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Theorem leibpi 25526
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 15032). (2) Using leibpilem2 25525 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 25522). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 25522, Abel's theorem (abelth2 25035) 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 25520) 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 12268 . . . . 5 0 = (ℤ‘0)
2 0zd 11981 . . . . 5 (⊤ → 0 ∈ ℤ)
3 eqidd 2823 . . . . 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 10623 . . . . . . . . 9 ((𝑘 ∈ ℕ0 ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
5 ioran 981 . . . . . . . . . 10 (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
6 neg1rr 11740 . . . . . . . . . . . . 13 -1 ∈ ℝ
7 leibpilem1 25524 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ ((𝑘 − 1) / 2) ∈ ℕ0))
87simprd 499 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ0)
9 reexpcl 13442 . . . . . . . . . . . . 13 ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ0) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
106, 8, 9sylancr 590 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
117simpld 498 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
1210, 11nndivred 11679 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
1312recnd 10658 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
145, 13sylan2b 596 . . . . . . . . 9 ((𝑘 ∈ ℕ0 ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
154, 14ifclda 4473 . . . . . . . 8 (𝑘 ∈ ℕ0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
1615adantl 485 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ ℂ)
1716fmpttd 6861 . . . . . 6 (⊤ → (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ0⟶ℂ)
1817ffvelrnda 6833 . . . . 5 ((⊤ ∧ 𝑗 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
19 2nn0 11902 . . . . . . . . . . . . . 14 2 ∈ ℕ0
2019a1i 11 . . . . . . . . . . . . 13 (⊤ → 2 ∈ ℕ0)
21 nn0mulcl 11921 . . . . . . . . . . . . 13 ((2 ∈ ℕ0𝑛 ∈ ℕ0) → (2 · 𝑛) ∈ ℕ0)
2220, 21sylan 583 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ0) → (2 · 𝑛) ∈ ℕ0)
23 nn0p1nn 11924 . . . . . . . . . . . 12 ((2 · 𝑛) ∈ ℕ0 → ((2 · 𝑛) + 1) ∈ ℕ)
2422, 23syl 17 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ0) → ((2 · 𝑛) + 1) ∈ ℕ)
2524nnrecred 11676 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ0) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ)
2625fmpttd 6861 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))):ℕ0⟶ℝ)
27 nn0mulcl 11921 . . . . . . . . . . . . . 14 ((2 ∈ ℕ0𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
2820, 27sylan 583 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
2928nn0red 11944 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℝ)
30 peano2nn0 11925 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
3130adantl 485 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
32 nn0mulcl 11921 . . . . . . . . . . . . . 14 ((2 ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℕ0)
3319, 31, 32sylancr 590 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℕ0)
3433nn0red 11944 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℝ)
35 1red 10631 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 1 ∈ ℝ)
36 nn0re 11894 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
3736adantl 485 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
3837lep1d 11560 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → 𝑘 ≤ (𝑘 + 1))
39 peano2re 10802 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
4037, 39syl 17 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℝ)
41 2re 11699 . . . . . . . . . . . . . . 15 2 ∈ ℝ
4241a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 2 ∈ ℝ)
43 2pos 11728 . . . . . . . . . . . . . . 15 0 < 2
4443a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < 2)
45 lemul2 11482 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
4637, 40, 42, 44, 45syl112anc 1371 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
4738, 46mpbid 235 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ≤ (2 · (𝑘 + 1)))
4829, 34, 35, 47leadd1dd 11243 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1))
49 nn0p1nn 11924 . . . . . . . . . . . . . 14 ((2 · 𝑘) ∈ ℕ0 → ((2 · 𝑘) + 1) ∈ ℕ)
5028, 49syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℕ)
5150nnred 11640 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℝ)
5250nngt0d 11674 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < ((2 · 𝑘) + 1))
53 nn0p1nn 11924 . . . . . . . . . . . . . 14 ((2 · (𝑘 + 1)) ∈ ℕ0 → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
5433, 53syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
5554nnred 11640 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℝ)
5654nngt0d 11674 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < ((2 · (𝑘 + 1)) + 1))
57 lerec 11512 . . . . . . . . . . . 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))))
5851, 52, 55, 56, 57syl22anc 837 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1))))
5948, 58mpbid 235 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1)))
