MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  leibpi Structured version   Visualization version   GIF version

Theorem leibpi 26999
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 15717). (2) Using leibpilem2 26998 to rewrite the series as a power series, it is the 𝑥 = 1 special case of the Taylor series for arctan (atantayl2 26995). (3) Although we cannot directly plug 𝑥 = 1 into atantayl2 26995, Abel's theorem (abelth2 26500) 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 26993) 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 12917 . . . . 5 0 = (ℤ‘0)
2 0zd 12622 . . . . 5 (⊤ → 0 ∈ ℤ)
3 eqidd 2735 . . . . 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 11251 . . . . . . . . 9 ((𝑘 ∈ ℕ0 ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
5 ioran 985 . . . . . . . . . 10 (¬ (𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘))
6 neg1rr 12378 . . . . . . . . . . . . 13 -1 ∈ ℝ
7 leibpilem1 26997 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ ((𝑘 − 1) / 2) ∈ ℕ0))
87simprd 495 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ0)
9 reexpcl 14115 . . . . . . . . . . . . 13 ((-1 ∈ ℝ ∧ ((𝑘 − 1) / 2) ∈ ℕ0) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
106, 8, 9sylancr 587 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
117simpld 494 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
1210, 11nndivred 12317 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℝ)
1312recnd 11286 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
145, 13sylan2b 594 . . . . . . . . 9 ((𝑘 ∈ ℕ0 ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
154, 14ifclda 4565 . . . . . . . 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)
1716fmpttd 7134 . . . . . 6 (⊤ → (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ0⟶ℂ)
1817ffvelcdmda 7103 . . . . 5 ((⊤ ∧ 𝑗 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
19 2nn0 12540 . . . . . . . . . . . . . 14 2 ∈ ℕ0
2019a1i 11 . . . . . . . . . . . . 13 (⊤ → 2 ∈ ℕ0)
21 nn0mulcl 12559 . . . . . . . . . . . . 13 ((2 ∈ ℕ0𝑛 ∈ ℕ0) → (2 · 𝑛) ∈ ℕ0)
2220, 21sylan 580 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ0) → (2 · 𝑛) ∈ ℕ0)
23 nn0p1nn 12562 . . . . . . . . . . . 12 ((2 · 𝑛) ∈ ℕ0 → ((2 · 𝑛) + 1) ∈ ℕ)
2422, 23syl 17 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ0) → ((2 · 𝑛) + 1) ∈ ℕ)
2524nnrecred 12314 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ0) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ)
2625fmpttd 7134 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))):ℕ0⟶ℝ)
27 nn0mulcl 12559 . . . . . . . . . . . . . 14 ((2 ∈ ℕ0𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
2820, 27sylan 580 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
2928nn0red 12585 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℝ)
30 peano2nn0 12563 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
3130adantl 481 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
32 nn0mulcl 12559 . . . . . . . . . . . . . 14 ((2 ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℕ0)
3319, 31, 32sylancr 587 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℕ0)
3433nn0red 12585 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · (𝑘 + 1)) ∈ ℝ)
35 1red 11259 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 1 ∈ ℝ)
36 nn0re 12532 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
3736adantl 481 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
3837lep1d 12196 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → 𝑘 ≤ (𝑘 + 1))
39 peano2re 11431 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
4037, 39syl 17 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℝ)
41 2re 12337 . . . . . . . . . . . . . . 15 2 ∈ ℝ
4241a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 2 ∈ ℝ)
43 2pos 12366 . . . . . . . . . . . . . . 15 0 < 2
4443a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < 2)
45 lemul2 12117 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
4637, 40, 42, 44, 45syl112anc 1373 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → (𝑘 ≤ (𝑘 + 1) ↔ (2 · 𝑘) ≤ (2 · (𝑘 + 1))))
4738, 46mpbid 232 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → (2 · 𝑘) ≤ (2 · (𝑘 + 1)))
4829, 34, 35, 47leadd1dd 11874 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1))
49 nn0p1nn 12562 . . . . . . . . . . . . . 14 ((2 · 𝑘) ∈ ℕ0 → ((2 · 𝑘) + 1) ∈ ℕ)
5028, 49syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℕ)
5150nnred 12278 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℝ)
5250nngt0d 12312 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < ((2 · 𝑘) + 1))
