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Theorem logtayl 25254
 Description: The Taylor series for -log(1 − 𝐴). (Contributed by Mario Carneiro, 1-Apr-2015.)
Assertion
Ref Expression
logtayl ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))) ⇝ -(log‘(1 − 𝐴)))
Distinct variable group:   𝐴,𝑘

Proof of Theorem logtayl
Dummy variables 𝑗 𝑚 𝑛 𝑟 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 12277 . . . 4 0 = (ℤ‘0)
2 0zd 11990 . . . 4 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ∈ ℤ)
3 eqeq1 2828 . . . . . . . 8 (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
4 oveq2 7157 . . . . . . . 8 (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
53, 4ifbieq2d 4475 . . . . . . 7 (𝑘 = 𝑛 → if(𝑘 = 0, 0, (1 / 𝑘)) = if(𝑛 = 0, 0, (1 / 𝑛)))
6 oveq2 7157 . . . . . . 7 (𝑘 = 𝑛 → (𝐴𝑘) = (𝐴𝑛))
75, 6oveq12d 7167 . . . . . 6 (𝑘 = 𝑛 → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
8 eqid 2824 . . . . . 6 (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘))) = (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))
9 ovex 7182 . . . . . 6 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)) ∈ V
107, 8, 9fvmpt 6759 . . . . 5 (𝑛 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
1110adantl 485 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
12 0cnd 10632 . . . . . 6 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → 0 ∈ ℂ)
13 simpr 488 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
14 elnn0 11896 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
1513, 14sylib 221 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → (𝑛 ∈ ℕ ∨ 𝑛 = 0))
1615ord 861 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 ∈ ℕ → 𝑛 = 0))
1716con1d 147 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 = 0 → 𝑛 ∈ ℕ))
1817imp 410 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
1918nnrecred 11685 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℝ)
2019recnd 10667 . . . . . 6 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℂ)
2112, 20ifclda 4484 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, (1 / 𝑛)) ∈ ℂ)
22 expcl 13452 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℂ)
2322adantlr 714 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℂ)
2421, 23mulcld 10659 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)) ∈ ℂ)
25 logtayllem 25253 . . . 4 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ∈ dom ⇝ )
261, 2, 11, 24, 25isumclim2 15113 . . 3 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ⇝ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
27 simpl 486 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ)
28 0cn 10631 . . . . . . . 8 0 ∈ ℂ
29 eqid 2824 . . . . . . . . 9 (abs ∘ − ) = (abs ∘ − )
3029cnmetdval 23379 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴(abs ∘ − )0) = (abs‘(𝐴 − 0)))
3127, 28, 30sylancl 589 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) = (abs‘(𝐴 − 0)))
32 subid1 10904 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
3332adantr 484 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 − 0) = 𝐴)
3433fveq2d 6665 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴 − 0)) = (abs‘𝐴))
3531, 34eqtrd 2859 . . . . . 6 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) = (abs‘𝐴))
36 simpr 488 . . . . . 6 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1)
3735, 36eqbrtrd 5074 . . . . 5 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) < 1)
38 cnxmet 23381 . . . . . . 7 (abs ∘ − ) ∈ (∞Met‘ℂ)
39 1xr 10698 . . . . . . 7 1 ∈ ℝ*
40 elbl3 23002 . . . . . . 7 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ*) ∧ (0 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
4138, 39, 40mpanl12 701 . . . . . 6 ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
4228, 27, 41sylancr 590 . . . . 5 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
4337, 42mpbird 260 . . . 4 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ (0(ball‘(abs ∘ − ))1))
44 tru 1542 . . . . . 6
45 eqid 2824 . . . . . . . 8 (0(ball‘(abs ∘ − ))1) = (0(ball‘(abs ∘ − ))1)
46 0cnd 10632 . . . . . . . 8 (⊤ → 0 ∈ ℂ)
4739a1i 11 . . . . . . . 8 (⊤ → 1 ∈ ℝ*)
48 ax-1cn 10593 . . . . . . . . . . . . 13 1 ∈ ℂ
49 blssm 23028 . . . . . . . . . . . . . . 15 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ*) → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
5038, 28, 39, 49mp3an 1458 . . . . . . . . . . . . . 14 (0(ball‘(abs ∘ − ))1) ⊆ ℂ
5150sseli 3949 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 𝑦 ∈ ℂ)
52 subcl 10883 . . . . . . . . . . . . 13 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − 𝑦) ∈ ℂ)
5348, 51, 52sylancr 590 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ∈ ℂ)
5451abscld 14796 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ ℝ)
5529cnmetdval 23379 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑦(abs ∘ − )0) = (abs‘(𝑦 − 0)))
5651, 28, 55sylancl 589 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) = (abs‘(𝑦 − 0)))
5751subid1d 10984 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦 − 0) = 𝑦)
5857fveq2d 6665 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(𝑦 − 0)) = (abs‘𝑦))
5956, 58eqtrd 2859 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) = (abs‘𝑦))
60 elbl3 23002 . . . . . . . . . . . . . . . . . . 19 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ*) ∧ (0 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
6138, 39, 60mpanl12 701 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
6228, 51, 61sylancr 590 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
6362ibi 270 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) < 1)
6459, 63eqbrtrrd 5076 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < 1)
6554, 64gtned 10773 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ≠ (abs‘𝑦))
66 abs1 14657 . . . . . . . . . . . . . . . 16 (abs‘1) = 1
67 fveq2 6661 . . . . . . . . . . . . . . . 16 (1 = 𝑦 → (abs‘1) = (abs‘𝑦))
6866, 67syl5eqr 2873 . . . . . . . . . . . . . . 15 (1 = 𝑦 → 1 = (abs‘𝑦))
6968necon3i 3046 . . . . . . . . . . . . . 14 (1 ≠ (abs‘𝑦) → 1 ≠ 𝑦)
7065, 69syl 17 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ≠ 𝑦)
71 subeq0 10910 . . . . . . . . . . . . . . 15 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((1 − 𝑦) = 0 ↔ 1 = 𝑦))
7271necon3bid 3058 . . . . . . . . . . . . . 14 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((1 − 𝑦) ≠ 0 ↔ 1 ≠ 𝑦))
