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Theorem logtayl 24620
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 11940 . . . 4 0 = (ℤ‘0)
2 0zd 11655 . . . 4 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ∈ ℤ)
3 eqeq1 2810 . . . . . . . 8 (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
4 oveq2 6882 . . . . . . . 8 (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
53, 4ifbieq2d 4304 . . . . . . 7 (𝑘 = 𝑛 → if(𝑘 = 0, 0, (1 / 𝑘)) = if(𝑛 = 0, 0, (1 / 𝑛)))
6 oveq2 6882 . . . . . . 7 (𝑘 = 𝑛 → (𝐴𝑘) = (𝐴𝑛))
75, 6oveq12d 6892 . . . . . 6 (𝑘 = 𝑛 → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
8 eqid 2806 . . . . . 6 (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘))) = (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))
9 ovex 6906 . . . . . 6 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)) ∈ V
107, 8, 9fvmpt 6503 . . . . 5 (𝑛 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
1110adantl 469 . . . 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 10318 . . . . . 6 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → 0 ∈ ℂ)
13 simpr 473 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
14 elnn0 11561 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
1513, 14sylib 209 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → (𝑛 ∈ ℕ ∨ 𝑛 = 0))
1615ord 882 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 ∈ ℕ → 𝑛 = 0))
1716con1d 141 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 = 0 → 𝑛 ∈ ℕ))
1817imp 395 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
1918nnrecred 11352 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℝ)
2019recnd 10353 . . . . . 6 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℂ)
2112, 20ifclda 4313 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 0, 0, (1 / 𝑛)) ∈ ℂ)
22 expcl 13101 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℂ)
2322adantlr 697 . . . . 5 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → (𝐴𝑛) ∈ ℂ)
2421, 23mulcld 10345 . . . 4 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ0) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)) ∈ ℂ)
25 logtayllem 24619 . . . 4 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ∈ dom ⇝ )
261, 2, 11, 24, 25isumclim2 14712 . . 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 470 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ)
28 0cn 10317 . . . . . . . 8 0 ∈ ℂ
29 eqid 2806 . . . . . . . . 9 (abs ∘ − ) = (abs ∘ − )
3029cnmetdval 22787 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴(abs ∘ − )0) = (abs‘(𝐴 − 0)))
3127, 28, 30sylancl 576 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) = (abs‘(𝐴 − 0)))
32 subid1 10586 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
3332adantr 468 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 − 0) = 𝐴)
3433fveq2d 6412 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴 − 0)) = (abs‘𝐴))
3531, 34eqtrd 2840 . . . . . 6 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) = (abs‘𝐴))
36 simpr 473 . . . . . 6 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1)
3735, 36eqbrtrd 4866 . . . . 5 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) < 1)
38 cnxmet 22789 . . . . . . 7 (abs ∘ − ) ∈ (∞Met‘ℂ)
39 1rp 12050 . . . . . . . 8 1 ∈ ℝ+
40 rpxr 12054 . . . . . . . 8 (1 ∈ ℝ+ → 1 ∈ ℝ*)
4139, 40ax-mp 5 . . . . . . 7 1 ∈ ℝ*
42 elbl3 22410 . . . . . . 7 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ*) ∧ (0 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
4338, 41, 42mpanl12 685 . . . . . 6 ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
4428, 27, 43sylancr 577 . . . . 5 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
4537, 44mpbird 248 . . . 4 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ (0(ball‘(abs ∘ − ))1))
46 tru 1642 . . . . . 6
47 eqid 2806 . . . . . . . 8 (0(ball‘(abs ∘ − ))1) = (0(ball‘(abs ∘ − ))1)
48 0cnd 10318 . . . . . . . 8 (⊤ → 0 ∈ ℂ)
4941a1i 11 . . . . . . . 8 (⊤ → 1 ∈ ℝ*)
50 ax-1cn 10279 . . . . . . . . . . . . 13 1 ∈ ℂ
51 blssm 22436 . . . . . . . . . . . . . . 15 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ*) → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
5238, 28, 41, 51mp3an 1578 . . . . . . . . . . . . . 14 (0(ball‘(abs ∘ − ))1) ⊆ ℂ
5352sseli 3794 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 𝑦 ∈ ℂ)
54 subcl 10565 . . . . . . . . . . . . 13 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − 𝑦) ∈ ℂ)
5550, 53, 54sylancr 577 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ∈ ℂ)
5653abscld 14398 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ ℝ)
5729cnmetdval 22787 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑦(abs ∘ − )0) = (abs‘(𝑦 − 0)))
5853, 28, 57sylancl 576 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) = (abs‘(𝑦 − 0)))
5953subid1d 10666 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦 − 0) = 𝑦)
6059fveq2d 6412 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(𝑦 − 0)) = (abs‘𝑦))
6158, 60eqtrd 2840 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) = (abs‘𝑦))
62 elbl3 22410 . . . . . . . . . . . . . . . . . . 19 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ*) ∧ (0 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
6338, 41, 62mpanl12 685 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
6428, 53, 63sylancr 577 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
6564ibi 258 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) < 1)
6661, 65eqbrtrrd 4868 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < 1)
6756, 66gtned 10457 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ≠ (abs‘𝑦))
68 abs1 14260 . . . . . . . . . . . . . . . 16 (abs‘1) = 1
69 fveq2 6408 . . . . . . . . . . . . . . . 16 (1 = 𝑦 → (abs‘1) = (abs‘𝑦))
7068, 69syl5eqr 2854 . . . . . . . . . . . . . . 15 (1 = 𝑦 → 1 = (abs‘𝑦))
7170necon3i 3010 . . . . . . . . . . . . . 14 (1 ≠ (abs‘𝑦) → 1 ≠ 𝑦)
7267, 71syl 17 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ≠ 𝑦)
