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Theorem leibpi 24469
 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 14263). (2) Using leibpilem2 24468 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 24465). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 24465, Abel's theorem (abelth2 24000) 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 24463) 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 11598 . . . . 5 0 = (ℤ‘0)
2 0zd 11266 . . . . 5 (⊤ → 0 ∈ ℤ)
3 eqidd 2611 . . . . 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 9912 . . . . . . . . 9 ((𝑘 ∈ ℕ0 ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
5 ioran 510 . . . . . . . . . 10 (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
6 neg1rr 11002 . . . . . . . . . . . . 13 -1 ∈ ℝ
7 leibpilem1 24467 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ ((𝑘 − 1) / 2) ∈ ℕ0))
87simprd 478 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ0)
9 reexpcl 12739 . . . . . . . . . . . . 13 ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ0) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
106, 8, 9sylancr 694 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
117simpld 474 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
1210, 11nndivred 10946 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
1312recnd 9947 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
145, 13sylan2b 491 . . . . . . . . 9 ((𝑘 ∈ ℕ0 ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
154, 14ifclda 4070 . . . . . . . 8 (𝑘 ∈ ℕ0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
1615adantl 481 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
17 eqid 2610 . . . . . . 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 6292 . . . . . 6 (⊤ → (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ0⟶ℂ)
1918ffvelrnda 6267 . . . . 5 ((⊤ ∧ 𝑗 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
20 2nn0 11186 . . . . . . . . . . . . . 14 2 ∈ ℕ0
2120a1i 11 . . . . . . . . . . . . 13 (⊤ → 2 ∈ ℕ0)
22 nn0mulcl 11206 . . . . . . . . . . . . 13 ((2 ∈ ℕ0𝑛 ∈ ℕ0) → (2 · 𝑛) ∈ ℕ0)
2321, 22sylan 487 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ0) → (2 · 𝑛) ∈ ℕ0)
24 nn0p1nn 11209 . . . . . . . . . . . 12 ((2 · 𝑛) ∈ ℕ0 → ((2 · 𝑛) + 1) ∈ ℕ)
2523, 24syl 17 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ0) → ((2 · 𝑛) + 1) ∈ ℕ)
2625nnrecred 10943 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ0) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ)
27 eqid 2610 . . . . . . . . . 10 (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))
2826, 27fmptd 6292 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))):ℕ0⟶ℝ)
29 nn0mulcl 11206 . . . . . . . . . . . . . 14 ((2 ∈ ℕ0𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
3021, 29sylan 487 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
3130nn0red 11229 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℝ)
32 peano2nn0 11210 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
3332adantl 481 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
34 nn0mulcl 11206 . . . . . . . . . . . . . 14 ((2 ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℕ0)
3520, 33, 34sylancr 694 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℕ0)
3635nn0red 11229 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℝ)
37 1red 9934 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 1 ∈ ℝ)
38 nn0re 11178 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
3938adantl 481 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
4039lep1d 10834 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → 𝑘 ≤ (𝑘 + 1))
41 peano2re 10088 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
4239, 41syl 17 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℝ)
43 2re 10967 . . . . . . . . . . . . . . 15 2 ∈ ℝ
4443a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 2 ∈ ℝ)
45 2pos 10989 . . . . . . . . . . . . . . 15 0 < 2
4645a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < 2)
47 lemul2 10755 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
4839, 42, 44, 46, 47syl112anc 1322 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
4940, 48mpbid 221 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ≤ (2 · (𝑘 + 1)))
5031, 36, 37, 49leadd1dd 10520 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1))
51 nn0p1nn 11209 . . . . . . . . . . . . . 14 ((2 · 𝑘) ∈ ℕ0 → ((2 · 𝑘) + 1) ∈ ℕ)
5230, 51syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℕ)
5352nnred 10912 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℝ)
5452nngt0d 10941 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < ((2 · 𝑘) + 1))
55 nn0p1nn 11209 . . . . . . . . . . . . . 14 ((2 · (𝑘 + 1)) ∈ ℕ0 → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
5635, 55syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
5756nnred 10912 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℝ)
5856nngt0d 10941 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < ((2 · (𝑘 + 1)) + 1))
59 lerec 10785 . . . . . . . . . . . 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 1319 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1))))
6150, 60mpbid 221 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1)))
62 oveq2 6557 . . . . . . . . . . . . . 14 (𝑛 = (𝑘 + 1) → (2 · 𝑛) = (2 · (𝑘 + 1)))
6362oveq1d 6564 . . . . . . . . . . . . 13 (𝑛 = (𝑘 + 1) → ((2 · 𝑛) + 1) = ((2 · (𝑘 + 1)) + 1))
6463oveq2d 6565 . . . . . . . . . . . 12 (𝑛 = (𝑘 + 1) → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
65 ovex 6577 . . . . . . . . . . . 12 (1 / ((2 · (𝑘 + 1)) + 1)) ∈ V
6664, 27, 65fvmpt 6191 . . . . . . . . . . 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 6557 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
6968oveq1d 6564 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((2 · 𝑛) + 1) = ((2 · 𝑘) + 1))
7069oveq2d 6565 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · 𝑘) + 1)))
71 ovex 6577 . . . . . . . . . . . 12 (1 / ((2 · 𝑘) + 1)) ∈ V