53 nn0p1nn 12562 . . . . . . . . . . . . . 14 ((2 · (𝑘 + 1)) ∈ ℕ0 → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
5433, 53syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℕ)
5554nnred 12278 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · (𝑘 + 1)) + 1) ∈ ℝ)
5654nngt0d 12312 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ0) → 0 < ((2 · (𝑘 + 1)) + 1))
57 lerec 12148 . . . . . . . . . . . 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 839 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → (((2 · 𝑘) + 1) ≤ ((2 · (𝑘 + 1)) + 1) ↔ (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1))))
5948, 58mpbid 232 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → (1 / ((2 · (𝑘 + 1)) + 1)) ≤ (1 / ((2 · 𝑘) + 1)))
60 oveq2 7438 . . . . . . . . . . . . . 14 (𝑛 = (𝑘 + 1) → (2 · 𝑛) = (2 · (𝑘 + 1)))
6160oveq1d 7445 . . . . . . . . . . . . 13 (𝑛 = (𝑘 + 1) → ((2 · 𝑛) + 1) = ((2 · (𝑘 + 1)) + 1))
6261oveq2d 7446 . . . . . . . . . . . 12 (𝑛 = (𝑘 + 1) → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · (𝑘 + 1)) + 1)))
63 eqid 2734 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))
64 ovex 7463 . . . . . . . . . . . 12 (1 / ((2 · (𝑘 + 1)) + 1)) ∈ V
6562, 63, 64fvmpt 7015 . . . . . . . . . . 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 7438 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (2 · 𝑛) = (2 · 𝑘))
6867oveq1d 7445 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((2 · 𝑛) + 1) = ((2 · 𝑘) + 1))
6968oveq2d 7446 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (1 / ((2 · 𝑛) + 1)) = (1 / ((2 · 𝑘) + 1)))
70 ovex 7463 . . . . . . . . . . . 12 (1 / ((2 · 𝑘) + 1)) ∈ V
7169, 63, 70fvmpt 7015 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
7271adantl 481 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
7359, 66, 723brtr4d 5179 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘(𝑘 + 1)) ≤ ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
74 nnuz 12918 . . . . . . . . . 10 ℕ = (ℤ‘1)
75 1zzd 12645 . . . . . . . . . 10 (⊤ → 1 ∈ ℤ)
76 ax-1cn 11210 . . . . . . . . . . 11 1 ∈ ℂ
77 divcnv 15885 . . . . . . . . . . 11 (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
7876, 77mp1i 13 . . . . . . . . . 10 (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
79 nn0ex 12529 . . . . . . . . . . . 12 0 ∈ V
8079mptex 7242 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ∈ V
8180a1i 11 . . . . . . . . . 10 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ∈ V)
82 oveq2 7438 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
83 eqid 2734 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
84 ovex 7463 . . . . . . . . . . . . 13 (1 / 𝑘) ∈ V
8582, 83, 84fvmpt 7015 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
8685adantl 481 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) = (1 / 𝑘))
87 nnrecre 12305 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
8887adantl 481 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
8986, 88eqeltrd 2838 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℝ)
90 nnnn0 12530 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
9190adantl 481 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0)
9291, 71syl 17 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) = (1 / ((2 · 𝑘) + 1)))
9390, 50sylan2 593 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℕ)
9493nnrecred 12314 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ)
9592, 94eqeltrd 2838 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ∈ ℝ)
96 nnre 12270 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
9796adantl 481 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
9819, 91, 27sylancr 587 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℕ0)
9998nn0red 12585 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℝ)
100 peano2re 11431 . . . . . . . . . . . . . 14 ((2 · 𝑘) ∈ ℝ → ((2 · 𝑘) + 1) ∈ ℝ)
10199, 100syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
102 nn0addge1 12569 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → 𝑘 ≤ (𝑘 + 𝑘))
10397, 91, 102syl2anc 584 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (𝑘 + 𝑘))
10497recnd 11286 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
1051042timesd 12506 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) = (𝑘 + 𝑘))
106103, 105breqtrrd 5175 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (2 · 𝑘))
10799lep1d 12196 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ≤ ((2 · 𝑘) + 1))
10897, 99, 101, 106, 107letrd 11415 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ ((2 · 𝑘) + 1))
109 nngt0 12294 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 0 < 𝑘)
110109adantl 481 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