7348, 51, 72sylancr 590 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 − 𝑦) ≠ 0 ↔ 1 ≠ 𝑦))
7470, 73mpbird 260 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ≠ 0)
7553, 74logcld 25165 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (log‘(1 − 𝑦)) ∈ ℂ)
7675negcld 10982 . . . . . . . . . 10 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -(log‘(1 − 𝑦)) ∈ ℂ)
7776adantl 485 . . . . . . . . 9 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → -(log‘(1 − 𝑦)) ∈ ℂ)
7877fmpttd 6870 . . . . . . . 8 (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))):(0(ball‘(abs ∘ − ))1)⟶ℂ)
7951absge0d 14804 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ≤ (abs‘𝑦))
8054rexrd 10689 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ ℝ*)
81 peano2re 10811 . . . . . . . . . . . . . . . 16 ((abs‘𝑦) ∈ ℝ → ((abs‘𝑦) + 1) ∈ ℝ)
8254, 81syl 17 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) + 1) ∈ ℝ)
8382rehalfcld 11881 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℝ)
8483rexrd 10689 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℝ*)
85 iccssxr 12817 . . . . . . . . . . . . . . 15 (0[,]+∞) ⊆ ℝ*
86 eqeq1 2828 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑗 → (𝑚 = 0 ↔ 𝑗 = 0))
87 oveq2 7157 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑗 → (1 / 𝑚) = (1 / 𝑗))
8886, 87ifbieq2d 4475 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑗 → if(𝑚 = 0, 0, (1 / 𝑚)) = if(𝑗 = 0, 0, (1 / 𝑗)))
89 eqid 2824 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))
90 c0ex 10633 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
91 ovex 7182 . . . . . . . . . . . . . . . . . . . . . 22 (1 / 𝑗) ∈ V
9290, 91ifex 4498 . . . . . . . . . . . . . . . . . . . . 21 if(𝑗 = 0, 0, (1 / 𝑗)) ∈ V
9388, 89, 92fvmpt 6759 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) = if(𝑗 = 0, 0, (1 / 𝑗)))
9493eqcomd 2830 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ℕ0 → if(𝑗 = 0, 0, (1 / 𝑗)) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗))
9594oveq1d 7164 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℕ0 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)) = (((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥𝑗)))
9695mpteq2ia 5143 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))) = (𝑗 ∈ ℕ0 ↦ (((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥𝑗)))
9796mpteq2i 5144 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)))) = (𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥𝑗))))
98 0cnd 10632 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → 0 ∈ ℂ)
99 nn0cn 11904 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
10099adantl 485 . . . . . . . . . . . . . . . . . . 19 ((⊤ ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℂ)
101 neqne 3022 . . . . . . . . . . . . . . . . . . 19 𝑚 = 0 → 𝑚 ≠ 0)
102 reccl 11303 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℂ ∧ 𝑚 ≠ 0) → (1 / 𝑚) ∈ ℂ)
103100, 101, 102syl2an 598 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑚 ∈ ℕ0) ∧ ¬ 𝑚 = 0) → (1 / 𝑚) ∈ ℂ)
10498, 103ifclda 4484 . . . . . . . . . . . . . . . . 17 ((⊤ ∧ 𝑚 ∈ ℕ0) → if(𝑚 = 0, 0, (1 / 𝑚)) ∈ ℂ)
105104fmpttd 6870 . . . . . . . . . . . . . . . 16 (⊤ → (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚))):ℕ0⟶ℂ)
106 recn 10625 . . . . . . . . . . . . . . . . . . . . . 22 (𝑟 ∈ ℝ → 𝑟 ∈ ℂ)
107 oveq1 7156 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑟 → (𝑥𝑗) = (𝑟𝑗))
108107oveq2d 7165 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑟 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))
109108mpteq2dv 5148 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑟 → (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗))))
110 eqid 2824 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)))) = (𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))
111 nn0ex 11900 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
112111mptex 6977 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗))) ∈ V
113109, 110, 112fvmpt 6759 . . . . . . . . . . . . . . . . . . . . . 22 (𝑟 ∈ ℂ → ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑟) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗))))
114106, 113syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 ∈ ℝ → ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑟) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗))))
115114eqcomd 2830 . . . . . . . . . . . . . . . . . . . 20 (𝑟 ∈ ℝ → (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗))) = ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑟))
116115seqeq3d 13381 . . . . . . . . . . . . . . . . . . 19 (𝑟 ∈ ℝ → seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) = seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑟)))
117116eleq1d 2900 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ ℝ → (seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ ↔ seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑟)) ∈ dom ⇝ ))
118117rabbiia 3457 . . . . . . . . . . . . . . . . 17 {𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ } = {𝑟 ∈ ℝ ∣ seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑟)) ∈ dom ⇝ }
119118supeq1i 8908 . . . . . . . . . . . . . . . 16 sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) = sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )
12097, 105, 119radcnvcl 25015 . . . . . . . . . . . . . . 15 (⊤ → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ (0[,]+∞))
12185, 120sseldi 3951 . . . . . . . . . . . . . 14 (⊤ → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
12244, 121mp1i 13 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
123 1re 10639 . . . . . . . . . . . . . . 15 1 ∈ ℝ
124 avglt1 11872 . . . . . . . . . . . . . . 15 (((abs‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (abs‘𝑦) < (((abs‘𝑦) + 1) / 2)))
12554, 123, 124sylancl 589 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) < 1 ↔ (abs‘𝑦) < (((abs‘𝑦) + 1) / 2)))
12664, 125mpbid 235 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < (((abs‘𝑦) + 1) / 2))
127 0red 10642 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ∈ ℝ)
128127, 54, 83, 79, 126lelttrd 10796 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 < (((abs‘𝑦) + 1) / 2))
129127, 83, 128ltled 10786 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ≤ (((abs‘𝑦) + 1) / 2))
13083, 129absidd 14782 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) = (((abs‘𝑦) + 1) / 2))
