . . . . . . . . . . . . . . 18 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℂ)
221 expcl 13078 . . . . . . . . . . . . . . . . . 18 (((1 − (1 / 𝑛)) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
222220, 221sylan 569 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
223222mul02d 10434 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (0 · ((1 − (1 / 𝑛))↑𝑘)) = 0)
224223ifeq1d 4243 . . . . . . . . . . . . . . 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))))
225219, 224eqtr4d 2808 . . . . . . . . . . . . . 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))))
226 ovif 6882 . . . . . . . . . . . . . 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)))
227225, 226syl6eqr 2823 . . . . . . . . . . . . 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)))
228 simpr 471 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
229 c0ex 10234 . . . . . . . . . . . . . . 15 0 ∈ V
230 ovex 6821 . . . . . . . . . . . . . . 15 ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ V
231229, 230ifex 4295 . . . . . . . . . . . . . 14 if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
232 eqid 2771 . . . . . . . . . . . . . . 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))))
233232fvmpt2 6431 . . . . . . . . . . . . . 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))))
234228, 231, 233sylancl 574 . . . . . . . . . . . . 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))))
235 ovex 6821 . . . . . . . . . . . . . . . 16 ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ V
236229, 235ifex 4295 . . . . . . . . . . . . . . 15 if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ V
23717fvmpt2 6431 . . . . . . . . . . . . . . 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)))
238228, 236, 237sylancl 574 . . . . . . . . . . . . . 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)))
239238oveq1d 6806 . . . . . . . . . . . . 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)))
240227, 234, 2393eqtr4d 2815 . . . . . . . . . . . 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)))
241240ralrimiva 3115 . . . . . . . . . . 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)))
242 nfv 1995 . . . . . . . . . . . 12 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))
243 nffvmpt1 6338 . . . . . . . . . . . . 13 𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
244 nffvmpt1 6338 . . . . . . . . . . . . . 14 𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)
245 nfcv 2913 . . . . . . . . . . . . . 14 𝑘 ·
246 nfcv 2913 . . . . . . . . . . . . . 14 𝑘((1 − (1 / 𝑛))↑𝑗)
247244, 245, 246nfov 6819 . . . . . . . . . . . . 13 𝑘(((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))
248243, 247nfeq 2925 . . . . . . . . . . . 12 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))
249 fveq2 6330 . . . . . . . . . . . . 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))
250 fveq2 6330 . . . . . . . . . . . . . 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))
251 oveq2 6799 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ((1 − (1 / 𝑛))↑𝑘) = ((1 − (1 / 𝑛))↑𝑗))
252250, 251oveq12d 6809 . . . . . . . . . . . . 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)))
253249, 252eqeq12d 2786 . . . . . . . . . . . 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))))
254242, 248, 253cbvral 3316 . . . . . . . . . . 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)))
255241, 254sylib 208 . . . . . . . . . 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)))
256255r19.21bi 3081 . . . . . . . . 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)))
257 0cnd 10233 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
258208, 212nndivred 11269 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℝ)
259258recnd 10268 . . . . . . . . . . . . . . 15 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℂ)
260203, 259mulcld 10260 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
2615, 260sylan2b 581 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
262257, 261ifclda 4259 . . . . . . . . . . . 12 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ ℂ)
263262, 232fmptd 6525 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))):ℕ0⟶ℂ)
264263ffvelrnda 6500 . . . . . . . . . 10 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) ∈ ℂ)
265256, 264eqeltrrd 2851 . . . . . . . . 9 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)) ∈ ℂ)
266 0nn0 11507 . . . . . . . . . . . 12 0 ∈ ℕ0
267266a1i 11 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℕ0)