11193nnred 12278 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ)
11293nngt0d 12312 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 < ((2 · 𝑘) + 1))
113 lerec 12148 . . . . . . . . . . . . 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 839 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 ≤ ((2 · 𝑘) + 1) ↔ (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘)))
115108, 114mpbid 232 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ≤ (1 / 𝑘))
116115, 92, 863brtr4d 5179 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘))
11793nnrpd 13072 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) + 1) ∈ ℝ+)
118117rpreccld 13084 . . . . . . . . . . . 12 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / ((2 · 𝑘) + 1)) ∈ ℝ+)
119118rpge0d 13078 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / ((2 · 𝑘) + 1)))
120119, 92breqtrrd 5175 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘))
12174, 75, 78, 81, 89, 95, 116, 120climsqz2 15674 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1))) ⇝ 0)
122 neg1cn 12377 . . . . . . . . . . . . 13 -1 ∈ ℂ
123122a1i 11 . . . . . . . . . . . 12 (⊤ → -1 ∈ ℂ)
124 expcl 14116 . . . . . . . . . . . 12 ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℂ)
125123, 124sylan 580 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℂ)
12650nncnd 12279 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ∈ ℂ)
12750nnne0d 12313 . . . . . . . . . . 11 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((2 · 𝑘) + 1) ≠ 0)
128125, 126, 127divrecd 12043 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) / ((2 · 𝑘) + 1)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
129 oveq2 7438 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (-1↑𝑛) = (-1↑𝑘))
130129, 68oveq12d 7448 . . . . . . . . . . . 12 (𝑛 = 𝑘 → ((-1↑𝑛) / ((2 · 𝑛) + 1)) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
131 eqid 2734 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))) = (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))
132 ovex 7463 . . . . . . . . . . . 12 ((-1↑𝑘) / ((2 · 𝑘) + 1)) ∈ V
133130, 131, 132fvmpt 7015 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
134133adantl 481 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))‘𝑘) = ((-1↑𝑘) / ((2 · 𝑘) + 1)))
13572oveq2d 7446 . . . . . . . . . 10 ((⊤ ∧ 𝑘 ∈ ℕ0) → ((-1↑𝑘) · ((𝑛 ∈ ℕ0 ↦ (1 / ((2 · 𝑛) + 1)))‘𝑘)) = ((-1↑𝑘) · (1 / ((2 · 𝑘) + 1))))
136128, 134, 1353eqtr4d 2784 . . . . . . . . 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 15717 . . . . . . . 8 (⊤ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ∈ dom ⇝ )
138 climdm 15586 . . . . . . . 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 218 . . . . . . 7 (⊤ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
140 eqid 2734 . . . . . . . 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 6919 . . . . . . . 8 ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))) ∈ V
142131, 140, 141leibpilem2 26998 . . . . . . 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 218 . . . . . 6 (⊤ → seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ ( ⇝ ‘seq0( + , (𝑛 ∈ ℕ0 ↦ ((-1↑𝑛) / ((2 · 𝑛) + 1))))))
144 seqex 14040 . . . . . . 7 seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ∈ V
145144, 141breldm 5921 . . . . . 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 15790 . . . 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 2734 . . . . . . . 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 26500 . . . . . . 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 13043 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
151150adantl 481 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+)
152151rpreccld 13084 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ+)
153152rpred 13074 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
154152rpge0d 13078 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ≤ (1 / 𝑛))
155 nnge1 12291 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 1 ≤ 𝑛)
156155adantl 481 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
157 nnre 12270 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
158157adantl 481 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
159158recnd 11286 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
160159mulridd 11275 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑛 · 1) = 𝑛)
161156, 160breqtrrd 5175 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ≤ (𝑛 · 1))
162 1red 11259 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
163 nngt0 12294 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 0 < 𝑛)
164163adantl 481 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