60 oveq2 7148 . . . . . . . . . . . . . 14 (𝑛 = (𝑘 + 1) → (2 · 𝑛) = (2 · (𝑘 + 1)))
6160oveq1d 7155 . . . . . . . . . . . . 13 (𝑛 = (𝑘 + 1) → ((2 · 𝑛) + 1) = ((2 · (𝑘 + 1)) + 1))
6261oveq2d 7156 . . . . . . . . . . . 12 (𝑛 = (𝑘 + 1) → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
63 eqid 2822 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))
64 ovex 7173 . . . . . . . . . . . 12 (1 / ((2 · (𝑘 + 1)) + 1)) ∈ V
6562, 63, 64fvmpt 6750 . . . . . . . . . . 11 ((𝑘 + 1) ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
6631, 65syl 17 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
67 oveq2 7148 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
6867oveq1d 7155 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((2 · 𝑛) + 1) = ((2 · 𝑘) + 1))
6968oveq2d 7156 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · 𝑘) + 1)))
70 ovex 7173 . . . . . . . . . . . 12 (1 / ((2 · 𝑘) + 1)) ∈ V
7169, 63, 70fvmpt 6750 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
7271adantl 485 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
7359, 66, 723brtr4d 5074 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) ≤ ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
74 nnuz 12269 . . . . . . . . . 10 ℕ = (ℤ‘1)
75 1zzd 12001 . . . . . . . . . 10 (⊤ → 1 ∈ ℤ)
76 ax-1cn 10584 . . . . . . . . . . 11 1 ∈ ℂ
77 divcnv 15199 . . . . . . . . . . 11 (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
7876, 77mp1i 13 . . . . . . . . . 10 (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
79 nn0ex 11891 . . . . . . . . . . . 12 0 ∈ V
8079mptex 6968 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ∈ V
8180a1i 11 . . . . . . . . . 10 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ∈ V)
82 oveq2 7148 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
83 eqid 2822 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
84 ovex 7173 . . . . . . . . . . . . 13 (1 / 𝑘) ∈ V
8582, 83, 84fvmpt 6750 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
8685adantl 485 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
87 nnrecre 11667 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
8887adantl 485 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
8986, 88eqeltrd 2914 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℝ)
90 nnnn0 11892 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
9190adantl 485 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
9291, 71syl 17 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
9390, 50sylan2 595 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℕ)
9493nnrecred 11676 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ)
9592, 94eqeltrd 2914 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ∈ ℝ)
96 nnre 11632 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
9796adantl 485 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
9819, 91, 27sylancr 590 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℕ0)
9998nn0red 11944 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℝ)
100 peano2re 10802 . . . . . . . . . . . . . 14 ((2 · 𝑘) ∈ ℝ → ((2 · 𝑘) + 1) ∈ ℝ)
10199, 100syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
102 nn0addge1 11931 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → 𝑘 ≤ (𝑘 + 𝑘))
10397, 91, 102syl2anc 587 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (𝑘 + 𝑘))
10497recnd 10658 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
1051042timesd 11868 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) = (𝑘 + 𝑘))
106103, 105breqtrrd 5070 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (2 · 𝑘))
10799lep1d 11560 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ≤ ((2 · 𝑘) + 1))
10897, 99, 101, 106, 107letrd 10786 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ ((2 · 𝑘) + 1))
109 nngt0 11656 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 0 < 𝑘)
110109adantl 485 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
11193nnred 11640 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
11293nngt0d 11674 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < ((2 · 𝑘) + 1))
113 lerec 11512 . . . . . . . . . . . . 13 (((𝑘 ∈ ℝ ∧ 0 < 𝑘) ∧ (((2 · 𝑘) + 1) ∈ ℝ ∧ 0 < ((2 · 𝑘) + 1))) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)))
11497, 110, 111, 112, 113syl22anc 837 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)))
115108, 114mpbid 235 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘))
116115, 92, 863brtr4d 5074 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))
11793nnrpd 12417 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ+)
118117rpreccld 12429 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ+)
119118rpge0d 12423 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / ((2 · 𝑘) + 1)))
120119, 92breqtrrd 5070 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
12174, 75, 78, 81, 89, 95, 116, 120climsqz2 14989 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ⇝ 0)
122 neg1cn 11739 . . . . . . . . . . . . 13 -1 ∈ ℂ
123122a1i 11 . . . . . . . . . . . 12 (⊤ → -1 ∈ ℂ)
124 expcl 13443 . . . . . . . . . . . 12 ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℂ)