7270, 27, 71fvmpt 6191 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
7372adantl 481 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
7461, 67, 733brtr4d 4615 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) ≤ ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
75 nnuz 11599 . . . . . . . . . 10 ℕ = (ℤ‘1)
76 1zzd 11285 . . . . . . . . . 10 (⊤ → 1 ∈ ℤ)
77 ax-1cn 9873 . . . . . . . . . . 11 1 ∈ ℂ
78 divcnv 14424 . . . . . . . . . . 11 (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
7977, 78mp1i 13 . . . . . . . . . 10 (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
80 nn0ex 11175 . . . . . . . . . . . 12 0 ∈ V
8180mptex 6390 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ∈ V
8281a1i 11 . . . . . . . . . 10 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ∈ V)
83 oveq2 6557 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
84 eqid 2610 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
85 ovex 6577 . . . . . . . . . . . . 13 (1 / 𝑘) ∈ V
8683, 84, 85fvmpt 6191 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
8786adantl 481 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
88 nnrecre 10934 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
8988adantl 481 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
9087, 89eqeltrd 2688 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℝ)
91 nnnn0 11176 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
9291adantl 481 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
9392, 72syl 17 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
9491, 52sylan2 490 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℕ)
9594nnrecred 10943 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ)
9693, 95eqeltrd 2688 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ∈ ℝ)
97 nnre 10904 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
9897adantl 481 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
9920, 92, 29sylancr 694 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℕ0)
10099nn0red 11229 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℝ)
101 peano2re 10088 . . . . . . . . . . . . . 14 ((2 · 𝑘) ∈ ℝ → ((2 · 𝑘) + 1) ∈ ℝ)
102100, 101syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
103 nn0addge1 11216 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → 𝑘 ≤ (𝑘 + 𝑘))
10498, 92, 103syl2anc 691 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (𝑘 + 𝑘))
10598recnd 9947 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
1061052timesd 11152 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) = (𝑘 + 𝑘))
107104, 106breqtrrd 4611 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (2 · 𝑘))
108100lep1d 10834 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ≤ ((2 · 𝑘) + 1))
10998, 100, 102, 107, 108letrd 10073 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ ((2 · 𝑘) + 1))
110 nngt0 10926 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 0 < 𝑘)
111110adantl 481 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
11294nnred 10912 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
11394nngt0d 10941 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < ((2 · 𝑘) + 1))
114 lerec 10785 . . . . . . . . . . . . 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 1319 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)))
116109, 115mpbid 221 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘))
117116, 93, 873brtr4d 4615 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))
11894nnrpd 11746 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ+)
119118rpreccld 11758 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ+)
120119rpge0d 11752 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / ((2 · 𝑘) + 1)))
121120, 93breqtrrd 4611 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
12275, 76, 79, 82, 90, 96, 117, 121climsqz2 14220 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ⇝ 0)
123 neg1cn 11001 . . . . . . . . . . . . 13 -1 ∈ ℂ
124123a1i 11 . . . . . . . . . . . 12 (⊤ → -1 ∈ ℂ)
125 expcl 12740 . . . . . . . . . . . 12 ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℂ)
126124, 125sylan 487 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℂ)
12752nncnd 10913 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℂ)
12852nnne0d 10942 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ≠ 0)
129126, 127, 128divrecd 10683 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) / ((2 · 𝑘) + 1)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
130 oveq2 6557 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (-1↑𝑛) = (-1↑𝑘))
131130, 69oveq12d 6567 . . . . . . . . . . . 12 (𝑛 = 𝑘 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
132 eqid 2610 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
133 ovex 6577 . . . . . . . . . . . 12 ((-1↑𝑘) / ((2 · 𝑘) + 1)) ∈ V
134131, 132, 133fvmpt 6191 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
135134adantl 481 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
13673oveq2d 6565 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) · ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
137129, 135, 1363eqtr4d 2654 . . . . . . . . 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 14263 . . . . . . . 8 (⊤ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝ )
139 climdm 14133 . . . . . . . 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 207 . . . . . . 7 (⊤ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
141 fvex 6113 . . . . . . . 8 ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ∈ V
142132, 17, 141leibpilem2 24468 . . . . . . 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 207 . . . . . 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 12665 . . . . . . 7 seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ V