165 ledivmul 12141 . . . . . . . . . . . 12 ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
166162, 162, 158, 164, 165syl112anc 1373 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → ((1 / 𝑛) ≤ 1 ↔ 1 ≤ (𝑛 · 1)))
167161, 166mpbird 257 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ≤ 1)
168 elicc01 13502 . . . . . . . . . 10 ((1 / 𝑛) ∈ (0[,]1) ↔ ((1 / 𝑛) ∈ ℝ ∧ 0 ≤ (1 / 𝑛) ∧ (1 / 𝑛) ≤ 1))
169153, 154, 167, 168syl3anbrc 1342 . . . . . . . . 9 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ (0[,]1))
170 iirev 24969 . . . . . . . . 9 ((1 / 𝑛) ∈ (0[,]1) → (1 − (1 / 𝑛)) ∈ (0[,]1))
171169, 170syl 17 . . . . . . . 8 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ (0[,]1))
172171fmpttd 7134 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶(0[,]1))
173 1cnd 11253 . . . . . . . . 9 (⊤ → 1 ∈ ℂ)
174 nnex 12269 . . . . . . . . . . 11 ℕ ∈ V
175174mptex 7242 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V
176175a1i 11 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ∈ V)
17789recnd 11286 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘) ∈ ℂ)
17882oveq2d 7446 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (1 − (1 / 𝑛)) = (1 − (1 / 𝑘)))
179 eqid 2734 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))
180 ovex 7463 . . . . . . . . . . . 12 (1 − (1 / 𝑘)) ∈ V
181178, 179, 180fvmpt 7015 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − (1 / 𝑘)))
18285oveq2d 7446 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)) = (1 − (1 / 𝑘)))
183181, 182eqtr4d 2777 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
184183adantl 481 . . . . . . . . 9 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘𝑘)))
18574, 75, 78, 173, 176, 177, 184climsubc2 15671 . . . . . . . 8 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ (1 − 0))
186 1m0e1 12384 . . . . . . . 8 (1 − 0) = 1
187185, 186breqtrdi 5188 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) ⇝ 1)
188 1elunit 13506 . . . . . . . 8 1 ∈ (0[,]1)
189188a1i 11 . . . . . . 7 (⊤ → 1 ∈ (0[,]1))
19074, 75, 149, 172, 187, 189climcncf 24939 . . . . . 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 2735 . . . . . . . 8 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))))
192 eqidd 2735 . . . . . . . 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 7437 . . . . . . . . . 10 (𝑥 = (1 − (1 / 𝑛)) → (𝑥𝑗) = ((1 − (1 / 𝑛))↑𝑗))
194193oveq2d 7446 . . . . . . . . 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 15735 . . . . . . . 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 7148 . . . . . . 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 12622 . . . . . . . . 9 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℤ)
1988adantll 714 . . . . . . . . . . . . . . . . . . . . . 22 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((𝑘 − 1) / 2) ∈ ℕ0)
1996, 198, 9sylancr 587 . . . . . . . . . . . . . . . . . . . . 21 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℝ)
200199recnd 11286 . . . . . . . . . . . . . . . . . . . 20 (((⊤ ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
201200adantllr 719 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (-1↑((𝑘 − 1) / 2)) ∈ ℂ)
202 1re 11258 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ
203 resubcl 11570 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ) → (1 − (1 / 𝑛)) ∈ ℝ)
204202, 153, 203sylancr 587 . . . . . . . . . . . . . . . . . . . . . 22 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℝ)
205204ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (1 − (1 / 𝑛)) ∈ ℝ)
206 simplr 769 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ0)
207205, 206reexpcld 14199 . . . . . . . . . . . . . . . . . . . 20 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℝ)
208207recnd 11286 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
209 nn0cn 12533 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
210209ad2antlr 727 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℂ)
21111adantll 714 . . . . . . . . . . . . . . . . . . . 20 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ∈ ℕ)
212211nnne0d 12313 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → 𝑘 ≠ 0)
213201, 208, 210, 212div12d 12076 . . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ ℂ)
215208, 214mulcomd 11279 . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) = (((-1↑((𝑘 − 1) / 2)) / 𝑘) · ((1 − (1 / 𝑛))↑𝑘)))
218217ifeq2da 4562 . . . . . . . . . . . . . . 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 11286 . . . . . . . . . . . . . . . . . 18 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ ℂ)