125123, 124sylan 583 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℂ)
12650nncnd 11641 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℂ)
12750nnne0d 11675 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ≠ 0)
128125, 126, 127divrecd 11408 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) / ((2 · 𝑘) + 1)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
129 oveq2 7148 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (-1↑𝑛) = (-1↑𝑘))
130129, 68oveq12d 7158 . . . . . . . . . . . 12 (𝑛 = 𝑘 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
131 eqid 2822 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
132 ovex 7173 . . . . . . . . . . . 12 ((-1↑𝑘) / ((2 · 𝑘) + 1)) ∈ V
133130, 131, 132fvmpt 6750 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
134133adantl 485 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
13572oveq2d 7156 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) · ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
136128, 134, 1353eqtr4d 2867 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) · ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)))
1371, 2, 26, 73, 121, 136iseralt 15032 . . . . . . . 8 (⊤ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝ )
138 climdm 14902 . . . . . . . 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))))))
139137, 138sylib 221 . . . . . . 7 (⊤ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
140 eqid 2822 . . . . . . . 8 (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))) = (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
141 fvex 6665 . . . . . . . 8 ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ∈ V
142131, 140, 141leibpilem2 25525 . . . . . . 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))))))
143139, 142sylib 221 . . . . . 6 (⊤ → seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
144 seqex 13366 . . . . . . 7 seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ V
145144, 141breldm 5754 . . . . . 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, 18, 146isumclim2 15104 . . . 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 2822 . . . . . . . 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)))
14917, 146, 148abelth2 25035 . . . . . . 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 12388 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
151150adantl 485 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+)
152151rpreccld 12429 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ+)
153152rpred 12419 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
154152rpge0d 12423 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ≤ (1 / 𝑛))
155 nnge1 11653 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 1 ≤ 𝑛)
156155adantl 485 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
157 nnre 11632 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
158157adantl 485 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
159158recnd 10658 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
160159mulid1d 10647 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑛 · 1) = 𝑛)
161156, 160breqtrrd 5070 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ (𝑛 · 1))
162 1red 10631 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
163 nngt0 11656 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 0 < 𝑛)
164163adantl 485 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
165 ledivmul 11505 . . . . . . . . . . . 12 ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
166162, 162, 158, 164, 165syl112anc 1371 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
167161, 166mpbird 260 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ≤ 1)
168 elicc01 12844 . . . . . . . . . 10 ((1 / 𝑛) ∈ (0[,]1) ↔ ((1 / 𝑛) ∈ ℝ ∧ 0 ≤ (1 / 𝑛) ∧ (1 / 𝑛) ≤ 1))
169153, 154, 167, 168syl3anbrc 1340 . . . . . . . . 9 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ (0[,]1))
170 iirev 23532 . . . . . . . . 9 ((1 / 𝑛) ∈ (0[,]1) → (1 − (1 / 𝑛)) ∈ (0[,]1))
171169, 170syl 17 . . . . . . . 8 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ (0[,]1))
172171fmpttd 6861 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶(0[,]1))
173 1cnd 10625 . . . . . . . . 9 (⊤ → 1 ∈ ℂ)
174 nnex 11631 . . . . . . . . . . 11 ℕ ∈ V
175174mptex 6968 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V
176175a1i 11 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V)
17789recnd 10658 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℂ)
17882oveq2d 7156 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (1 − (1 / 𝑛)) = (1 − (1 / 𝑘)))
179 eqid 2822 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))
180 ovex 7173 . . . . . . . . . . . 12 (1 − (1 / 𝑘)) ∈ V
181178, 179, 180fvmpt 6750 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − (1 / 𝑘)))
18285oveq2d 7156 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)) = (1 − (1 / 𝑘)))
183181, 182eqtr4d 2860 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
184183adantl 485 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