13144, 105mp1i 13 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚))):ℕ0⟶ℂ)
13283recnd 10667 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℂ)
133 oveq1 7156 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = (((abs‘𝑦) + 1) / 2) → (𝑥𝑗) = ((((abs‘𝑦) + 1) / 2)↑𝑗))
134133oveq2d 7165 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (((abs‘𝑦) + 1) / 2) → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))
135134mpteq2dv 5148 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (((abs‘𝑦) + 1) / 2) → (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗))))
136111mptex 6977 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗))) ∈ V
137135, 110, 136fvmpt 6759 . . . . . . . . . . . . . . . . . 18 ((((abs‘𝑦) + 1) / 2) ∈ ℂ → ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘(((abs‘𝑦) + 1) / 2)) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗))))
138132, 137syl 17 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘(((abs‘𝑦) + 1) / 2)) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗))))
139138seqeq3d 13381 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘(((abs‘𝑦) + 1) / 2))) = seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))))
140 avglt2 11873 . . . . . . . . . . . . . . . . . . . 20 (((abs‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (((abs‘𝑦) + 1) / 2) < 1))
14154, 123, 140sylancl 589 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) < 1 ↔ (((abs‘𝑦) + 1) / 2) < 1))
14264, 141mpbid 235 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) < 1)
143130, 142eqbrtrd 5074 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) < 1)
144 logtayllem 25253 . . . . . . . . . . . . . . . . 17 (((((abs‘𝑦) + 1) / 2) ∈ ℂ ∧ (abs‘(((abs‘𝑦) + 1) / 2)) < 1) → seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))) ∈ dom ⇝ )
145132, 143, 144syl2anc 587 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))) ∈ dom ⇝ )
146139, 145eqeltrd 2916 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘(((abs‘𝑦) + 1) / 2))) ∈ dom ⇝ )
14797, 131, 119, 132, 146radcnvle 25018 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))
148130, 147eqbrtrrd 5076 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))
14980, 84, 122, 126, 148xrltletrd 12551 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))
150 0re 10641 . . . . . . . . . . . . 13 0 ∈ ℝ
151 elico2 12798 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*) → ((abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )) ↔ ((abs‘𝑦) ∈ ℝ ∧ 0 ≤ (abs‘𝑦) ∧ (abs‘𝑦) < sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))))
152150, 122, 151sylancr 590 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )) ↔ ((abs‘𝑦) ∈ ℝ ∧ 0 ≤ (abs‘𝑦) ∧ (abs‘𝑦) < sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))))
15354, 79, 149, 152mpbir3and 1339 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))
154 absf 14697 . . . . . . . . . . . 12 abs:ℂ⟶ℝ
155 ffn 6503 . . . . . . . . . . . 12 (abs:ℂ⟶ℝ → abs Fn ℂ)
156 elpreima 6819 . . . . . . . . . . . 12 (abs Fn ℂ → (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↔ (𝑦 ∈ ℂ ∧ (abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))))
157154, 155, 156mp2b 10 . . . . . . . . . . 11 (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↔ (𝑦 ∈ ℂ ∧ (abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))))
15851, 153, 157sylanbrc 586 . . . . . . . . . 10 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))))
159 cnvimass 5936 . . . . . . . . . . . . . . . . 17 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ⊆ dom abs
160154fdmi 6514 . . . . . . . . . . . . . . . . 17 dom abs = ℂ
161159, 160sseqtri 3989 . . . . . . . . . . . . . . . 16 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ⊆ ℂ
162161sseli 3949 . . . . . . . . . . . . . . 15 (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) → 𝑦 ∈ ℂ)
163 oveq1 7156 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
164163oveq2d 7165 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)))
165164mpteq2dv 5148 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗))))
166111mptex 6977 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗))) ∈ V
167165, 110, 166fvmpt 6759 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℂ → ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗))))
168167adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗))))
169168fveq1d 6663 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑛) = ((𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)))‘𝑛))
170 eqeq1 2828 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑛 → (𝑗 = 0 ↔ 𝑛 = 0))
171 oveq2 7157 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑛 → (1 / 𝑗) = (1 / 𝑛))
172170, 171ifbieq2d 4475 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → if(𝑗 = 0, 0, (1 / 𝑗)) = if(𝑛 = 0, 0, (1 / 𝑛)))
173 oveq2 7157 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → (𝑦𝑗) = (𝑦𝑛))
174172, 173oveq12d 7167 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑛 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
175 eqid 2824 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗))) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)))
176 ovex 7182 . . . . . . . . . . . . . . . . . . 19 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) ∈ V
177174, 175, 176fvmpt 6759 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ0 → ((𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
178177adantl 485 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → ((𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
179169, 178eqtr2d 2860 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑛))
180179sumeq2dv 15060 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℂ → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = Σ𝑛 ∈ ℕ0 (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑛))
181162, 180syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = Σ𝑛 ∈ ℕ0 (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑛))
182181mpteq2ia 5143 . . . . . . . . . . . . 13 (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))) = (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑛))
183 eqid 2824 . . . . . . . . . . . . 13 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) = (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))