220 expcl 14116 . . . . . . . . . . . . . . . . . 18 (((1 − (1 / 𝑛)) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
221219, 220sylan 580 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1 − (1 / 𝑛))↑𝑘) ∈ ℂ)
222221mul02d 11456 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (0 · ((1 − (1 / 𝑛))↑𝑘)) = 0)
223222ifeq1d 4549 . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . 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 7530 . . . . . . . . . . . . . 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 2792 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = (if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) · ((1 − (1 / 𝑛))↑𝑘)))
227 simpr 484 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
228 c0ex 11252 . . . . . . . . . . . . . . 15 0 ∈ V
229 ovex 7463 . . . . . . . . . . . . . . 15 ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ V
230228, 229ifex 4580 . . . . . . . . . . . . . 14 if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
231 eqid 2734 . . . . . . . . . . . . . . 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 7026 . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
234 ovex 7463 . . . . . . . . . . . . . . . 16 ((-1↑((𝑘 − 1) / 2)) / 𝑘) ∈ V
235228, 234ifex 4580 . . . . . . . . . . . . . . 15 if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)) ∈ V
236140fvmpt2 7026 . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))
238237oveq1d 7445 . . . . . . . . . . . . 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 2784 . . . . . . . . . . . 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 3143 . . . . . . . . . . 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 1911 . . . . . . . . . . . 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 6917 . . . . . . . . . . . . 13 𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
243 nffvmpt1 6917 . . . . . . . . . . . . . 14 𝑘((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗)
244 nfcv 2902 . . . . . . . . . . . . . 14 𝑘 ·
245 nfcv 2902 . . . . . . . . . . . . . 14 𝑘((1 − (1 / 𝑛))↑𝑗)
246243, 244, 245nfov 7460 . . . . . . . . . . . . 13 𝑘(((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗))
247242, 246nfeq 2916 . . . . . . . . . . . 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 6906 . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → ((1 − (1 / 𝑛))↑𝑘) = ((1 − (1 / 𝑛))↑𝑗))
251249, 250oveq12d 7448 . . . . . . . . . . . . 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 2750 . . . . . . . . . . . 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 3303 . . . . . . . . . . 11 (∀𝑘 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑘) · ((1 − (1 / 𝑛))↑𝑘)) ↔ ∀𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
254240, 253sylib 218 . . . . . . . . . 10 ((⊤ ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = (((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) · ((1 − (1 / 𝑛))↑𝑗)))
255254r19.21bi 3248 . . . . . . . . 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 11251 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → 0 ∈ ℂ)
257207, 211nndivred 12317 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℝ)
258257recnd 11286 . . . . . . . . . . . . . . 15 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → (((1 − (1 / 𝑛))↑𝑘) / 𝑘) ∈ ℂ)
259201, 258mulcld 11278 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ (¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
2605, 259sylan2b 594 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) ∧ ¬ (𝑘 = 0 ∨ 2 ∥ 𝑘)) → ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)) ∈ ℂ)
261256, 260ifclda 4565 . . . . . . . . . . . 12 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ ℂ)
262261fmpttd 7134 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))):ℕ0⟶ℂ)
263262ffvelcdmda 7103 . . . . . . . . . 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 2839 . . . . . . . . 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 12538 . . . . . . . . . . . 12 0 ∈ ℕ0
266265a1i 11 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → 0 ∈ ℕ0)
267 0p1e1 12385 . . . . . . . . . . . . 13 (0 + 1) = 1
268 seqeq1 14041 . . . . . . . . . . . . 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 12645 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℤ)
271 elnnuz 12919 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ‘1))
272 nnne0 12297 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
273272neneqd 2942 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ ℕ → ¬ 𝑘 = 0)
274 biorf 936 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘 = 0 → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
275273, 274syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ → (2 ∥ 𝑘 ↔ (𝑘 = 0 ∨ 2 ∥ 𝑘)))