18574, 75, 78, 173, 176, 177, 184climsubc2 14986 . . . . . . . 8 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ (1 − 0))
186 1m0e1 11746 . . . . . . . 8 (1 − 0) = 1
187185, 186breqtrdi 5083 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ 1)
188 1elunit 12848 . . . . . . . 8 1 ∈ (0[,]1)
189188a1i 11 . . . . . . 7 (⊤ → 1 ∈ (0[,]1))
19074, 75, 149, 172, 187, 189climcncf 23503 . . . . . 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))
191 eqidd 2823 . . . . . . . 8 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))))
192 eqidd 2823 . . . . . . . 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))))
193 oveq1 7147 . . . . . . . . . 10 (𝑥 = (1 − (1 / 𝑛)) → (𝑥𝑗) = ((1 − (1 / 𝑛))↑𝑗))
194193oveq2d 7156 . . . . . . . . 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)))
195194sumeq2sdv 15052 . . . . . . . 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)))
196171, 191, 192, 195fmptco 6873 . . . . . . 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))))
197 0zd 11981 . . . . . . . . 9 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℤ)
1988adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ0)
1996, 198, 9sylancr 590 . . . . . . . . . . . . . . . . . . . . 21 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
200199recnd 10658 . . . . . . . . . . . . . . . . . . . 20 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
201200adantllr 718 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
202 1re 10630 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ
203 resubcl 10939 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ) → (1 − (1 / 𝑛)) ∈ ℝ)
204202, 153, 203sylancr 590 . . . . . . . . . . . . . . . . . . . . . 22 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℝ)
205204ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (1 − (1 / 𝑛)) ∈ ℝ)
206 simplr 768 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ0)
207205, 206reexpcld 13523 . . . . . . . . . . . . . . . . . . . 20 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℝ)
208207recnd 10658 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
209 nn0cn 11895 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
210209ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℂ)
21111adantll 713 . . . . . . . . . . . . . . . . . . . 20 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
212211nnne0d 11675 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ≠ 0)
213201, 208, 210, 212div12d 11441 . . . . . . . . . . . . . . . . . 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)))
21413adantll 713 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
215208, 214mulcomd 10651 . . . . . . . . . . . . . . . . . 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)))
216213, 215eqtrd 2857 . . . . . . . . . . . . . . . . 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)))
2175, 216sylan2b 596 . . . . . . . . . . . . . . . 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)))
218217ifeq2da 4470 . . . . . . . . . . . . . . 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))))
219204recnd 10658 . . . . . . . . . . . . . . . . . 18 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℂ)
220 expcl 13443 . . . . . . . . . . . . . . . . . 18 (((1 − (1 / 𝑛)) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
221219, 220sylan 583 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
222221mul02d 10827 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (0 · ((1 − (1 / 𝑛))↑𝑘)) = 0)
223222ifeq1d 4457 . . . . . . . . . . . . . . 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))))
224218, 223eqtr4d 2860 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), (0 · ((1 − (1 / 𝑛))↑𝑘)), (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘))))
225 ovif 7235 . . . . . . . . . . . . . 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)))
226224, 225eqtr4di 2875 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = (if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) · ((1 − (1 / 𝑛))↑𝑘)))
227 simpr 488 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
228 c0ex 10624 . . . . . . . . . . . . . . 15 0 ∈ V
229 ovex 7173 . . . . . . . . . . . . . . 15 ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ V
230228, 229ifex 4487 . . . . . . . . . . . . . 14 if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
231 eqid 2822 . . . . . . . . . . . . . . 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))))
232231fvmpt2 6761 . . . . . . . . . . . . . 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))))
233227, 230, 232sylancl 589 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
234 ovex 7173 . . . . . . . . . . . . . . . 16 ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ V
235228, 234ifex 4487 . . . . . . . . . . . . . . 15 if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ V
236140fvmpt2 6761 . . . . . . . . . . . . . . 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)))
237227, 235, 236sylancl 589 . . . . . . . . . . . . . 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)))
238237oveq1d 7155 . . . . . . . . . . . . 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)))