184 eqid 2824 . . . . . . . . . . . . 13 if(sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ, (((abs‘𝑧) + sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )) / 2), ((abs‘𝑧) + 1)) = if(sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ, (((abs‘𝑧) + sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )) / 2), ((abs‘𝑧) + 1))
18597, 182, 105, 119, 183, 184psercn 25024 . . . . . . . . . . . 12 (⊤ → (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))) ∈ ((abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))–cn→ℂ))
186 cncff 23501 . . . . . . . . . . . 12 ((𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))) ∈ ((abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))–cn→ℂ) → (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))):(abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))⟶ℂ)
187185, 186syl 17 . . . . . . . . . . 11 (⊤ → (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))):(abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))⟶ℂ)
188187fvmptelrn 6868 . . . . . . . . . 10 ((⊤ ∧ 𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))) → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) ∈ ℂ)
189158, 188sylan2 595 . . . . . . . . 9 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) ∈ ℂ)
190189fmpttd 6870 . . . . . . . 8 (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))):(0(ball‘(abs ∘ − ))1)⟶ℂ)
191 cnelprrecn 10628 . . . . . . . . . . . . 13 ℂ ∈ {ℝ, ℂ}
192191a1i 11 . . . . . . . . . . . 12 (⊤ → ℂ ∈ {ℝ, ℂ})
19375adantl 485 . . . . . . . . . . . 12 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → (log‘(1 − 𝑦)) ∈ ℂ)
194 ovexd 7184 . . . . . . . . . . . 12 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → ((1 / (1 − 𝑦)) · -1) ∈ V)
19529cnmetdval 23379 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℂ ∧ (1 − 𝑦) ∈ ℂ) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘(1 − (1 − 𝑦))))
19648, 53, 195sylancr 590 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘(1 − (1 − 𝑦))))
197 nncan 10913 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − (1 − 𝑦)) = 𝑦)
19848, 51, 197sylancr 590 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − (1 − 𝑦)) = 𝑦)
199198fveq2d 6665 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(1 − (1 − 𝑦))) = (abs‘𝑦))
200196, 199eqtrd 2859 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘𝑦))
201200, 64eqbrtrd 5074 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) < 1)
202 elbl 22998 . . . . . . . . . . . . . . . 16 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ*) → ((1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 − 𝑦) ∈ ℂ ∧ (1(abs ∘ − )(1 − 𝑦)) < 1)))
20338, 48, 39, 202mp3an 1458 . . . . . . . . . . . . . . 15 ((1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 − 𝑦) ∈ ℂ ∧ (1(abs ∘ − )(1 − 𝑦)) < 1))
20453, 201, 203sylanbrc 586 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1))
205204adantl 485 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → (1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1))
206 neg1cn 11748 . . . . . . . . . . . . . 14 -1 ∈ ℂ
207206a1i 11 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → -1 ∈ ℂ)
208 eqid 2824 . . . . . . . . . . . . . . . . . 18 (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
209208dvlog2lem 25246 . . . . . . . . . . . . . . . . 17 (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
210209sseli 3949 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
211210eldifad 3931 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ∈ ℂ)
212 eqid 2824 . . . . . . . . . . . . . . . . 17 (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
213212logdmn0 25234 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (ℂ ∖ (-∞(,]0)) → 𝑥 ≠ 0)
214210, 213syl 17 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ≠ 0)
215211, 214logcld 25165 . . . . . . . . . . . . . 14 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑥) ∈ ℂ)
216215adantl 485 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(ball‘(abs ∘ − ))1)) → (log‘𝑥) ∈ ℂ)
217 ovexd 7184 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(ball‘(abs ∘ − ))1)) → (1 / 𝑥) ∈ V)
218 simpr 488 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
21948, 218, 52sylancr 590 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑦 ∈ ℂ) → (1 − 𝑦) ∈ ℂ)
220206a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑦 ∈ ℂ) → -1 ∈ ℂ)
221 1cnd 10634 . . . . . . . . . . . . . . . 16 ((⊤ ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
222 0cnd 10632 . . . . . . . . . . . . . . . 16 ((⊤ ∧ 𝑦 ∈ ℂ) → 0 ∈ ℂ)
223 1cnd 10634 . . . . . . . . . . . . . . . . 17 (⊤ → 1 ∈ ℂ)
224192, 223dvmptc 24564 . . . . . . . . . . . . . . . 16 (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 1)) = (𝑦 ∈ ℂ ↦ 0))
225192dvmptid 24563 . . . . . . . . . . . . . . . 16 (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1))
226192, 221, 222, 224, 218, 221, 225dvmptsub 24573 . . . . . . . . . . . . . . 15 (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (1 − 𝑦))) = (𝑦 ∈ ℂ ↦ (0 − 1)))
227 df-neg 10871 . . . . . . . . . . . . . . . 16 -1 = (0 − 1)
228227mpteq2i 5144 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℂ ↦ -1) = (𝑦 ∈ ℂ ↦ (0 − 1))
229226, 228syl6eqr 2877 . . . . . . . . . . . . . 14 (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (1 − 𝑦))) = (𝑦 ∈ ℂ ↦ -1))
23050a1i 11 . . . . . . . . . . . . . 14 (⊤ → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
231 eqid 2824 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
232231cnfldtopon 23391 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
233232toponrestid 21529 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
234231cnfldtopn 23390 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
235234blopn 23110 . . . . . . . . . . . . . . . 16 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ*) → (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld))
23638, 28, 39, 235mp3an 1458 . . . . . . . . . . . . . . 15 (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld)
237236a1i 11 . . . . . . . . . . . . . 14 (⊤ → (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld))
238192, 219, 220, 229, 230, 233, 231, 237dvmptres 24569 . . . . . . . . . . . . 13 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -1))
239208dvlog2 25247 . . . . . . . . . . . . . 14 (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑥))
240 logf1o 25159 . . . . . . . . . . . . . . . . . . . 20 log:(ℂ ∖ {0})–1-1-onto→ran log
241 f1of 6606 . . . . . . . . . . . . . . . . . . . 20 (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