145144, 141breldm 5251 . . . . . 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 14331 . . . 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 2610 . . . . . . . 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 24000 . . . . . . 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 11718 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
151150adantl 481 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+)
152151rpreccld 11758 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ+)
153152rpred 11748 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
154152rpge0d 11752 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ≤ (1 / 𝑛))
155 nnge1 10923 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 1 ≤ 𝑛)
156155adantl 481 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
157 nnre 10904 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
158157adantl 481 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
159158recnd 9947 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
160159mulid1d 9936 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑛 · 1) = 𝑛)
161156, 160breqtrrd 4611 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ (𝑛 · 1))
162 1red 9934 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
163 nngt0 10926 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 0 < 𝑛)
164163adantl 481 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
165 ledivmul 10778 . . . . . . . . . . . 12 ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
166162, 162, 158, 164, 165syl112anc 1322 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
167161, 166mpbird 246 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ≤ 1)
168 0re 9919 . . . . . . . . . . 11 0 ∈ ℝ
169 1re 9918 . . . . . . . . . . 11 1 ∈ ℝ
170168, 169elicc2i 12110 . . . . . . . . . 10 ((1 / 𝑛) ∈ (0[,]1) ↔ ((1 / 𝑛) ∈ ℝ ∧ 0 ≤ (1 / 𝑛) ∧ (1 / 𝑛) ≤ 1))
171153, 154, 167, 170syl3anbrc 1239 . . . . . . . . 9 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ (0[,]1))
172 iirev 22536 . . . . . . . . 9 ((1 / 𝑛) ∈ (0[,]1) → (1 − (1 / 𝑛)) ∈ (0[,]1))
173171, 172syl 17 . . . . . . . 8 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ (0[,]1))
174 eqid 2610 . . . . . . . 8 (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))
175173, 174fmptd 6292 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶(0[,]1))
176 1cnd 9935 . . . . . . . . 9 (⊤ → 1 ∈ ℂ)
177 nnex 10903 . . . . . . . . . . 11 ℕ ∈ V
178177mptex 6390 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V
179178a1i 11 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V)
18090recnd 9947 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℂ)
18183oveq2d 6565 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (1 − (1 / 𝑛)) = (1 − (1 / 𝑘)))
182 ovex 6577 . . . . . . . . . . . 12 (1 − (1 / 𝑘)) ∈ V
183181, 174, 182fvmpt 6191 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − (1 / 𝑘)))
18486oveq2d 6565 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)) = (1 − (1 / 𝑘)))
185183, 184eqtr4d 2647 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
186185adantl 481 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
18775, 76, 79, 176, 179, 180, 186climsubc2 14217 . . . . . . . 8 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ (1 − 0))
188 1m0e1 11008 . . . . . . . 8 (1 − 0) = 1
189187, 188syl6breq 4624 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ 1)
190 1elunit 12162 . . . . . . . 8 1 ∈ (0[,]1)
191190a1i 11 . . . . . . 7 (⊤ → 1 ∈ (0[,]1))
19275, 76, 149, 175, 189, 191climcncf 22511 . . . . . 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 2611 . . . . . . . 8 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))))
194 eqidd 2611 . . . . . . . 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 6556 . . . . . . . . . 10 (𝑥 = (1 − (1 / 𝑛)) → (𝑥𝑗) = ((1 − (1 / 𝑛))↑𝑗))
196195oveq2d 6565 . . . . . . . . 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 14282 . . . . . . . 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 6303 . . . . . . 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 11266 . . . . . . . . 9 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℤ)
2008adantll 746 . . . . . . . . . . . . . . . . . . . . . 22 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ0)
2016, 200, 9sylancr 694 . . . . . . . . . . . . . . . . . . . . 21 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
202201recnd 9947 . . . . . . . . . . . . . . . . . . . 20 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
203202adantllr 751 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
204 resubcl 10224 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ) → (1 − (1 / 𝑛)) ∈ ℝ)
205169, 153, 204sylancr 694 . . . . . . . . . . . . . . . . . . . . . 22 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℝ)
206205ad2antrr 758 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (1 − (1 / 𝑛)) ∈ ℝ)
207 simplr 788 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ0)
208206, 207reexpcld 12887 . . . . . . . . . . . . . . . . . . . 20 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℝ)
209208recnd 9947 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
210 nn0cn 11179 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
211210ad2antlr 759 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℂ)
21211adantll 746 . . . . . . . . . . . . . . . . . . . 20 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
213212nnne0d 10942 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ≠ 0)
214203, 209, 211, 213div12d 10716 . . . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
216209, 215mulcomd 9940 . . . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . . . 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 4067 . . . . . . . . . . . . . . 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 9947 . . . . . . . . . . . . . . . . . 18 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℂ)
221 expcl 12740 . . . . . . . . . . . . . . . . . 18 (((1 − (1 / 𝑛)) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
222220, 221sylan 487 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