276275bicomd 223 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ → ((𝑘 = 0 ∨ 2 ∥ 𝑘) ↔ 2 ∥ 𝑘))
277276ifbid 4553 . . . . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
279228, 229ifex 4580 . . . . . . . . . . . . . . . . . . . . 21 if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) ∈ V
280 eqid 2734 . . . . . . . . . . . . . . . . . . . . . 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 7026 . . . . . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))
283277, 278, 2823eqtr4d 2784 . . . . . . . . . . . . . . . . . . 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 3060 . . . . . . . . . . . . . . . . . 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 1911 . . . . . . . . . . . . . . . . . 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 6917 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗)
288242, 287nfeq 2916 . . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . 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 2750 . . . . . . . . . . . . . . . . . 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 3303 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ ℕ ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑘) ↔ ∀𝑗 ∈ ℕ ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
292285, 291sylib 218 . . . . . . . . . . . . . . . 16 ((⊤ ∧ 𝑛 ∈ ℕ) → ∀𝑗 ∈ ℕ ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
293292r19.21bi 3248 . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑛 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗) = ((𝑘 ∈ ℕ ↦ if(2 ∥ 𝑘, 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))))‘𝑗))
295270, 294seqfeq 14064 . . . . . . . . . . . . 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 15466 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) = (1 − (1 / 𝑛)))
297 ltsubrp 13068 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ ∧ (1 / 𝑛) ∈ ℝ+) → (1 − (1 / 𝑛)) < 1)
298202, 152, 297sylancr 587 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) < 1)
299296, 298eqbrtrd 5169 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑛 ∈ ℕ) → (abs‘(1 − (1 / 𝑛))) < 1)
300280atantayl2 26995 . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . 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 5169 . . . . . . . . . . . 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 5182 . . . . . . . . . . 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 15688 . . . . . . . . . 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 12621 . . . . . . . . . . . . . 14 0 ∈ ℤ
306 seq1 14051 . . . . . . . . . . . . . 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 4536 . . . . . . . . . . . . . . . 16 ((𝑘 = 0 ∨ 2 ∥ 𝑘) → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
309308orcs 875 . . . . . . . . . . . . . . 15 (𝑘 = 0 → if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘))) = 0)
310309, 231, 228fvmpt 7015 . . . . . . . . . . . . . 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 2762 . . . . . . . . . . . 12 (seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) · (((1 − (1 / 𝑛))↑𝑘) / 𝑘)))))‘0) = 0
313312oveq2i 7441 . . . . . . . . . . 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 26968 . . . . . . . . . . . . . 14 ((1 − (1 / 𝑛)) ∈ ℝ → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
315204, 314syl 17 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℝ)
316315recnd 11286 . . . . . . . . . . . 12 ((⊤ ∧ 𝑛 ∈ ℕ) → (arctan‘(1 − (1 / 𝑛))) ∈ ℂ)
317316addridd 11458 . . . . . . . . . . 11 ((⊤ ∧ 𝑛 ∈ ℕ) → ((arctan‘(1 − (1 / 𝑛))) + 0) = (arctan‘(1 − (1 / 𝑛))))
318313, 317eqtrid 2786 . . . . . . . . . 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 5173 . . . . . . . . 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 15789 . . . . . . . 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 5247 . . . . . . 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 2774 . . . . . 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 7437 . . . . . . . . . . . 12 (𝑥 = 1 → (𝑥𝑗) = (1↑𝑗))
324 nn0z 12635 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ0𝑗 ∈ ℤ)
325 1exp 14128 . . . . . . . . . . . . 13 (𝑗 ∈ ℤ → (1↑𝑗) = 1)
326324, 325syl 17 . . . . . . . . . . . 12 (𝑗 ∈ ℕ0 → (1↑𝑗) = 1)
327323, 326sylan9eq 2794 . . . . . . . . . . 11 ((𝑥 = 1 ∧ 𝑗 ∈ ℕ0) → (𝑥𝑗) = 1)
328327oveq2d 7446 . . . . . . . . . 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 1543 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘))):ℕ0⟶ℂ
330329ffvelcdmi 7102 . . . . . . . . . . . 12 (𝑗 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ ℂ)
331330mulridd 11275 . . . . . . . . . . 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 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))