73 subeq0 10592 . . . . . . . . . . . . . . 15 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((1 − 𝑦) = 0 ↔ 1 = 𝑦))
7473necon3bid 3022 . . . . . . . . . . . . . 14 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((1 − 𝑦) ≠ 0 ↔ 1 ≠ 𝑦))
7550, 53, 74sylancr 577 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 − 𝑦) ≠ 0 ↔ 1 ≠ 𝑦))
7672, 75mpbird 248 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ≠ 0)
7755, 76logcld 24531 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (log‘(1 − 𝑦)) ∈ ℂ)
7877negcld 10664 . . . . . . . . . 10 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -(log‘(1 − 𝑦)) ∈ ℂ)
7978adantl 469 . . . . . . . . 9 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → -(log‘(1 − 𝑦)) ∈ ℂ)
8079fmpttd 6607 . . . . . . . 8 (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))):(0(ball‘(abs ∘ − ))1)⟶ℂ)
8153absge0d 14406 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ≤ (abs‘𝑦))
8256rexrd 10374 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ ℝ*)
83 peano2re 10494 . . . . . . . . . . . . . . . 16 ((abs‘𝑦) ∈ ℝ → ((abs‘𝑦) + 1) ∈ ℝ)
8456, 83syl 17 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) + 1) ∈ ℝ)
8584rehalfcld 11546 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℝ)
8685rexrd 10374 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℝ*)
87 iccssxr 12474 . . . . . . . . . . . . . . 15 (0[,]+∞) ⊆ ℝ*
88 eqeq1 2810 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑗 → (𝑚 = 0 ↔ 𝑗 = 0))
89 oveq2 6882 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑗 → (1 / 𝑚) = (1 / 𝑗))
9088, 89ifbieq2d 4304 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑗 → if(𝑚 = 0, 0, (1 / 𝑚)) = if(𝑗 = 0, 0, (1 / 𝑗)))
91 eqid 2806 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))
92 c0ex 10319 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
93 ovex 6906 . . . . . . . . . . . . . . . . . . . . . 22 (1 / 𝑗) ∈ V
9492, 93ifex 4327 . . . . . . . . . . . . . . . . . . . . 21 if(𝑗 = 0, 0, (1 / 𝑗)) ∈ V
9590, 91, 94fvmpt 6503 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) = if(𝑗 = 0, 0, (1 / 𝑗)))
9695eqcomd 2812 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ℕ0 → if(𝑗 = 0, 0, (1 / 𝑗)) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗))
9796oveq1d 6889 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℕ0 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)) = (((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥𝑗)))
9897mpteq2ia 4934 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))) = (𝑗 ∈ ℕ0 ↦ (((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥𝑗)))
9998mpteq2i 4935 . . . . . . . . . . . . . . . 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))))
100 0cnd 10318 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → 0 ∈ ℂ)
101 nn0cn 11569 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
102101adantl 469 . . . . . . . . . . . . . . . . . . 19 ((⊤ ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℂ)
103 df-ne 2979 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ≠ 0 ↔ ¬ 𝑚 = 0)
104103biimpri 219 . . . . . . . . . . . . . . . . . . 19 𝑚 = 0 → 𝑚 ≠ 0)
105 reccl 10977 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℂ ∧ 𝑚 ≠ 0) → (1 / 𝑚) ∈ ℂ)
106102, 104, 105syl2an 585 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑚 ∈ ℕ0) ∧ ¬ 𝑚 = 0) → (1 / 𝑚) ∈ ℂ)
107100, 106ifclda 4313 . . . . . . . . . . . . . . . . 17 ((⊤ ∧ 𝑚 ∈ ℕ0) → if(𝑚 = 0, 0, (1 / 𝑚)) ∈ ℂ)
108107fmpttd 6607 . . . . . . . . . . . . . . . 16 (⊤ → (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚))):ℕ0⟶ℂ)
109 recn 10311 . . . . . . . . . . . . . . . . . . . . . 22 (𝑟 ∈ ℝ → 𝑟 ∈ ℂ)
110 oveq1 6881 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑟 → (𝑥𝑗) = (𝑟𝑗))
111110oveq2d 6890 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑟 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))
112111mpteq2dv 4939 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑟 → (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗))))
113 eqid 2806 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)))) = (𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))
114 nn0ex 11565 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
115114mptex 6711 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗))) ∈ V
116112, 113, 115fvmpt 6503 . . . . . . . . . . . . . . . . . . . . . 22 (𝑟 ∈ ℂ → ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑟) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗))))
117109, 116syl 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))))
118117eqcomd 2812 . . . . . . . . . . . . . . . . . . . 20 (𝑟 ∈ ℝ → (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗))) = ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑟))
119118seqeq3d 13032 . . . . . . . . . . . . . . . . . . 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)))
120119eleq1d 2870 . . . . . . . . . . . . . . . . . 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 ⇝ ))
121120rabbiia 3374 . . . . . . . . . . . . . . . . 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 ⇝ }
122121supeq1i 8592 . . . . . . . . . . . . . . . 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*, < )
12399, 108, 122radcnvcl 24385 . . . . . . . . . . . . . . 15 (⊤ → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ (0[,]+∞))
12487, 123sseldi 3796 . . . . . . . . . . . . . 14 (⊤ → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
12546, 124mp1i 13 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
126 1re 10325 . . . . . . . . . . . . . . 15 1 ∈ ℝ
127 avglt1 11537 . . . . . . . . . . . . . . 15 (((abs‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (abs‘𝑦) < (((abs‘𝑦) + 1) / 2)))
12856, 126, 127sylancl 576 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) < 1 ↔ (abs‘𝑦) < (((abs‘𝑦) + 1) / 2)))
12966, 128mpbid 223 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < (((abs‘𝑦) + 1) / 2))
130 0red 10328 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ∈ ℝ)