242240, 241ax-mp 5 . . . . . . . . . . . . . . . . . . 19 log:(ℂ ∖ {0})⟶ran log
243212logdmss 25236 . . . . . . . . . . . . . . . . . . . 20 (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
244209, 243sstri 3962 . . . . . . . . . . . . . . . . . . 19 (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
245 fssres 6534 . . . . . . . . . . . . . . . . . . 19 ((log:(ℂ ∖ {0})⟶ran log ∧ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})) → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log)
246242, 244, 245mp2an 691 . . . . . . . . . . . . . . . . . 18 (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log
247246a1i 11 . . . . . . . . . . . . . . . . 17 (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log)
248247feqmptd 6724 . . . . . . . . . . . . . . . 16 (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥)))
249 fvres 6680 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥) = (log‘𝑥))
250249mpteq2ia 5143 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))
251248, 250syl6eq 2875 . . . . . . . . . . . . . . 15 (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥)))
252251oveq2d 7165 . . . . . . . . . . . . . 14 (⊤ → (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (ℂ D (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))))
253239, 252syl5reqr 2874 . . . . . . . . . . . . 13 (⊤ → (ℂ D (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑥)))
254 fveq2 6661 . . . . . . . . . . . . 13 (𝑥 = (1 − 𝑦) → (log‘𝑥) = (log‘(1 − 𝑦)))
255 oveq2 7157 . . . . . . . . . . . . 13 (𝑥 = (1 − 𝑦) → (1 / 𝑥) = (1 / (1 − 𝑦)))
256192, 192, 205, 207, 216, 217, 238, 253, 254, 255dvmptco 24578 . . . . . . . . . . . 12 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ ((1 / (1 − 𝑦)) · -1)))
257192, 193, 194, 256dvmptneg 24572 . . . . . . . . . . 11 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -((1 / (1 − 𝑦)) · -1)))
25853, 74reccld 11407 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 / (1 − 𝑦)) ∈ ℂ)
259 mulcom 10621 . . . . . . . . . . . . . . . 16 (((1 / (1 − 𝑦)) ∈ ℂ ∧ -1 ∈ ℂ) → ((1 / (1 − 𝑦)) · -1) = (-1 · (1 / (1 − 𝑦))))
260258, 206, 259sylancl 589 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 / (1 − 𝑦)) · -1) = (-1 · (1 / (1 − 𝑦))))
261258mulm1d 11090 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (-1 · (1 / (1 − 𝑦))) = -(1 / (1 − 𝑦)))
262260, 261eqtrd 2859 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 / (1 − 𝑦)) · -1) = -(1 / (1 − 𝑦)))
263262negeqd 10878 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -((1 / (1 − 𝑦)) · -1) = --(1 / (1 − 𝑦)))
264258negnegd 10986 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → --(1 / (1 − 𝑦)) = (1 / (1 − 𝑦)))
265263, 264eqtrd 2859 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -((1 / (1 − 𝑦)) · -1) = (1 / (1 − 𝑦)))
266265mpteq2ia 5143 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -((1 / (1 − 𝑦)) · -1)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦)))
267257, 266syl6eq 2875 . . . . . . . . . 10 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
268267dmeqd 5761 . . . . . . . . 9 (⊤ → dom (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
269 dmmptg 6083 . . . . . . . . . 10 (∀𝑦 ∈ (0(ball‘(abs ∘ − ))1)(1 / (1 − 𝑦)) ∈ V → dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))) = (0(ball‘(abs ∘ − ))1))
270 ovexd 7184 . . . . . . . . . 10 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 / (1 − 𝑦)) ∈ V)
271269, 270mprg 3147 . . . . . . . . 9 dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))) = (0(ball‘(abs ∘ − ))1)
272268, 271syl6eq 2875 . . . . . . . 8 (⊤ → dom (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (0(ball‘(abs ∘ − ))1))
273 sumex 15044 . . . . . . . . . . . 12 Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) ∈ V
274273a1i 11 . . . . . . . . . . 11 ((⊤ ∧ 𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))) → Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) ∈ V)
275 fveq2 6661 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑛) = (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑘))
276275cbvsumv 15053 . . . . . . . . . . . . . 14 Σ𝑛 ∈ ℕ0 (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑛) = Σ𝑘 ∈ ℕ0 (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑘)
277181, 276syl6eq 2875 . . . . . . . . . . . . 13 (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = Σ𝑘 ∈ ℕ0 (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑘))
278277mpteq2ia 5143 . . . . . . . . . . . 12 (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))) = (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑘 ∈ ℕ0 (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑘))
279 eqid 2824 . . . . . . . . . . . 12 (0(ball‘(abs ∘ − ))(((abs‘𝑧) + if(sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ, (((abs‘𝑧) + sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )) / 2), ((abs‘𝑧) + 1))) / 2)) = (0(ball‘(abs ∘ − ))(((abs‘𝑧) + if(sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ, (((abs‘𝑧) + sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )) / 2), ((abs‘𝑧) + 1))) / 2))
28097, 278, 105, 119, 183, 184, 279pserdv2 25028 . . . . . . . . . . 11 (⊤ → (ℂ D (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))) = (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1)))))
281158ssriv 3957 . . . . . . . . . . . 12 (0(ball‘(abs ∘ − ))1) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))
282281a1i 11 . . . . . . . . . . 11 (⊤ → (0(ball‘(abs ∘ − ))1) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))))
283192, 188, 274, 280, 282, 233, 231, 237dvmptres 24569 . . . . . . . . . 10 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1)))))
284 nnnn0 11901 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
285284adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
286 eqeq1 2828 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (𝑚 = 0 ↔ 𝑛 = 0))
287 oveq2 7157 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
288286, 287ifbieq2d 4475 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → if(𝑚 = 0, 0, (1 / 𝑚)) = if(𝑛 = 0, 0, (1 / 𝑛)))
289 ovex 7182 . . . . . . . . . . . . . . . . . . . . 21 (1 / 𝑛) ∈ V
29090, 289ifex 4498 . . . . . . . . . . . . . . . . . . . 20 if(𝑛 = 0, 0, (1 / 𝑛)) ∈ V
291288, 89, 290fvmpt 6759 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)))
292285, 291syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)))