333328, 332eqtrd 2774 . . . . . . . . 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 15734 . . . . . . . 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 15720 . . . . . . . 8 Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) ∈ V
336334, 148, 335fvmpt 7015 . . . . . . 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 5178 . . . . 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 2734 . . . . . . . . 9 (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
340 eqid 2734 . . . . . . . . 9 {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} = {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
341339, 340atancn 26993 . . . . . . . 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 13535 . . . . . . . . 9 (0[,]1) ⊆ ℝ
344339, 340ressatans 26991 . . . . . . . . 9 ℝ ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
345343, 344sstri 4004 . . . . . . . 8 (0[,]1) ⊆ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}
346 fss 6752 . . . . . . . 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 586 . . . . . . 7 (⊤ → (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛))):ℕ⟶{𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
348344, 202sselii 3991 . . . . . . . 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 24939 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) ⇝ ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1))
351345, 171sselid 3992 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℕ) → (1 − (1 / 𝑛)) ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})
352 cncff 24932 . . . . . . . . . 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 6976 . . . . . . . 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 6925 . . . . . . . . 9 (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘𝑘) = (arctan‘𝑘))
356355mpteq2ia 5250 . . . . . . . 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, 356eqtrdi 2790 . . . . . . 7 (⊤ → (arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) = (𝑘 ∈ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))} ↦ (arctan‘𝑘)))
358 fveq2 6906 . . . . . . 7 (𝑘 = (1 − (1 / 𝑛)) → (arctan‘𝑘) = (arctan‘(1 − (1 / 𝑛))))
359351, 191, 357, 358fmptco 7148 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))}) ∘ (𝑛 ∈ ℕ ↦ (1 − (1 / 𝑛)))) = (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))))
360 fvres 6925 . . . . . . . 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 26985 . . . . . . 7 (arctan‘1) = (π / 4)
363361, 362eqtrdi 2790 . . . . . 6 (⊤ → ((arctan ↾ {𝑥 ∈ ℂ ∣ (1 + (𝑥↑2)) ∈ (ℂ ∖ (-∞(,]0))})‘1) = (π / 4))
364350, 359, 3633brtr3d 5178 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (arctan‘(1 − (1 / 𝑛)))) ⇝ (π / 4))
365 climuni 15584 . . . . 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 584 . . . 4 (⊤ → Σ𝑗 ∈ ℕ0 ((𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))‘𝑗) = (π / 4))
367147, 366breqtrd 5173 . . 3 (⊤ → seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
368367mptru 1543 . 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 7463 . . 3 (π / 4) ∈ V
371369, 140, 370leibpilem2 26998 . 2 (seq0( + , 𝐹) ⇝ (π / 4) ↔ seq0( + , (𝑘 ∈ ℕ0 ↦ if((𝑘 = 0 ∨ 2 ∥ 𝑘), 0, ((-1↑((𝑘 − 1) / 2)) / 𝑘)))) ⇝ (π / 4))
372368, 371mpbir 231 1 seq0( + , 𝐹) ⇝ (π / 4)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847   = wceq 1536  wtru 1537  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  cdif 3959  wss 3962  ifcif 4530   class class class wbr 5147  cmpt 5230  dom cdm 5688  cres 5690  ccom 5692  wf 6558  cfv 6562  (class class class)co 7430  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157  -∞cmnf 11290   < clt 11292  cle 11293  cmin 11489  -cneg 11490   / cdiv 11917  cn 12263  2c2 12318  4c4 12320  0cn0 12523  cz 12610  cuz 12875  +crp 13031  (,]cioc 13384  [,]cicc 13386  seqcseq 14038  cexp 14098  abscabs 15269  cli 15516  Σcsu 15718  πcpi 16098  cdvds 16286  cnccncf 24915  arctancatan 26921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ioc 13388  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15102  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-sum 15719  df-ef 16099  df-sin 16101  df-cos 16102  df-tan 16103  df-pi 16104  df-dvds 16287  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lp 23159  df-perf 23160  df-cn 23250  df-cnp 23251  df-t1 23337  df-haus 23338  df-cmp 23410  df-tx 23585  df-hmeo 23778  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917  df-limc 25915  df-dv 25916  df-ulm 26434  df-log 26612  df-atan 26924
This theorem is referenced by:  leibpisum  27000
  Copyright terms: Public domain W3C validator