131130, 56, 85, 81, 129lelttrd 10480 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 < (((abs‘𝑦) + 1) / 2))
132130, 85, 131ltled 10470 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ≤ (((abs‘𝑦) + 1) / 2))
13385, 132absidd 14384 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) = (((abs‘𝑦) + 1) / 2))
13446, 108mp1i 13 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚))):ℕ0⟶ℂ)
13585recnd 10353 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℂ)
136 oveq1 6881 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = (((abs‘𝑦) + 1) / 2) → (𝑥𝑗) = ((((abs‘𝑦) + 1) / 2)↑𝑗))
137136oveq2d 6890 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = (((abs‘𝑦) + 1) / 2) → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))
138137mpteq2dv 4939 . . . . . . . . . . . . . . . . . . 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))))
139114mptex 6711 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗))) ∈ V
140138, 113, 139fvmpt 6503 . . . . . . . . . . . . . . . . . 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))))
141135, 140syl 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))))
142141seqeq3d 13032 . . . . . . . . . . . . . . . 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)))))
143 avglt2 11538 . . . . . . . . . . . . . . . . . . . 20 (((abs‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (((abs‘𝑦) + 1) / 2) < 1))
14456, 126, 143sylancl 576 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) < 1 ↔ (((abs‘𝑦) + 1) / 2) < 1))
14566, 144mpbid 223 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) < 1)
146133, 145eqbrtrd 4866 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) < 1)
147 logtayllem 24619 . . . . . . . . . . . . . . . . 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 ⇝ )
148135, 146, 147syl2anc 575 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))) ∈ dom ⇝ )
149142, 148eqeltrd 2885 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘(((abs‘𝑦) + 1) / 2))) ∈ dom ⇝ )
15099, 134, 122, 135, 149radcnvle 24388 . . . . . . . . . . . . . 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*, < ))
151133, 150eqbrtrrd 4868 . . . . . . . . . . . . 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*, < ))
15282, 86, 125, 129, 151xrltletrd 12210 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))
153 0re 10327 . . . . . . . . . . . . 13 0 ∈ ℝ
154 elico2 12455 . . . . . . . . . . . . 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*, < ))))
155153, 125, 154sylancr 577 . . . . . . . . . . . 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*, < ))))
15656, 81, 152, 155mpbir3and 1435 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))
157 absf 14300 . . . . . . . . . . . 12 abs:ℂ⟶ℝ
158 ffn 6256 . . . . . . . . . . . 12 (abs:ℂ⟶ℝ → abs Fn ℂ)
159 elpreima 6559 . . . . . . . . . . . 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*, < )))))
160157, 158, 159mp2b 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*, < ))))
16153, 156, 160sylanbrc 574 . . . . . . . . . 10 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))))
162 cnvimass 5695 . . . . . . . . . . . . . . . . . 18 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ⊆ dom abs
163157fdmi 6266 . . . . . . . . . . . . . . . . . 18 dom abs = ℂ
164162, 163sseqtri 3834 . . . . . . . . . . . . . . . . 17 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) ⊆ ℂ
165164sseli 3794 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))) → 𝑦 ∈ ℂ)
166 oveq1 6881 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑦 → (𝑥𝑗) = (𝑦𝑗))
167166oveq2d 6890 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑦 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)))
168167mpteq2dv 4939 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗))))
169114mptex 6711 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗))) ∈ V
170168, 113, 169fvmpt 6503 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℂ → ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗))))
171170adantr 468 . . . . . . . . . . . . . . . . . . 19 ((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))))
172171fveq1d 6410 . . . . . . . . . . . . . . . . . 18 ((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))
173 eqeq1 2810 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑛 → (𝑗 = 0 ↔ 𝑛 = 0))
174 oveq2 6882 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑛 → (1 / 𝑗) = (1 / 𝑛))
175173, 174ifbieq2d 4304 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑛 → if(𝑗 = 0, 0, (1 / 𝑗)) = if(𝑛 = 0, 0, (1 / 𝑛)))
176 oveq2 6882 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑛 → (𝑦𝑗) = (𝑦𝑛))
177175, 176oveq12d 6892 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑛 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
178 eqid 2806 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗))) = (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)))
179 ovex 6906 . . . . . . . . . . . . . . . . . . . 20 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) ∈ V
180177, 178, 179fvmpt 6503 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ0 → ((𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
181180adantl 469 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → ((𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦𝑗)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
182172, 181eqtr2d 2841 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥𝑗))))‘𝑦)‘𝑛))
183182sumeq2dv 14656 . . . . . . . . . . . . . . . 16 (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))
184165, 183syl 17 . . . . . . . . . . . . . . 15 (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))
185184mpteq2ia 4934 . . . . . . . . . . . . . 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))) = (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))
186 eqid 2806 . . . . . . . . . . . . . 14 (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*, < )))