223222mul02d 10113 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (0 · ((1 − (1 / 𝑛))↑𝑘)) = 0)
224223ifeq1d 4054 . . . . . . . . . . . . . . 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 2647 . . . . . . . . . . . . . 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 6635 . . . . . . . . . . . . . 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 2662 . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
229 c0ex 9913 . . . . . . . . . . . . . . 15 0 ∈ V
230 ovex 6577 . . . . . . . . . . . . . . 15 ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ V
231229, 230ifex 4106 . . . . . . . . . . . . . 14 if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
232 eqid 2610 . . . . . . . . . . . . . . 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 6200 . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . 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 6577 . . . . . . . . . . . . . . . 16 ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ V
236229, 235ifex 4106 . . . . . . . . . . . . . . 15 if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ V
23717fvmpt2 6200 . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . 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 6564 . . . . . . . . . . . . 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 2654 . . . . . . . . . . . 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 2949 . . . . . . . . . . 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 1830 . . . . . . . . . . . 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 6111 . . . . . . . . . . . . 13 𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
244 nffvmpt1 6111 . . . . . . . . . . . . . 14 𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)
245 nfcv 2751 . . . . . . . . . . . . . 14 𝑘 ·
246 nfcv 2751 . . . . . . . . . . . . . 14 𝑘((1 − (1 / 𝑛))↑𝑗)
247244, 245, 246nfov 6575 . . . . . . . . . . . . 13 𝑘(((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))
248243, 247nfeq 2762 . . . . . . . . . . . 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 6103 . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ((1 − (1 / 𝑛))↑𝑘) = ((1 − (1 / 𝑛))↑𝑗))
252250, 251oveq12d 6567 . . . . . . . . . . . . 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 2625 . . . . . . . . . . . 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 3143 . . . . . . . . . . 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 207 . . . . . . . . . 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 2916 . . . . . . . . 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 9912 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
258208, 212nndivred 10946 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℝ)
259258recnd 9947 . . . . . . . . . . . . . . 15 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℂ)
260203, 259mulcld 9939 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
2615, 260sylan2b 491 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
262257, 261ifclda 4070 . . . . . . . . . . . 12 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ ℂ)
263262, 232fmptd 6292 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))):ℕ0⟶ℂ)
264263ffvelrnda 6267 . . . . . . . . . 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 2689 . . . . . . . . 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 11184 . . . . . . . . . . . 12 0 ∈ ℕ0
267266a1i 11 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℕ0)
268 0p1e1 11009 . . . . . . . . . . . . 13 (0 + 1) = 1
269 seqeq1 12666 . . . . . . . . . . . . 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 11285 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
272 elnnuz 11600 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ‘1))
273 nnne0 10930 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
274273neneqd 2787 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ ℕ → ¬ 𝑘 = 0)
275 biorf 419 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘 = 0 → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
276274, 275syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
277276bicomd 212 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ 2 ∥ 𝑘))
278277ifbid 4058 . . . . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . . . . . 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 4106 . . . . . . . . . . . . . . . . . . . . 21 if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
281 eqid 2610 . . . . . . . . . . . . . . . . . . . . . 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 6200 . . . . . . . . . . . . . . . . . . . . 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 703 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
284278, 279, 2833eqtr4d 2654 . . . . . . . . . . . . . . . . . . 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 2906 . . . . . . . . . . . . . . . . . 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 1830 . . . . . . . . . . . . . . . . . 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 6111 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
289243, 288nfeq 2762 . . . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . 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 2625 . . . . . . . . . . . . . . . . . 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 3143 . . . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . . . . 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 2916 . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . 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 12688 . . . . . . . . . . . . 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 14018 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) = (1 − (1 / 𝑛)))
298 ltsubrp 11742 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ+) → (1 − (1 / 𝑛)) < 1)
299169, 152, 298sylancr 694 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) < 1)
300297, 299eqbrtrd 4605 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) < 1)