268 0p1e1 11332 . . . . . . . . . . . . 13 (0 + 1) = 1
269 seqeq1 13004 . . . . . . . . . . . . 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))))))
270268, 269ax-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)))))
271 1zzd 11608 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
272 elnnuz 11924 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ‘1))
273 nnne0 11253 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
274273neneqd 2948 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ ℕ → ¬ 𝑘 = 0)
275 biorf 922 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘 = 0 → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
276274, 275syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
277276bicomd 213 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ 2 ∥ 𝑘))
278277ifbid 4247 . . . . . . . . . . . . . . . . . . . 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))))
27991, 231, 233sylancl 574 . . . . . . . . . . . . . . . . . . . 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))))
280229, 230ifex 4295 . . . . . . . . . . . . . . . . . . . . 21 if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
281 eqid 2771 . . . . . . . . . . . . . . . . . . . . . 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))))
282281fvmpt2 6431 . . . . . . . . . . . . . . . . . . . . 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))))
283280, 282mpan2 671 . . . . . . . . . . . . . . . . . . . 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))))
284278, 279, 2833eqtr4d 2815 . . . . . . . . . . . . . . . . . . 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))
285284rgen 3071 . . . . . . . . . . . . . . . . . 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)
286285a1i 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))
287 nfv 1995 . . . . . . . . . . . . . . . . . 18 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)
288 nffvmpt1 6338 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
289243, 288nfeq 2925 . . . . . . . . . . . . . . . . . 18 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)
290 fveq2 6330 . . . . . . . . . . . . . . . . . . 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))
291249, 290eqeq12d 2786 . . . . . . . . . . . . . . . . . 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)))
292287, 289, 291cbvral 3316 . . . . . . . . . . . . . . . . 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))
293286, 292sylib 208 . . . . . . . . . . . . . . . 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))
294293r19.21bi 3081 . . . . . . . . . . . . . . 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))
295272, 294sylan2br 582 . . . . . . . . . . . . . 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))
296271, 295seqfeq 13026 . . . . . . . . . . . . 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))))))
297153, 162, 167abssubge0d 14371 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) = (1 − (1 / 𝑛)))
298 ltsubrp 12062 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ+) → (1 − (1 / 𝑛)) < 1)
299169, 152, 298sylancr 575 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) < 1)
300297, 299eqbrtrd 4808 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) < 1)
301281atantayl2 24879 . . . . . . . . . . . . . 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))))
302220, 300, 301syl2anc 573 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑛 ∈ ℕ) → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 / 𝑛))))
303296, 302eqbrtrd 4808 . . . . . . . . . . . 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))))
304270, 303syl5eqbr 4821 . . . . . . . . . . 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))))
3051, 267, 264, 304clim2ser2 14587 . . . . . . . . . 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)))
306 0z 11588 . . . . . . . . . . . . . 14 0 ∈ ℤ
307 seq1 13014 . . . . . . . . . . . . . 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))
308306, 307ax-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)
309 iftrue 4231 . . . . . . . . . . . . . . . 16 ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
310309orcs 864 . . . . . . . . . . . . . . 15 (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
311310, 232, 229fvmpt 6422 . . . . . . . . . . . . . 14 (0 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0) = 0)
312266, 311ax-mp 5 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0) = 0
313308, 312eqtri 2793 . . . . . . . . . . . 12 (seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = 0