187 eqid 2806 . . . . . . . . . . . . . 14 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))
18899, 185, 108, 122, 186, 187psercn 24394 . . . . . . . . . . . . 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))) ∈ ((abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))–cn→ℂ))
189 cncff 22909 . . . . . . . . . . . . 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))) ∈ ((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)
190188, 189syl 17 . . . . . . . . . . . 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*, < )))⟶ℂ)
191 eqid 2806 . . . . . . . . . . . . 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 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
192191fmpt 6602 . . . . . . . . . . . 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)) ∈ 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)
193190, 192sylibr 225 . . . . . . . . . . 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)) ∈ ℂ)
194193r19.21bi 3120 . . . . . . . . . 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)
195161, 194sylan2 582 . . . . . . . . 9 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) ∈ ℂ)
196195fmpttd 6607 . . . . . . . 8 (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))):(0(ball‘(abs ∘ − ))1)⟶ℂ)
197 cnelprrecn 10314 . . . . . . . . . . . . 13 ℂ ∈ {ℝ, ℂ}
198197a1i 11 . . . . . . . . . . . 12 (⊤ → ℂ ∈ {ℝ, ℂ})
19977adantl 469 . . . . . . . . . . . 12 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → (log‘(1 − 𝑦)) ∈ ℂ)
200 ovexd 6908 . . . . . . . . . . . 12 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → ((1 / (1 − 𝑦)) · -1) ∈ V)
20129cnmetdval 22787 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℂ ∧ (1 − 𝑦) ∈ ℂ) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘(1 − (1 − 𝑦))))
20250, 55, 201sylancr 577 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘(1 − (1 − 𝑦))))
203 nncan 10595 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − (1 − 𝑦)) = 𝑦)
20450, 53, 203sylancr 577 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − (1 − 𝑦)) = 𝑦)
205204fveq2d 6412 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(1 − (1 − 𝑦))) = (abs‘𝑦))
206202, 205eqtrd 2840 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘𝑦))
207206, 66eqbrtrd 4866 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) < 1)
208 elbl 22406 . . . . . . . . . . . . . . . 16 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ*) → ((1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 − 𝑦) ∈ ℂ ∧ (1(abs ∘ − )(1 − 𝑦)) < 1)))
20938, 50, 41, 208mp3an 1578 . . . . . . . . . . . . . . 15 ((1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 − 𝑦) ∈ ℂ ∧ (1(abs ∘ − )(1 − 𝑦)) < 1))
21055, 207, 209sylanbrc 574 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1))
211210adantl 469 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → (1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1))
212 neg1cn 11406 . . . . . . . . . . . . . 14 -1 ∈ ℂ
213212a1i 11 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → -1 ∈ ℂ)
214 eqid 2806 . . . . . . . . . . . . . . . . . 18 (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
215214dvlog2lem 24612 . . . . . . . . . . . . . . . . 17 (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
216215sseli 3794 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
217216eldifad 3781 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ∈ ℂ)
218 eqid 2806 . . . . . . . . . . . . . . . . 17 (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
219218logdmn0 24600 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (ℂ ∖ (-∞(,]0)) → 𝑥 ≠ 0)
220216, 219syl 17 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ≠ 0)
221217, 220logcld 24531 . . . . . . . . . . . . . 14 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑥) ∈ ℂ)
222221adantl 469 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(ball‘(abs ∘ − ))1)) → (log‘𝑥) ∈ ℂ)
223 ovexd 6908 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(ball‘(abs ∘ − ))1)) → (1 / 𝑥) ∈ V)
224 simpr 473 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
22550, 224, 54sylancr 577 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑦 ∈ ℂ) → (1 − 𝑦) ∈ ℂ)
226212a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑦 ∈ ℂ) → -1 ∈ ℂ)
227 1cnd 10320 . . . . . . . . . . . . . . . 16 ((⊤ ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
228 0cnd 10318 . . . . . . . . . . . . . . . 16 ((⊤ ∧ 𝑦 ∈ ℂ) → 0 ∈ ℂ)
229 1cnd 10320 . . . . . . . . . . . . . . . . 17 (⊤ → 1 ∈ ℂ)
230198, 229dvmptc 23935 . . . . . . . . . . . . . . . 16 (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 1)) = (𝑦 ∈ ℂ ↦ 0))
231198dvmptid 23934 . . . . . . . . . . . . . . . 16 (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1))
232198, 227, 228, 230, 224, 227, 231dvmptsub 23944 . . . . . . . . . . . . . . 15 (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (1 − 𝑦))) = (𝑦 ∈ ℂ ↦ (0 − 1)))
233 df-neg 10554 . . . . . . . . . . . . . . . 16 -1 = (0 − 1)
234233mpteq2i 4935 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℂ ↦ -1) = (𝑦 ∈ ℂ ↦ (0 − 1))
235232, 234syl6eqr 2858 . . . . . . . . . . . . . 14 (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (1 − 𝑦))) = (𝑦 ∈ ℂ ↦ -1))
23652a1i 11 . . . . . . . . . . . . . 14 (⊤ → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
237 eqid 2806 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
238237cnfldtopon 22799 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
239238toponrestid 20939 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
240237cnfldtopn 22798 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
241240blopn 22518 . . . . . . . . . . . . . . . 16 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ*) → (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld))
24238, 28, 41, 241mp3an 1578 . . . . . . . . . . . . . . 15 (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld)
243242a1i 11 . . . . . . . . . . . . . 14 (⊤ → (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld))