239226, 233, 2383eqtr4d 2867 . . . . . . . . . . . 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)))
240239ralrimiva 3174 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)))
241 nfv 1915 . . . . . . . . . . . 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))
242 nffvmpt1 6663 . . . . . . . . . . . . 13 𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
243 nffvmpt1 6663 . . . . . . . . . . . . . 14 𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)
244 nfcv 2979 . . . . . . . . . . . . . 14 𝑘 ·
245 nfcv 2979 . . . . . . . . . . . . . 14 𝑘((1 − (1 / 𝑛))↑𝑗)
246243, 244, 245nfov 7170 . . . . . . . . . . . . 13 𝑘(((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))
247242, 246nfeq 2992 . . . . . . . . . . . 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))
248 fveq2 6652 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
249 fveq2 6652 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
250 oveq2 7148 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ((1 − (1 / 𝑛))↑𝑘) = ((1 − (1 / 𝑛))↑𝑗))
251249, 250oveq12d 7158 . . . . . . . . . . . . 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)))
252248, 251eqeq12d 2838 . . . . . . . . . . . 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))))
253241, 247, 252cbvralw 3415 . . . . . . . . . . 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)))
254240, 253sylib 221 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
255254r19.21bi 3198 . . . . . . . . 9 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
256 0cnd 10623 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
257207, 211nndivred 11679 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℝ)
258257recnd 10658 . . . . . . . . . . . . . . 15 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℂ)
259201, 258mulcld 10650 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
2605, 259sylan2b 596 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
261256, 260ifclda 4473 . . . . . . . . . . . 12 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ ℂ)
262261fmpttd 6861 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))):ℕ0⟶ℂ)
263262ffvelrnda 6833 . . . . . . . . . 10 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) ∈ ℂ)
264255, 263eqeltrrd 2915 . . . . . . . . 9 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)) ∈ ℂ)
265 0nn0 11900 . . . . . . . . . . . 12 0 ∈ ℕ0
266265a1i 11 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℕ0)
267 0p1e1 11747 . . . . . . . . . . . . 13 (0 + 1) = 1
268 seqeq1 13367 . . . . . . . . . . . . 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))))))
269267, 268ax-mp 5 . . . . . . . . . . . 12 seq(0 + 1)( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) = seq1( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))
270 1zzd 12001 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
271 elnnuz 12270 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ‘1))
272 nnne0 11659 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
273272neneqd 3016 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ ℕ → ¬ 𝑘 = 0)
274 biorf 934 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘 = 0 → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
275273, 274syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
276275bicomd 226 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ 2 ∥ 𝑘))
277276ifbid 4461 . . . . . . . . . . . . . . . . . . . 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))))
27890, 230, 232sylancl 589 . . . . . . . . . . . . . . . . . . . 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))))
279228, 229ifex 4487 . . . . . . . . . . . . . . . . . . . . 21 if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
280 eqid 2822 . . . . . . . . . . . . . . . . . . . . . 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))))
281280fvmpt2 6761 . . . . . . . . . . . . . . . . . . . . 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))))
282279, 281mpan2 690 . . . . . . . . . . . . . . . . . . . 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))))
283277, 278, 2823eqtr4d 2867 . . . . . . . . . . . . . . . . . . 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))
284283rgen 3140 . . . . . . . . . . . . . . . . . 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)
285284a1i 11 . . . . . . . . . . . . . . . . 17 ((⊤ ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘))
286 nfv 1915 . . . . . . . . . . . . . . . . . 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)
287 nffvmpt1 6663 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
288242, 287nfeq 2992 . . . . . . . . . . . . . . . . . 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)
289 fveq2 6652 . . . . . . . . . . . . . . . . . . 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))
290248, 289eqeq12d 2838 . . . . . . . . . . . . . . . . . 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)))
291286, 288, 290cbvralw 3415 . . . . . . . . . . . . . . . . 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))