314313oveq2i 6802 . . . . . . . . . . 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)
315 atanrecl 24852 . . . . . . . . . . . . . 14 ((1 − (1 / 𝑛)) ∈ ℝ → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
316205, 315syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
317316recnd 10268 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℂ)
318317addid1d 10436 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + 0) = (arctan‘(1 − (1 / 𝑛))))
319314, 318syl5eq 2817 . . . . . . . . . 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))))
320305, 319breqtrd 4812 . . . . . . . . 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))))
3211, 199, 256, 265, 320isumclim 14689 . . . . . . . 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))))
322321mpteq2dva 4878 . . . . . . 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)))))
323198, 322eqtrd 2805 . . . . . 6 (⊤ → ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗))) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))))
324 oveq1 6798 . . . . . . . . . . . 12 (𝑥 = 1 → (𝑥𝑗) = (1↑𝑗))
325 nn0z 11600 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ0𝑗 ∈ ℤ)
326 1exp 13089 . . . . . . . . . . . . 13 (𝑗 ∈ ℤ → (1↑𝑗) = 1)
327325, 326syl 17 . . . . . . . . . . . 12 (𝑗 ∈ ℕ0 → (1↑𝑗) = 1)
328324, 327sylan9eq 2825 . . . . . . . . . . 11 ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (𝑥𝑗) = 1)
329328oveq2d 6807 . . . . . . . . . 10 ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗)) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · 1))
33018trud 1641 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ0⟶ℂ
331330ffvelrni 6499 . . . . . . . . . . . 12 (𝑗 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
332331mulid1d 10257 . . . . . . . . . . 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))
333332adantl 467 . . . . . . . . . 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))
334329, 333eqtrd 2805 . . . . . . . . 9 ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗)) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
335334sumeq2dv 14634 . . . . . . . 8 (𝑥 = 1 → Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗)) = Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
336 sumex 14619 . . . . . . . 8 Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ V
337335, 148, 336fvmpt 6422 . . . . . . 7 (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗)))‘1) = Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
338190, 337mp1i 13 . . . . . 6 (⊤ → ((𝑥 ∈ (0[,]1) ↦ Σ𝑗 ∈ ℕ0 (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · (𝑥𝑗)))‘1) = Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
339192, 323, 3383brtr3d 4817 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
340 eqid 2771 . . . . . . . . 9 (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
341 eqid 2771 . . . . . . . . 9 {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} = {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
342340, 341atancn 24877 . . . . . . . 8 (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}–cn→ℂ)
343342a1i 11 . . . . . . 7 (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}–cn→ℂ))
344 unitssre 12519 . . . . . . . . 9 (0[,]1) ⊆ ℝ
345340, 341ressatans 24875 . . . . . . . . 9 ℝ ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
346344, 345sstri 3761 . . . . . . . 8 (0[,]1) ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
347 fss 6194 . . . . . . . 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))})
348175, 346, 347sylancl 574 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
349345, 169sselii 3749 . . . . . . . 8 1 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
350349a1i 11 . . . . . . 7 (⊤ → 1 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
35175, 76, 343, 348, 189, 350climcncf 22916 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) ⇝ ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1))
352346, 173sseldi 3750 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
353 cncff 22909 . . . . . . . . . 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)
354342, 353mp1i 13 . . . . . . . . 9 (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}):{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}⟶ℂ)
355354feqmptd 6389 . . . . . . . 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)))
356 fvres 6346 . . . . . . . . 9 (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘𝑘) = (arctan‘𝑘))
357356mpteq2ia 4874 . . . . . . . 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))