244198, 225, 226, 235, 236, 239, 237, 243dvmptres 23940 . . . . . . . . . . . . 13 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -1))
245214dvlog2 24613 . . . . . . . . . . . . . 14 (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑥))
246 logf1o 24525 . . . . . . . . . . . . . . . . . . . 20 log:(ℂ ∖ {0})–1-1-onto→ran log
247 f1of 6353 . . . . . . . . . . . . . . . . . . . 20 (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
248246, 247ax-mp 5 . . . . . . . . . . . . . . . . . . 19 log:(ℂ ∖ {0})⟶ran log
249218logdmss 24602 . . . . . . . . . . . . . . . . . . . 20 (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
250215, 249sstri 3807 . . . . . . . . . . . . . . . . . . 19 (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
251 fssres 6285 . . . . . . . . . . . . . . . . . . 19 ((log:(ℂ ∖ {0})⟶ran log ∧ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})) → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log)
252248, 250, 251mp2an 675 . . . . . . . . . . . . . . . . . 18 (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log
253252a1i 11 . . . . . . . . . . . . . . . . 17 (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log)
254253feqmptd 6470 . . . . . . . . . . . . . . . 16 (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥)))
255 fvres 6427 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥) = (log‘𝑥))
256255mpteq2ia 4934 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))
257254, 256syl6eq 2856 . . . . . . . . . . . . . . 15 (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥)))
258257oveq2d 6890 . . . . . . . . . . . . . 14 (⊤ → (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (ℂ D (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))))
259245, 258syl5reqr 2855 . . . . . . . . . . . . 13 (⊤ → (ℂ D (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑥)))
260 fveq2 6408 . . . . . . . . . . . . 13 (𝑥 = (1 − 𝑦) → (log‘𝑥) = (log‘(1 − 𝑦)))
261 oveq2 6882 . . . . . . . . . . . . 13 (𝑥 = (1 − 𝑦) → (1 / 𝑥) = (1 / (1 − 𝑦)))
262198, 198, 211, 213, 222, 223, 244, 259, 260, 261dvmptco 23949 . . . . . . . . . . . 12 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ ((1 / (1 − 𝑦)) · -1)))
263198, 199, 200, 262dvmptneg 23943 . . . . . . . . . . 11 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -((1 / (1 − 𝑦)) · -1)))
26455, 76reccld 11079 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 / (1 − 𝑦)) ∈ ℂ)
265 mulcom 10307 . . . . . . . . . . . . . . . 16 (((1 / (1 − 𝑦)) ∈ ℂ ∧ -1 ∈ ℂ) → ((1 / (1 − 𝑦)) · -1) = (-1 · (1 / (1 − 𝑦))))
266264, 212, 265sylancl 576 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 / (1 − 𝑦)) · -1) = (-1 · (1 / (1 − 𝑦))))
267264mulm1d 10767 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (-1 · (1 / (1 − 𝑦))) = -(1 / (1 − 𝑦)))
268266, 267eqtrd 2840 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 / (1 − 𝑦)) · -1) = -(1 / (1 − 𝑦)))
269268negeqd 10560 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -((1 / (1 − 𝑦)) · -1) = --(1 / (1 − 𝑦)))
270264negnegd 10668 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → --(1 / (1 − 𝑦)) = (1 / (1 − 𝑦)))
271269, 270eqtrd 2840 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -((1 / (1 − 𝑦)) · -1) = (1 / (1 − 𝑦)))
272271mpteq2ia 4934 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -((1 / (1 − 𝑦)) · -1)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦)))
273263, 272syl6eq 2856 . . . . . . . . . 10 (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
274273dmeqd 5527 . . . . . . . . 9 (⊤ → dom (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
275 dmmptg 5846 . . . . . . . . . 10 (∀𝑦 ∈ (0(ball‘(abs ∘ − ))1)(1 / (1 − 𝑦)) ∈ V → dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))) = (0(ball‘(abs ∘ − ))1))
276 ovexd 6908 . . . . . . . . . 10 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 / (1 − 𝑦)) ∈ V)
277275, 276mprg 3114 . . . . . . . . 9 dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))) = (0(ball‘(abs ∘ − ))1)
278274, 277syl6eq 2856 . . . . . . . 8 (⊤ → dom (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (0(ball‘(abs ∘ − ))1))
279 sumex 14641 . . . . . . . . . . . 12 Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) ∈ V
280279a1i 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)
281 fveq2 6408 . . . . . . . . . . . . . . 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))
282281cbvsumv 14649 . . . . . . . . . . . . . 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)
283184, 282syl6eq 2856 . . . . . . . . . . . . 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))
284283mpteq2ia 4934 . . . . . . . . . . . 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))
285 eqid 2806 . . . . . . . . . . . 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))
28699, 284, 108, 122, 186, 187, 285pserdv2 24398 . . . . . . . . . . 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)))))
287161ssriv 3802 . . . . . . . . . . . 12 (0(ball‘(abs ∘ − ))1) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < )))
288287a1i 11 . . . . . . . . . . 11 (⊤ → (0(ball‘(abs ∘ − ))1) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ0 ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟𝑗)))) ∈ dom ⇝ }, ℝ*, < ))))
289198, 194, 280, 286, 288, 239, 237, 243dvmptres 23940 . . . . . . . . . 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)))))
290 nnnn0 11566 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
291290adantl 469 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
292 eqeq1 2810 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (𝑚 = 0 ↔ 𝑛 = 0))
293 oveq2 6882 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
294292, 293ifbieq2d 4304 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → if(𝑚 = 0, 0, (1 / 𝑚)) = if(𝑛 = 0, 0, (1 / 𝑛)))
295 ovex 6906 . . . . . . . . . . . . . . . . . . . . 21 (1 / 𝑛) ∈ V