292285, 291sylib 221 . . . . . . . . . . . . . . . 16 ((⊤ ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
293292r19.21bi 3198 . . . . . . . . . . . . . . 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))
294271, 293sylan2br 597 . . . . . . . . . . . . . 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))
295270, 294seqfeq 13391 . . . . . . . . . . . . 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))))))
296153, 162, 167abssubge0d 14782 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) = (1 − (1 / 𝑛)))
297 ltsubrp 12413 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ+) → (1 − (1 / 𝑛)) < 1)
298202, 152, 297sylancr 590 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) < 1)
299296, 298eqbrtrd 5064 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) < 1)
300280atantayl2 25522 . . . . . . . . . . . . . 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))))
301219, 299, 300syl2anc 587 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑛 ∈ ℕ) → seq1( + , (𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ (arctan‘(1 − (1 / 𝑛))))
302295, 301eqbrtrd 5064 . . . . . . . . . . . 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))))
303269, 302eqbrtrid 5077 . . . . . . . . . . 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))))
3041, 266, 263, 303clim2ser2 15003 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))) ⇝ ((arctan‘(1 − (1 / 𝑛))) + (seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0)))
305 0z 11980 . . . . . . . . . . . . . 14 0 ∈ ℤ
306 seq1 13377 . . . . . . . . . . . . . 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))
307305, 306ax-mp 5 . . . . . . . . . . . . 13 (seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0)
308 iftrue 4445 . . . . . . . . . . . . . . . 16 ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
309308orcs 872 . . . . . . . . . . . . . . 15 (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
310309, 231, 228fvmpt 6750 . . . . . . . . . . . . . 14 (0 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0) = 0)
311265, 310ax-mp 5 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘0) = 0
312307, 311eqtri 2845 . . . . . . . . . . . 12 (seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = 0
313312oveq2i 7151 . . . . . . . . . . 11 ((arctan‘(1 − (1 / 𝑛))) + (seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0)) = ((arctan‘(1 − (1 / 𝑛))) + 0)
314 atanrecl 25495 . . . . . . . . . . . . . 14 ((1 − (1 / 𝑛)) ∈ ℝ → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
315204, 314syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
316315recnd 10658 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℂ)
317316addid1d 10829 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + 0) = (arctan‘(1 − (1 / 𝑛))))
318313, 317syl5eq 2869 . . . . . . . . . 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))))
319304, 318breqtrd 5068 . . . . . . . . 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))))
3201, 197, 255, 264, 319isumclim 15103 . . . . . . . 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))))
321320mpteq2dva 5137 . . . . . . 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)))))
322196, 321eqtrd 2857 . . . . . 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)))))
323 oveq1 7147 . . . . . . . . . . . 12 (𝑥 = 1 → (𝑥𝑗) = (1↑𝑗))
324 nn0z 11993 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ0𝑗 ∈ ℤ)
325 1exp 13454 . . . . . . . . . . . . 13 (𝑗 ∈ ℤ → (1↑𝑗) = 1)
326324, 325syl 17 . . . . . . . . . . . 12 (𝑗 ∈ ℕ0 → (1↑𝑗) = 1)
327323, 326sylan9eq 2877 . . . . . . . . . . 11 ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (𝑥𝑗) = 1)
328327oveq2d 7156 . . . . . . . . . 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))
32917mptru 1545 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ0⟶ℂ
330329ffvelrni 6832 . . . . . . . . . . . 12 (𝑗 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
331330mulid1d 10647 . . . . . . . . . . 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))
332331adantl 485 . . . . . . . . . 10 ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · 1) = ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
333328, 332eqtrd 2857 . . . . . . . . 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))
334333sumeq2dv 15051 . . . . . . . 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))
335 sumex 15035 . . . . . . . 8 Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ V
336334, 148, 335fvmpt 6750 . . . . . . 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))
337188, 336mp1i 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))
338190, 322, 3373brtr3d 5073 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗))
339 eqid 2822 . . . . . . . . 9 (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
340 eqid 2822 . . . . . . . . 9 {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} = {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
341339, 340atancn 25520 . . . . . . . 8 (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}–cn→ℂ)
342341a1i 11 . . . . . . 7 (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∈ ({𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}–cn→ℂ))