293 nnne0 11668 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
294293adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
295294neneqd 3019 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
296295iffalsed 4461 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, 0, (1 / 𝑛)) = (1 / 𝑛))
297292, 296eqtrd 2859 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = (1 / 𝑛))
298297oveq2d 7165 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) = (𝑛 · (1 / 𝑛)))
299 nncn 11642 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
300299adantl 485 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
301300, 294recidd 11409 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · (1 / 𝑛)) = 1)
302298, 301eqtrd 2859 . . . . . . . . . . . . . . 15 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) = 1)
303302oveq1d 7164 . . . . . . . . . . . . . 14 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (1 · (𝑦↑(𝑛 − 1))))
304 nnm1nn0 11935 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
305 expcl 13452 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℂ ∧ (𝑛 − 1) ∈ ℕ0) → (𝑦↑(𝑛 − 1)) ∈ ℂ)
30651, 304, 305syl2an 598 . . . . . . . . . . . . . . 15 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑦↑(𝑛 − 1)) ∈ ℂ)
307306mulid2d 10657 . . . . . . . . . . . . . 14 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (1 · (𝑦↑(𝑛 − 1))) = (𝑦↑(𝑛 − 1)))
308303, 307eqtrd 2859 . . . . . . . . . . . . 13 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (𝑦↑(𝑛 − 1)))
309308sumeq2dv 15060 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)))
310 nnuz 12278 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
311 1e0p1 12137 . . . . . . . . . . . . . . . 16 1 = (0 + 1)
312311fveq2i 6664 . . . . . . . . . . . . . . 15 (ℤ‘1) = (ℤ‘(0 + 1))
313310, 312eqtri 2847 . . . . . . . . . . . . . 14 ℕ = (ℤ‘(0 + 1))
314 oveq1 7156 . . . . . . . . . . . . . . 15 (𝑛 = (1 + 𝑚) → (𝑛 − 1) = ((1 + 𝑚) − 1))
315314oveq2d 7165 . . . . . . . . . . . . . 14 (𝑛 = (1 + 𝑚) → (𝑦↑(𝑛 − 1)) = (𝑦↑((1 + 𝑚) − 1)))
316 1zzd 12010 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ∈ ℤ)
317 0zd 11990 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ∈ ℤ)
3181, 313, 315, 316, 317, 306isumshft 15194 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)) = Σ𝑚 ∈ ℕ0 (𝑦↑((1 + 𝑚) − 1)))
319 pncan2 10891 . . . . . . . . . . . . . . . 16 ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((1 + 𝑚) − 1) = 𝑚)
32048, 99, 319sylancr 590 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → ((1 + 𝑚) − 1) = 𝑚)
321320oveq2d 7165 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (𝑦↑((1 + 𝑚) − 1)) = (𝑦𝑚))
322321sumeq2i 15056 . . . . . . . . . . . . 13 Σ𝑚 ∈ ℕ0 (𝑦↑((1 + 𝑚) − 1)) = Σ𝑚 ∈ ℕ0 (𝑦𝑚)
323318, 322syl6eq 2875 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)) = Σ𝑚 ∈ ℕ0 (𝑦𝑚))
324 geoisum 15233 . . . . . . . . . . . . 13 ((𝑦 ∈ ℂ ∧ (abs‘𝑦) < 1) → Σ𝑚 ∈ ℕ0 (𝑦𝑚) = (1 / (1 − 𝑦)))
32551, 64, 324syl2anc 587 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑚 ∈ ℕ0 (𝑦𝑚) = (1 / (1 − 𝑦)))
326309, 323, 3253eqtrd 2863 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (1 / (1 − 𝑦)))
327326mpteq2ia 5143 . . . . . . . . . 10 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦)))
328283, 327syl6eq 2875 . . . . . . . . 9 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
329267, 328eqtr4d 2862 . . . . . . . 8 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))))
330 1rp 12390 . . . . . . . . . 10 1 ∈ ℝ+
331 blcntr 23023 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ+) → 0 ∈ (0(ball‘(abs ∘ − ))1))
33238, 28, 330, 331mp3an 1458 . . . . . . . . 9 0 ∈ (0(ball‘(abs ∘ − ))1)
333332a1i 11 . . . . . . . 8 (⊤ → 0 ∈ (0(ball‘(abs ∘ − ))1))
334 oveq2 7157 . . . . . . . . . . . . . . . 16 (𝑦 = 0 → (1 − 𝑦) = (1 − 0))
335 1m0e1 11755 . . . . . . . . . . . . . . . 16 (1 − 0) = 1
336334, 335syl6eq 2875 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (1 − 𝑦) = 1)
337336fveq2d 6665 . . . . . . . . . . . . . 14 (𝑦 = 0 → (log‘(1 − 𝑦)) = (log‘1))
338 log1 25180 . . . . . . . . . . . . . 14 (log‘1) = 0
339337, 338syl6eq 2875 . . . . . . . . . . . . 13 (𝑦 = 0 → (log‘(1 − 𝑦)) = 0)
340339negeqd 10878 . . . . . . . . . . . 12 (𝑦 = 0 → -(log‘(1 − 𝑦)) = -0)
341 neg0 10930 . . . . . . . . . . . 12 -0 = 0
342340, 341syl6eq 2875 . . . . . . . . . . 11 (𝑦 = 0 → -(log‘(1 − 𝑦)) = 0)
343 eqid 2824 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))
344342, 343, 90fvmpt 6759 . . . . . . . . . 10 (0 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = 0)
345332, 344mp1i 13 . . . . . . . . 9 (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = 0)
346 oveq1 7156 . . . . . . . . . . . . . . 15 (0 = if(𝑛 = 0, 0, (1 / 𝑛)) → (0 · (𝑦𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
347346eqeq1d 2826 . . . . . . . . . . . . . 14 (0 = if(𝑛 = 0, 0, (1 / 𝑛)) → ((0 · (𝑦𝑛)) = 0 ↔ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = 0))
348 oveq1 7156 . . . . . . . . . . . . . . 15 ((1 / 𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)) → ((1 / 𝑛) · (𝑦𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
349348eqeq1d 2826 . . . . . . . . . . . . . 14 ((1 / 𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)) → (((1 / 𝑛) · (𝑦𝑛)) = 0 ↔ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = 0))
350 simpll 766 . . . . . . . . . . . . . . . . 17 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → 𝑦 = 0)
351350, 28eqeltrdi 2924 . . . . . . . . . . . . . . . 16 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → 𝑦 ∈ ℂ)
352 simplr 768 . . . . . . . . . . . . . . . 16 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → 𝑛 ∈ ℕ0)
353351, 352expcld 13515 . . . . . . . . . . . . . . 15 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → (𝑦𝑛) ∈ ℂ)
354353mul02d 10836 . . . . . . . . . . . . . 14 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → (0 · (𝑦𝑛)) = 0)
355 simpll 766 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑦 = 0)
356355oveq1d 7164 . . . . . . . . . . . . . . . . 17 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (𝑦𝑛) = (0↑𝑛))
357 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
358357, 14sylib 221 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) → (𝑛 ∈ ℕ ∨ 𝑛 = 0))
359358ord 861 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 ∈ ℕ → 𝑛 = 0))
360359con1d 147 . . . . . . . . . . . . . . . . . . 19 ((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 = 0 → 𝑛 ∈ ℕ))