29692, 295ifex 4327 . . . . . . . . . . . . . . . . . . . 20 if(𝑛 = 0, 0, (1 / 𝑛)) ∈ V
297294, 91, 296fvmpt 6503 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)))
298291, 297syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)))
299 nnne0 11339 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
300299adantl 469 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
301300neneqd 2983 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
302301iffalsed 4290 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, 0, (1 / 𝑛)) = (1 / 𝑛))
303298, 302eqtrd 2840 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = (1 / 𝑛))
304303oveq2d 6890 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) = (𝑛 · (1 / 𝑛)))
305 nncn 11313 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
306305adantl 469 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
307306, 300recidd 11081 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · (1 / 𝑛)) = 1)
308304, 307eqtrd 2840 . . . . . . . . . . . . . . 15 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) = 1)
309308oveq1d 6889 . . . . . . . . . . . . . 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))))
310 nnm1nn0 11600 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
311 expcl 13101 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℂ ∧ (𝑛 − 1) ∈ ℕ0) → (𝑦↑(𝑛 − 1)) ∈ ℂ)
31253, 310, 311syl2an 585 . . . . . . . . . . . . . . 15 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑦↑(𝑛 − 1)) ∈ ℂ)
313312mulid2d 10343 . . . . . . . . . . . . . 14 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (1 · (𝑦↑(𝑛 − 1))) = (𝑦↑(𝑛 − 1)))
314309, 313eqtrd 2840 . . . . . . . . . . . . 13 ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (𝑦↑(𝑛 − 1)))
315314sumeq2dv 14656 . . . . . . . . . . . 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)))
316 nnuz 11941 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
317 1e0p1 11801 . . . . . . . . . . . . . . . 16 1 = (0 + 1)
318317fveq2i 6411 . . . . . . . . . . . . . . 15 (ℤ‘1) = (ℤ‘(0 + 1))
319316, 318eqtri 2828 . . . . . . . . . . . . . 14 ℕ = (ℤ‘(0 + 1))
320 oveq1 6881 . . . . . . . . . . . . . . 15 (𝑛 = (1 + 𝑚) → (𝑛 − 1) = ((1 + 𝑚) − 1))
321320oveq2d 6890 . . . . . . . . . . . . . 14 (𝑛 = (1 + 𝑚) → (𝑦↑(𝑛 − 1)) = (𝑦↑((1 + 𝑚) − 1)))
322 1zzd 11674 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ∈ ℤ)
323 0zd 11655 . . . . . . . . . . . . . 14 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ∈ ℤ)
3241, 319, 321, 322, 323, 312isumshft 14793 . . . . . . . . . . . . 13 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)) = Σ𝑚 ∈ ℕ0 (𝑦↑((1 + 𝑚) − 1)))
325 pncan2 10573 . . . . . . . . . . . . . . . 16 ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((1 + 𝑚) − 1) = 𝑚)
32650, 101, 325sylancr 577 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → ((1 + 𝑚) − 1) = 𝑚)
327326oveq2d 6890 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (𝑦↑((1 + 𝑚) − 1)) = (𝑦𝑚))
328327sumeq2i 14652 . . . . . . . . . . . . 13 Σ𝑚 ∈ ℕ0 (𝑦↑((1 + 𝑚) − 1)) = Σ𝑚 ∈ ℕ0 (𝑦𝑚)
329324, 328syl6eq 2856 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)) = Σ𝑚 ∈ ℕ0 (𝑦𝑚))
330 geoisum 14830 . . . . . . . . . . . . 13 ((𝑦 ∈ ℂ ∧ (abs‘𝑦) < 1) → Σ𝑚 ∈ ℕ0 (𝑦𝑚) = (1 / (1 − 𝑦)))
33153, 66, 330syl2anc 575 . . . . . . . . . . . 12 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑚 ∈ ℕ0 (𝑦𝑚) = (1 / (1 − 𝑦)))
332315, 329, 3313eqtrd 2844 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (1 / (1 − 𝑦)))
333332mpteq2ia 4934 . . . . . . . . . 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)))
334289, 333syl6eq 2856 . . . . . . . . 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))))
335273, 334eqtr4d 2843 . . . . . . . 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)))))
336 blcntr 22431 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ+) → 0 ∈ (0(ball‘(abs ∘ − ))1))
33738, 28, 39, 336mp3an 1578 . . . . . . . . 9 0 ∈ (0(ball‘(abs ∘ − ))1)
338337a1i 11 . . . . . . . 8 (⊤ → 0 ∈ (0(ball‘(abs ∘ − ))1))
339 oveq2 6882 . . . . . . . . . . . . . . . 16 (𝑦 = 0 → (1 − 𝑦) = (1 − 0))
340 1m0e1 11413 . . . . . . . . . . . . . . . 16 (1 − 0) = 1
341339, 340syl6eq 2856 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (1 − 𝑦) = 1)
342341fveq2d 6412 . . . . . . . . . . . . . 14 (𝑦 = 0 → (log‘(1 − 𝑦)) = (log‘1))
343 log1 24546 . . . . . . . . . . . . . 14 (log‘1) = 0
344342, 343syl6eq 2856 . . . . . . . . . . . . 13 (𝑦 = 0 → (log‘(1 − 𝑦)) = 0)
345344negeqd 10560 . . . . . . . . . . . 12 (𝑦 = 0 → -(log‘(1 − 𝑦)) = -0)
346 neg0 10612 . . . . . . . . . . . 12 -0 = 0
347345, 346syl6eq 2856 . . . . . . . . . . 11 (𝑦 = 0 → -(log‘(1 − 𝑦)) = 0)
348 eqid 2806 . . . . . . . . . . 11 (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))
349347, 348, 92fvmpt 6503 . . . . . . . . . 10 (0 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = 0)
350337, 349mp1i 13 . . . . . . . . 9 (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = 0)
351 oveq1 6881 . . . . . . . . . . . . . . 15 (0 = if(𝑛 = 0, 0, (1 / 𝑛)) → (0 · (𝑦𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
352351eqeq1d 2808 . . . . . . . . . . . . . 14 (0 = if(𝑛 = 0, 0, (1 / 𝑛)) → ((0 · (𝑦𝑛)) = 0 ↔ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = 0))
353 oveq1 6881 . . . . . . . . . . . . . . 15 ((1 / 𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)) → ((1 / 𝑛) · (𝑦𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))
354353eqeq1d 2808 . . . . . . . . . . . . . 14 ((1 / 𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)) → (((1 / 𝑛) · (𝑦𝑛)) = 0 ↔ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = 0))
355 simpll 774 . . . . . . . . . . . . . . . . 17 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → 𝑦 = 0)
356355, 28syl6eqel 2893 . . . . . . . . . . . . . . . 16 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → 𝑦 ∈ ℂ)
357 simplr 776 . . . . . . . . . . . . . . . 16 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → 𝑛 ∈ ℕ0)