301281atantayl2 24465 . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . 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 4605 . . . . . . . . . . . 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 4618 . . . . . . . . . . 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 14234 . . . . . . . . . 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 11265 . . . . . . . . . . . . . 14 0 ∈ ℤ
307 seq1 12676 . . . . . . . . . . . . . 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 4042 . . . . . . . . . . . . . . . 16 ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
310309orcs 408 . . . . . . . . . . . . . . 15 (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
311310, 232, 229fvmpt 6191 . . . . . . . . . . . . . 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 2632 . . . . . . . . . . . 12 (seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = 0
314313oveq2i 6560 . . . . . . . . . . 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 24438 . . . . . . . . . . . . . 14 ((1 − (1 / 𝑛)) ∈ ℝ → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
316205, 315syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
317316recnd 9947 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℂ)
318317addid1d 10115 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + 0) = (arctan‘(1 − (1 / 𝑛))))
319314, 318syl5eq 2656 . . . . . . . . . 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 4609 . . . . . . . . 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 14330 . . . . . . . 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 4672 . . . . . . 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 2644 . . . . . 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 6556 . . . . . . . . . . . 12 (𝑥 = 1 → (𝑥𝑗) = (1↑𝑗))
325 nn0z 11277 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ0𝑗 ∈ ℤ)
326 1exp 12751 . . . . . . . . . . . . 13 (𝑗 ∈ ℤ → (1↑𝑗) = 1)
327325, 326syl 17 . . . . . . . . . . . 12 (𝑗 ∈ ℕ0 → (1↑𝑗) = 1)
328324, 327sylan9eq 2664 . . . . . . . . . . 11 ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (𝑥𝑗) = 1)
329328oveq2d 6565 . . . . . . . . . 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 1484 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ0⟶ℂ
331330ffvelrni 6266 . . . . . . . . . . . 12 (𝑗 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
332331mulid1d 9936 . . . . . . . . . . 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 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))
334329, 333eqtrd 2644 . . . . . . . . 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 14281 . . . . . . . 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 14266 . . . . . . . 8 Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ V
337335, 148, 336fvmpt 6191 . . . . . . 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 4614 . . . . 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 2610 . . . . . . . . 9 (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
341 eqid 2610 . . . . . . . . 9 {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} = {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
342340, 341atancn 24463 . . . . . . . 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 12190 . . . . . . . . 9 (0[,]1) ⊆ ℝ
345340, 341ressatans 24461 . . . . . . . . 9 ℝ ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
346344, 345sstri 3577 . . . . . . . 8 (0[,]1) ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
347 fss 5969 . . . . . . . 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 693 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
349345, 169sselii 3565 . . . . . . . 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 22511 . . . . . 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 3566 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
353 cncff 22504 . . . . . . . . . 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 6159 . . . . . . . 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 6117 . . . . . . . . 9 (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘𝑘) = (arctan‘𝑘))
357356mpteq2ia 4668 . . . . . . . 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 2660 . . . . . . 7 (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ (arctan‘𝑘)))
359 fveq2 6103 . . . . . . 7 (𝑘 = (1 − (1 / 𝑛)) → (arctan‘𝑘) = (arctan‘(1 − (1 / 𝑛))))
360352, 193, 358, 359fmptco 6303 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))))
361 fvres 6117 . . . . . . . 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 24455 . . . . . . 7 (arctan‘1) = (π / 4)
364362, 363syl6eq 2660 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (π / 4))
365351, 360, 3643brtr3d 4614 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ (π / 4))
366 climuni 14131 . . . . 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 691 . . . 4 (⊤ → Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4))
368147, 367breqtrd 4609 . . 3 (⊤ → seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
369368trud 1484 . 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 6577 . . 3 (π / 4) ∈ V
372370, 17, 371leibpilem2 24468 . 2 (seq0( + , 𝐹) ⇝ (π / 4) ↔ seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
373369, 372mpbir 220 1 seq0( + , 𝐹) ⇝ (π / 4)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  -∞cmnf 9951   < clt 9953   ≤ cle 9954   − cmin 10145  -cneg 10146   / cdiv 10563  ℕcn 10897  2c2 10947  4c4 10949  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ℝ+crp 11708  (,]cioc 12047  [,]cicc 12049  seqcseq 12663  ↑cexp 12722  abscabs 13822   ⇝ cli 14063  Σcsu 14264  πcpi 14636   ∥ cdvds 14821  –cn→ccncf 22487  arctancatan 24391 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-tan 14641  df-pi 14642  df-dvds 14822  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-t1 20928  df-haus 20929  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-ulm 23935  df-log 24107  df-atan 24394 This theorem is referenced by:  leibpisum  24470
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