361360imp 410 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
3623610expd 13508 . . . . . . . . . . . . . . . . 17 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (0↑𝑛) = 0)
363356, 362eqtrd 2859 . . . . . . . . . . . . . . . 16 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (𝑦𝑛) = 0)
364363oveq2d 7165 . . . . . . . . . . . . . . 15 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · (𝑦𝑛)) = ((1 / 𝑛) · 0))
365361nnrecred 11685 . . . . . . . . . . . . . . . . 17 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℝ)
366365recnd 10667 . . . . . . . . . . . . . . . 16 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℂ)
367366mul01d 10837 . . . . . . . . . . . . . . 15 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · 0) = 0)
368364, 367eqtrd 2859 . . . . . . . . . . . . . 14 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · (𝑦𝑛)) = 0)
369347, 349, 354, 368ifbothda 4487 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = 0)
370369sumeq2dv 15060 . . . . . . . . . . . 12 (𝑦 = 0 → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = Σ𝑛 ∈ ℕ0 0)
3711eqimssi 4011 . . . . . . . . . . . . . 14 0 ⊆ (ℤ‘0)
372371orci 862 . . . . . . . . . . . . 13 (ℕ0 ⊆ (ℤ‘0) ∨ ℕ0 ∈ Fin)
373 sumz 15079 . . . . . . . . . . . . 13 ((ℕ0 ⊆ (ℤ‘0) ∨ ℕ0 ∈ Fin) → Σ𝑛 ∈ ℕ0 0 = 0)
374372, 373ax-mp 5 . . . . . . . . . . . 12 Σ𝑛 ∈ ℕ0 0 = 0
375370, 374syl6eq 2875 . . . . . . . . . . 11 (𝑦 = 0 → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = 0)
376 eqid 2824 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
377375, 376, 90fvmpt 6759 . . . . . . . . . 10 (0 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))‘0) = 0)
378332, 377mp1i 13 . . . . . . . . 9 (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))‘0) = 0)
379345, 378eqtr4d 2862 . . . . . . . 8 (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))‘0))
38045, 46, 47, 78, 190, 272, 329, 333, 379dv11cn 24607 . . . . . . 7 (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))))
381380fveq1d 6663 . . . . . 6 (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘𝐴) = ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))‘𝐴))
38244, 381mp1i 13 . . . . 5 (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘𝐴) = ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))‘𝐴))
383 oveq2 7157 . . . . . . . 8 (𝑦 = 𝐴 → (1 − 𝑦) = (1 − 𝐴))
384383fveq2d 6665 . . . . . . 7 (𝑦 = 𝐴 → (log‘(1 − 𝑦)) = (log‘(1 − 𝐴)))
385384negeqd 10878 . . . . . 6 (𝑦 = 𝐴 → -(log‘(1 − 𝑦)) = -(log‘(1 − 𝐴)))
386 negex 10882 . . . . . 6 -(log‘(1 − 𝐴)) ∈ V
387385, 343, 386fvmpt 6759 . . . . 5 (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘𝐴) = -(log‘(1 − 𝐴)))
388 oveq1 7156 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑛) = (𝐴𝑛))
389388oveq2d 7165 . . . . . . 7 (𝑦 = 𝐴 → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
390389sumeq2sdv 15061 . . . . . 6 (𝑦 = 𝐴 → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
391 sumex 15044 . . . . . 6 Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)) ∈ V
392390, 376, 391fvmpt 6759 . . . . 5 (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))‘𝐴) = Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
393382, 387, 3923eqtr3d 2867 . . . 4 (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → -(log‘(1 − 𝐴)) = Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
39443, 393syl 17 . . 3 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(log‘(1 − 𝐴)) = Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
39526, 394breqtrrd 5080 . 2 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ⇝ -(log‘(1 − 𝐴)))
396 seqex 13375 . . . 4 seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ∈ V
397396a1i 11 . . 3 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ∈ V)
398 seqex 13375 . . . 4 seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))) ∈ V
399398a1i 11 . . 3 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))) ∈ V)
400 1zzd 12010 . . 3 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℤ)
401 elnnuz 12279 . . . . . 6 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
402 fvres 6680 . . . . . 6 (𝑛 ∈ (ℤ‘1) → ((seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ↾ (ℤ‘1))‘𝑛) = (seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘))))‘𝑛))
403401, 402sylbi 220 . . . . 5 (𝑛 ∈ ℕ → ((seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ↾ (ℤ‘1))‘𝑛) = (seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘))))‘𝑛))
404403eqcomd 2830 . . . 4 (𝑛 ∈ ℕ → (seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘))))‘𝑛) = ((seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ↾ (ℤ‘1))‘𝑛))
405 addid2 10821 . . . . . . . 8 (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
406405adantl 485 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
407 0cnd 10632 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ∈ ℂ)
408 1eluzge0 12289 . . . . . . . 8 1 ∈ (ℤ‘0)
409408a1i 11 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ (ℤ‘0))
410 0cnd 10632 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) ∧ 𝑘 = 0) → 0 ∈ ℂ)
411 nn0cn 11904 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
412411adantl 485 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
413 neqne 3022 . . . . . . . . . . . 12 𝑘 = 0 → 𝑘 ≠ 0)
414 reccl 11303 . . . . . . . . . . . 12 ((𝑘 ∈ ℂ ∧ 𝑘 ≠ 0) → (1 / 𝑘) ∈ ℂ)
415412, 413, 414syl2an 598 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 = 0) → (1 / 𝑘) ∈ ℂ)
416410, 415ifclda 4484 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 0, 0, (1 / 𝑘)) ∈ ℂ)
417 expcl 13452 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
418417adantlr 714 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
419416, 418mulcld 10659 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)) ∈ ℂ)
420419fmpttd 6870 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘))):ℕ0⟶ℂ)
421 1nn0 11910 . . . . . . . 8 1 ∈ ℕ0
422 ffvelrn 6840 . . . . . . . 8 (((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘))):ℕ0⟶ℂ ∧ 1 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘1) ∈ ℂ)
423420, 421, 422sylancl 589 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘1) ∈ ℂ)
424 elfz1eq 12922 . . . . . . . . . 10 (𝑛 ∈ (0...0) → 𝑛 = 0)
425 1m1e0 11706 . . . . . . . . . . 11 (1 − 1) = 0
426425oveq2i 7160 . . . . . . . . . 10 (0...(1 − 1)) = (0...0)
427424, 426eleq2s 2934 . . . . . . . . 9 (𝑛 ∈ (0...(1 − 1)) → 𝑛 = 0)
428427fveq2d 6665 . . . . . . . 8 (𝑛 ∈ (0...(1 − 1)) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘0))