358356, 357expcld 13231 . . . . . . . . . . . . . . 15 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → (𝑦𝑛) ∈ ℂ)
359358mul02d 10519 . . . . . . . . . . . . . 14 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → (0 · (𝑦𝑛)) = 0)
360 simpll 774 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑦 = 0)
361360oveq1d 6889 . . . . . . . . . . . . . . . . 17 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (𝑦𝑛) = (0↑𝑛))
362 simpr 473 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
363362, 14sylib 209 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) → (𝑛 ∈ ℕ ∨ 𝑛 = 0))
364363ord 882 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 ∈ ℕ → 𝑛 = 0))
365364con1d 141 . . . . . . . . . . . . . . . . . . 19 ((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 = 0 → 𝑛 ∈ ℕ))
366365imp 395 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
3673660expd 13247 . . . . . . . . . . . . . . . . 17 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (0↑𝑛) = 0)
368361, 367eqtrd 2840 . . . . . . . . . . . . . . . 16 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (𝑦𝑛) = 0)
369368oveq2d 6890 . . . . . . . . . . . . . . 15 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · (𝑦𝑛)) = ((1 / 𝑛) · 0))
370366nnrecred 11352 . . . . . . . . . . . . . . . . 17 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℝ)
371370recnd 10353 . . . . . . . . . . . . . . . 16 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℂ)
372371mul01d 10520 . . . . . . . . . . . . . . 15 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · 0) = 0)
373369, 372eqtrd 2840 . . . . . . . . . . . . . 14 (((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · (𝑦𝑛)) = 0)
374352, 354, 359, 373ifbothda 4316 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑛 ∈ ℕ0) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = 0)
375374sumeq2dv 14656 . . . . . . . . . . . 12 (𝑦 = 0 → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = Σ𝑛 ∈ ℕ0 0)
3761eqimssi 3856 . . . . . . . . . . . . . 14 0 ⊆ (ℤ‘0)
377376orci 883 . . . . . . . . . . . . 13 (ℕ0 ⊆ (ℤ‘0) ∨ ℕ0 ∈ Fin)
378 sumz 14676 . . . . . . . . . . . . 13 ((ℕ0 ⊆ (ℤ‘0) ∨ ℕ0 ∈ Fin) → Σ𝑛 ∈ ℕ0 0 = 0)
379377, 378ax-mp 5 . . . . . . . . . . . 12 Σ𝑛 ∈ ℕ0 0 = 0
380375, 379syl6eq 2856 . . . . . . . . . . 11 (𝑦 = 0 → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = 0)
381 eqid 2806 . . . . . . . . . . 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)))
382380, 381, 92fvmpt 6503 . . . . . . . . . 10 (0 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))‘0) = 0)
383337, 382mp1i 13 . . . . . . . . 9 (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)))‘0) = 0)
384350, 383eqtr4d 2843 . . . . . . . 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))
38547, 48, 49, 80, 196, 278, 335, 338, 384dv11cn 23978 . . . . . . 7 (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛))))
386385fveq1d 6410 . . . . . 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))
38746, 386mp1i 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))
388 oveq2 6882 . . . . . . . 8 (𝑦 = 𝐴 → (1 − 𝑦) = (1 − 𝐴))
389388fveq2d 6412 . . . . . . 7 (𝑦 = 𝐴 → (log‘(1 − 𝑦)) = (log‘(1 − 𝐴)))
390389negeqd 10560 . . . . . 6 (𝑦 = 𝐴 → -(log‘(1 − 𝑦)) = -(log‘(1 − 𝐴)))
391 negex 10564 . . . . . 6 -(log‘(1 − 𝐴)) ∈ V
392390, 348, 391fvmpt 6503 . . . . 5 (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘𝐴) = -(log‘(1 − 𝐴)))
393 oveq1 6881 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑛) = (𝐴𝑛))
394393oveq2d 6890 . . . . . . 7 (𝑦 = 𝐴 → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
395394sumeq2sdv 14658 . . . . . 6 (𝑦 = 𝐴 → Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦𝑛)) = Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
396 sumex 14641 . . . . . 6 Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)) ∈ V
397395, 381, 396fvmpt 6503 . . . . 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)))
398387, 392, 3973eqtr3d 2848 . . . 4 (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → -(log‘(1 − 𝐴)) = Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
39945, 398syl 17 . . 3 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(log‘(1 − 𝐴)) = Σ𝑛 ∈ ℕ0 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))
40026, 399breqtrrd 4872 . 2 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ⇝ -(log‘(1 − 𝐴)))
401 seqex 13026 . . . 4 seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ∈ V
402401a1i 11 . . 3 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ∈ V)
403 seqex 13026 . . . 4 seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))) ∈ V
404403a1i 11 . . 3 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))) ∈ V)
405 1zzd 11674 . . 3 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℤ)
406 elnnuz 11942 . . . . . 6 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
407 fvres 6427 . . . . . 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))
408406, 407sylbi 208 . . . . 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))
409408eqcomd 2812 . . . 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))
410 addid2 10504 . . . . . . . 8 (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
411410adantl 469 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
412 0cnd 10318 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ∈ ℂ)
413 1eluzge0 11950 . . . . . . . 8 1 ∈ (ℤ‘0)
414413a1i 11 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ (ℤ‘0))
415 0cnd 10318 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) ∧ 𝑘 = 0) → 0 ∈ ℂ)
416 nn0cn 11569 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
417416adantl 469 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
418 df-ne 2979 . . . . . . . . . . . . 13 (𝑘 ≠ 0 ↔ ¬ 𝑘 = 0)
419418biimpri 219 . . . . . . . . . . . 12 𝑘 = 0 → 𝑘 ≠ 0)
420 reccl 10977 . . . . . . . . . . . 12 ((𝑘 ∈ ℂ ∧ 𝑘 ≠ 0) → (1 / 𝑘) ∈ ℂ)
421417, 419, 420syl2an 585 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 = 0) → (1 / 𝑘) ∈ ℂ)