343 unitssre 12877 . . . . . . . . 9 (0[,]1) ⊆ ℝ
344339, 340ressatans 25518 . . . . . . . . 9 ℝ ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
345343, 344sstri 3951 . . . . . . . 8 (0[,]1) ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
346 fss 6508 . . . . . . . 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))})
347172, 345, 346sylancl 589 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
348344, 202sselii 3939 . . . . . . . 8 1 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
349348a1i 11 . . . . . . 7 (⊤ → 1 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
35074, 75, 342, 347, 187, 349climcncf 23503 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) ⇝ ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1))
351345, 171sseldi 3940 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
352 cncff 23496 . . . . . . . . . 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)
353341, 352mp1i 13 . . . . . . . . 9 (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}):{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}⟶ℂ)
354353feqmptd 6715 . . . . . . . 8 (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘𝑘)))
355 fvres 6671 . . . . . . . . 9 (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘𝑘) = (arctan‘𝑘))
356355mpteq2ia 5133 . . . . . . . 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))
357354, 356syl6eq 2873 . . . . . . 7 (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ (arctan‘𝑘)))
358 fveq2 6652 . . . . . . 7 (𝑘 = (1 − (1 / 𝑛)) → (arctan‘𝑘) = (arctan‘(1 − (1 / 𝑛))))
359351, 191, 357, 358fmptco 6873 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))))
360 fvres 6671 . . . . . . . 8 (1 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (arctan‘1))
361348, 360mp1i 13 . . . . . . 7 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (arctan‘1))
362 atan1 25512 . . . . . . 7 (arctan‘1) = (π / 4)
363361, 362syl6eq 2873 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (π / 4))
364350, 359, 3633brtr3d 5073 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ (π / 4))
365 climuni 14900 . . . . 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))
366338, 364, 365syl2anc 587 . . . 4 (⊤ → Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4))
367147, 366breqtrd 5068 . . 3 (⊤ → seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
368367mptru 1545 . 2 seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4)
369 leibpi.1 . . 3 𝐹 = (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
370 ovex 7173 . . 3 (π / 4) ∈ V
371369, 140, 370leibpilem2 25525 . 2 (seq0( + , 𝐹) ⇝ (π / 4) ↔ seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
372368, 371mpbir 234 1 seq0( + , 𝐹) ⇝ (π / 4)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wo 844   = wceq 1538  wtru 1539  wcel 2114  wral 3130  {crab 3134  Vcvv 3469  cdif 3905  wss 3908  ifcif 4439   class class class wbr 5042  cmpt 5122  dom cdm 5532  cres 5534  ccom 5536  wf 6330  cfv 6334  (class class class)co 7140  cc 10524  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  -∞cmnf 10662   < clt 10664  cle 10665  cmin 10859  -cneg 10860   / cdiv 11286  cn 11625  2c2 11680  4c4 11682  0cn0 11885  cz 11969  cuz 12231  +crp 12377  (,]cioc 12727  [,]cicc 12729  seqcseq 13364  cexp 13425  abscabs 14584  cli 14832  Σcsu 15033  πcpi 15411  cdvds 15598  cnccncf 23479  arctancatan 25448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605  ax-mulf 10606
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-of 7394  df-om 7566  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-xnn0 11956  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13426  df-fac 13630  df-bc 13659  df-hash 13687  df-shft 14417  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-limsup 14819  df-clim 14836  df-rlim 14837  df-sum 15034  df-ef 15412  df-sin 15414  df-cos 15415  df-tan 15416  df-pi 15417  df-dvds 15599  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-mulr 16570  df-starv 16571  df-sca 16572  df-vsca 16573  df-ip 16574  df-tset 16575  df-ple 16576  df-ds 16578  df-unif 16579  df-hom 16580  df-cco 16581  df-rest 16687  df-topn 16688  df-0g 16706  df-gsum 16707  df-topgen 16708  df-pt 16709  df-prds 16712  df-xrs 16766  df-qtop 16771  df-imas 16772  df-xps 16774  df-mre 16848  df-mrc 16849  df-acs 16851  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-submnd 17948  df-mulg 18216  df-cntz 18438  df-cmn 18899  df-psmet 20081  df-xmet 20082  df-met 20083  df-bl 20084  df-mopn 20085  df-fbas 20086  df-fg 20087  df-cnfld 20090  df-top 21497  df-topon 21514  df-topsp 21536  df-bases 21549  df-cld 21622  df-ntr 21623  df-cls 21624  df-nei 21701  df-lp 21739  df-perf 21740  df-cn 21830  df-cnp 21831  df-t1 21917  df-haus 21918  df-cmp 21990  df-tx 22165  df-hmeo 22358  df-fil 22449  df-fm 22541  df-flim 22542  df-flf 22543  df-xms 22925  df-ms 22926  df-tms 22927  df-cncf 23481  df-limc 24467  df-dv 24468  df-ulm 24970  df-log 25146  df-atan 25451
This theorem is referenced by:  leibpisum  25527
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