429 0nn0 11909 . . . . . . . . . 10 0 ∈ ℕ0
430 iftrue 4456 . . . . . . . . . . . 12 (𝑘 = 0 → if(𝑘 = 0, 0, (1 / 𝑘)) = 0)
431 oveq2 7157 . . . . . . . . . . . 12 (𝑘 = 0 → (𝐴𝑘) = (𝐴↑0))
432430, 431oveq12d 7167 . . . . . . . . . . 11 (𝑘 = 0 → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)) = (0 · (𝐴↑0)))
433 ovex 7182 . . . . . . . . . . 11 (0 · (𝐴↑0)) ∈ V
434432, 8, 433fvmpt 6759 . . . . . . . . . 10 (0 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘0) = (0 · (𝐴↑0)))
435429, 434ax-mp 5 . . . . . . . . 9 ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘0) = (0 · (𝐴↑0))
436 expcl 13452 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 0 ∈ ℕ0) → (𝐴↑0) ∈ ℂ)
43727, 429, 436sylancl 589 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑0) ∈ ℂ)
438437mul02d 10836 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (0 · (𝐴↑0)) = 0)
439435, 438syl5eq 2871 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘0) = 0)
440428, 439sylan9eqr 2881 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ (0...(1 − 1))) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = 0)
441406, 407, 409, 423, 440seqid 13420 . . . . . 6 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ↾ (ℤ‘1)) = seq1( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))))
442293adantl 485 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
443442neneqd 3019 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
444443iffalsed 4461 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, 0, (1 / 𝑛)) = (1 / 𝑛))
445444oveq1d 7164 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)) = ((1 / 𝑛) · (𝐴𝑛)))
446284, 23sylan2 595 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℂ)
447299adantl 485 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
448446, 447, 442divrec2d 11418 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝐴𝑛) / 𝑛) = ((1 / 𝑛) · (𝐴𝑛)))
449445, 448eqtr4d 2862 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)) = ((𝐴𝑛) / 𝑛))
450284, 11sylan2 595 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
451 id 22 . . . . . . . . . . . 12 (𝑘 = 𝑛𝑘 = 𝑛)
4526, 451oveq12d 7167 . . . . . . . . . . 11 (𝑘 = 𝑛 → ((𝐴𝑘) / 𝑘) = ((𝐴𝑛) / 𝑛))
453 eqid 2824 . . . . . . . . . . 11 (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘)) = (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))
454 ovex 7182 . . . . . . . . . . 11 ((𝐴𝑛) / 𝑛) ∈ V
455452, 453, 454fvmpt 6759 . . . . . . . . . 10 (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))‘𝑛) = ((𝐴𝑛) / 𝑛))
456455adantl 485 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))‘𝑛) = ((𝐴𝑛) / 𝑛))
457449, 450, 4563eqtr4d 2869 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = ((𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))‘𝑛))
458401, 457sylan2br 597 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ (ℤ‘1)) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = ((𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))‘𝑛))
459400, 458seqfeq 13400 . . . . . 6 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))))
460441, 459eqtrd 2859 . . . . 5 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ↾ (ℤ‘1)) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))))
461460fveq1d 6663 . . . 4 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ↾ (ℤ‘1))‘𝑛) = (seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘)))‘𝑛))
462404, 461sylan9eqr 2881 . . 3 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘))))‘𝑛) = (seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘)))‘𝑛))
463310, 397, 399, 400, 462climeq 14924 . 2 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ⇝ -(log‘(1 − 𝐴)) ↔ seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))) ⇝ -(log‘(1 − 𝐴))))
464395, 463mpbid 235 1 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))) ⇝ -(log‘(1 − 𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ⊤wtru 1539   ∈ wcel 2115   ≠ wne 3014  {crab 3137  Vcvv 3480   ∖ cdif 3916   ⊆ wss 3919  ifcif 4450  {csn 4550  {cpr 4552   class class class wbr 5052   ↦ cmpt 5132  ◡ccnv 5541  dom cdm 5542  ran crn 5543   ↾ cres 5544   “ cima 5545   ∘ ccom 5546   Fn wfn 6338  ⟶wf 6339  –1-1-onto→wf1o 6342  ‘cfv 6343  (class class class)co 7149  Fincfn 8505  supcsup 8901  ℂcc 10533  ℝcr 10534  0cc0 10535  1c1 10536   + caddc 10538   · cmul 10540  +∞cpnf 10670  -∞cmnf 10671  ℝ*cxr 10672   < clt 10673   ≤ cle 10674   − cmin 10868  -cneg 10869   / cdiv 11295  ℕcn 11634  2c2 11689  ℕ0cn0 11894  ℤ≥cuz 12240  ℝ+crp 12386  (,]cioc 12736  [,)cico 12737  [,]cicc 12738  ...cfz 12894  seqcseq 13373  ↑cexp 13434  abscabs 14593   ⇝ cli 14841  Σcsu 15042  TopOpenctopn 16695  ∞Metcxmet 20530  ballcbl 20532  ℂfldccnfld 20545  –cn→ccncf 23484   D cdv 24469  logclog 25149 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613  ax-addf 10614  ax-mulf 10615 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-fi 8872  df-sup 8903  df-inf 8904  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-q 12346  df-rp 12387  df-xneg 12504  df-xadd 12505  df-xmul 12506  df-ioo 12739  df-ioc 12740  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-fl 13166  df-mod 13242  df-seq 13374  df-exp 13435  df-fac 13639  df-bc 13668  df-hash 13696  df-shft 14426  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-limsup 14828  df-clim 14845  df-rlim 14846  df-sum 15043  df-ef 15421  df-sin 15423  df-cos 15424  df-tan 15425  df-pi 15426  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-hom 16589  df-cco 16590  df-rest 16696  df-topn 16697  df-0g 16715  df-gsum 16716  df-topgen 16717  df-pt 16718  df-prds 16721  df-xrs 16775  df-qtop 16780  df-imas 16781  df-xps 16783  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-mulg 18225  df-cntz 18447  df-cmn 18908  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-fbas 20542  df-fg 20543  df-cnfld 20546  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-ntr 21628  df-cls 21629  df-nei 21706  df-lp 21744  df-perf 21745  df-cn 21835  df-cnp 21836  df-haus 21923  df-cmp 21995  df-tx 22170  df-hmeo 22363  df-fil 22454  df-fm 22546  df-flim 22547  df-flf 22548  df-xms 22930  df-ms 22931  df-tms 22932  df-cncf 23486  df-limc 24472  df-dv 24473  df-ulm 24975  df-log 25151 This theorem is referenced by:  logtaylsum  25255  logtayl2  25256  atantayl  25526  stirlinglem5  42646
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