358355, 357syl6eq 2821 . . . . . . 7 (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ (arctan‘𝑘)))
359 fveq2 6330 . . . . . . 7 (𝑘 = (1 − (1 / 𝑛)) → (arctan‘𝑘) = (arctan‘(1 − (1 / 𝑛))))
360352, 193, 358, 359fmptco 6537 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))))
361 fvres 6346 . . . . . . . 8 (1 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (arctan‘1))
362349, 361mp1i 13 . . . . . . 7 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (arctan‘1))
363 atan1 24869 . . . . . . 7 (arctan‘1) = (π / 4)
364362, 363syl6eq 2821 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (π / 4))
365351, 360, 3643brtr3d 4817 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ (π / 4))
366 climuni 14484 . . . . 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))
367339, 365, 366syl2anc 573 . . . 4 (⊤ → Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4))
368147, 367breqtrd 4812 . . 3 (⊤ → seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
369368trud 1641 . 2 seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4)
370 leibpi.1 . . 3 𝐹 = (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
371 ovex 6821 . . 3 (π / 4) ∈ V
372370, 17, 371leibpilem2 24882 . 2 (seq0( + , 𝐹) ⇝ (π / 4) ↔ seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
373369, 372mpbir 221 1 seq0( + , 𝐹) ⇝ (π / 4)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wo 836   = wceq 1631  wtru 1632  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  cdif 3720  wss 3723  ifcif 4225   class class class wbr 4786  cmpt 4863  dom cdm 5249  cres 5251  ccom 5253  wf 6025  cfv 6029  (class class class)co 6791  cc 10134  cr 10135  0cc0 10136  1c1 10137   + caddc 10139   · cmul 10141  -∞cmnf 10272   < clt 10274  cle 10275  cmin 10466  -cneg 10467   / cdiv 10884  cn 11220  2c2 11270  4c4 11272  0cn0 11492  cz 11577  cuz 11886  +crp 12028  (,]cioc 12374  [,]cicc 12376  seqcseq 13001  cexp 13060  abscabs 14175  cli 14416  Σcsu 14617  πcpi 14996  cdvds 15182  cnccncf 22892  arctancatan 24805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-inf2 8700  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213  ax-pre-sup 10214  ax-addf 10215  ax-mulf 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-isom 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-of 7042  df-om 7211  df-1st 7313  df-2nd 7314  df-supp 7445  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-2o 7712  df-oadd 7715  df-er 7894  df-map 8009  df-pm 8010  df-ixp 8061  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-fsupp 8430  df-fi 8471  df-sup 8502  df-inf 8503  df-oi 8569  df-card 8963  df-cda 9190  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-nn 11221  df-2 11279  df-3 11280  df-4 11281  df-5 11282  df-6 11283  df-7 11284  df-8 11285  df-9 11286  df-n0 11493  df-xnn0 11564  df-z 11578  df-dec 11694  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12144  df-xadd 12145  df-xmul 12146  df-ioo 12377  df-ioc 12378  df-ico 12379  df-icc 12380  df-fz 12527  df-fzo 12667  df-fl 12794  df-mod 12870  df-seq 13002  df-exp 13061  df-fac 13258  df-bc 13287  df-hash 13315  df-shft 14008  df-cj 14040  df-re 14041  df-im 14042  df-sqrt 14176  df-abs 14177  df-limsup 14403  df-clim 14420  df-rlim 14421  df-sum 14618  df-ef 14997  df-sin 14999  df-cos 15000  df-tan 15001  df-pi 15002  df-dvds 15183  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16155  df-mulr 16156  df-starv 16157  df-sca 16158  df-vsca 16159  df-ip 16160  df-tset 16161  df-ple 16162  df-ds 16165  df-unif 16166  df-hom 16167  df-cco 16168  df-rest 16284  df-topn 16285  df-0g 16303  df-gsum 16304  df-topgen 16305  df-pt 16306  df-prds 16309  df-xrs 16363  df-qtop 16368  df-imas 16369  df-xps 16371  df-mre 16447  df-mrc 16448  df-acs 16450  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-cntz 17950  df-cmn 18395  df-psmet 19946  df-xmet 19947  df-met 19948  df-bl 19949  df-mopn 19950  df-fbas 19951  df-fg 19952  df-cnfld 19955  df-top 20912  df-topon 20929  df-topsp 20951  df-bases 20964  df-cld 21037  df-ntr 21038  df-cls 21039  df-nei 21116  df-lp 21154  df-perf 21155  df-cn 21245  df-cnp 21246  df-t1 21332  df-haus 21333  df-cmp 21404  df-tx 21579  df-hmeo 21772  df-fil 21863  df-fm 21955  df-flim 21956  df-flf 21957  df-xms 22338  df-ms 22339  df-tms 22340  df-cncf 22894  df-limc 23843  df-dv 23844  df-ulm 24344  df-log 24517  df-atan 24808
This theorem is referenced by:  leibpisum  24884
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