422415, 421ifclda 4313 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 0, 0, (1 / 𝑘)) ∈ ℂ)
423 expcl 13101 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
424423adantlr 697 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
425422, 424mulcld 10345 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)) ∈ ℂ)
426425fmpttd 6607 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘))):ℕ0⟶ℂ)
427 1nn0 11575 . . . . . . . 8 1 ∈ ℕ0
428 ffvelrn 6579 . . . . . . . 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)
429426, 427, 428sylancl 576 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘1) ∈ ℂ)
430 elfz1eq 12575 . . . . . . . . . 10 (𝑛 ∈ (0...0) → 𝑛 = 0)
431 1m1e0 11373 . . . . . . . . . . 11 (1 − 1) = 0
432431oveq2i 6885 . . . . . . . . . 10 (0...(1 − 1)) = (0...0)
433430, 432eleq2s 2903 . . . . . . . . 9 (𝑛 ∈ (0...(1 − 1)) → 𝑛 = 0)
434433fveq2d 6412 . . . . . . . 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))
435 0nn0 11574 . . . . . . . . . 10 0 ∈ ℕ0
436 iftrue 4285 . . . . . . . . . . . 12 (𝑘 = 0 → if(𝑘 = 0, 0, (1 / 𝑘)) = 0)
437 oveq2 6882 . . . . . . . . . . . 12 (𝑘 = 0 → (𝐴𝑘) = (𝐴↑0))
438436, 437oveq12d 6892 . . . . . . . . . . 11 (𝑘 = 0 → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)) = (0 · (𝐴↑0)))
439 ovex 6906 . . . . . . . . . . 11 (0 · (𝐴↑0)) ∈ V
440438, 8, 439fvmpt 6503 . . . . . . . . . 10 (0 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘0) = (0 · (𝐴↑0)))
441435, 440ax-mp 5 . . . . . . . . 9 ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘0) = (0 · (𝐴↑0))
442 expcl 13101 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 0 ∈ ℕ0) → (𝐴↑0) ∈ ℂ)
44327, 435, 442sylancl 576 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑0) ∈ ℂ)
444443mul02d 10519 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (0 · (𝐴↑0)) = 0)
445441, 444syl5eq 2852 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘0) = 0)
446434, 445sylan9eqr 2862 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ (0...(1 − 1))) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = 0)
447411, 412, 414, 429, 446seqid 13069 . . . . . 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)))))
448299adantl 469 . . . . . . . . . . . . 13 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
449448neneqd 2983 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
450449iffalsed 4290 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, 0, (1 / 𝑛)) = (1 / 𝑛))
451450oveq1d 6889 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)) = ((1 / 𝑛) · (𝐴𝑛)))
452290, 23sylan2 582 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℂ)
453305adantl 469 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
454452, 453, 448divrec2d 11090 . . . . . . . . . 10 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝐴𝑛) / 𝑛) = ((1 / 𝑛) · (𝐴𝑛)))
455451, 454eqtr4d 2843 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)) = ((𝐴𝑛) / 𝑛))
456290, 11sylan2 582 . . . . . . . . 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)))
457 id 22 . . . . . . . . . . . 12 (𝑘 = 𝑛𝑘 = 𝑛)
4586, 457oveq12d 6892 . . . . . . . . . . 11 (𝑘 = 𝑛 → ((𝐴𝑘) / 𝑘) = ((𝐴𝑛) / 𝑛))
459 eqid 2806 . . . . . . . . . . 11 (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘)) = (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))
460 ovex 6906 . . . . . . . . . . 11 ((𝐴𝑛) / 𝑛) ∈ V
461458, 459, 460fvmpt 6503 . . . . . . . . . 10 (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))‘𝑛) = ((𝐴𝑛) / 𝑛))
462461adantl 469 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))‘𝑛) = ((𝐴𝑛) / 𝑛))
463455, 456, 4623eqtr4d 2850 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = ((𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))‘𝑛))
464406, 463sylan2br 584 . . . . . . 7 (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ (ℤ‘1)) → ((𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))‘𝑛) = ((𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))‘𝑛))
465405, 464seqfeq 13049 . . . . . 6 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))))
466447, 465eqtrd 2840 . . . . 5 ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (seq0( + , (𝑘 ∈ ℕ0 ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴𝑘)))) ↾ (ℤ‘1)) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))))
467466fveq1d 6410 . . . 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))
468409, 467sylan9eqr 2862 . . 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))
469316, 402, 404, 405, 468climeq 14521 . 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))))
470400, 469mpbid 223 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 197  wa 384  wo 865  w3a 1100   = wceq 1637  wtru 1638  wcel 2156  wne 2978  wral 3096  {crab 3100  Vcvv 3391  cdif 3766  wss 3769  ifcif 4279  {csn 4370  {cpr 4372   class class class wbr 4844  cmpt 4923  ccnv 5310  dom cdm 5311  ran crn 5312  cres 5313  cima 5314  ccom 5315   Fn wfn 6096  wf 6097  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6874  Fincfn 8192  supcsup 8585  cc 10219  cr 10220  0cc0 10221  1c1 10222   + caddc 10224   · cmul 10226  +∞cpnf 10356  -∞cmnf 10357  *cxr 10358   < clt 10359  cle 10360  cmin 10551  -cneg 10552   / cdiv 10969  cn 11305  2c2 11356  0cn0 11559  cuz 11904  +crp 12046  (,]cioc 12394  [,)cico 12395  [,]cicc 12396  ...cfz 12549  seqcseq 13024  cexp 13083  abscabs 14197  cli 14438  Σcsu 14639  TopOpenctopn 16287  ∞Metcxmt 19939  ballcbl 19941  fldccnfld 19954  cnccncf 22892   D cdv 23841  logclog 24515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-pre-sup 10299  ax-addf 10300  ax-mulf 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-of 7127  df-om 7296  df-1st 7398  df-2nd 7399  df-supp 7530  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-map 8094  df-pm 8095  df-ixp 8146  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-fsupp 8515  df-fi 8556  df-sup 8587  df-inf 8588  df-oi 8654  df-card 9048  df-cda 9275  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-div 10970  df-nn 11306  df-2 11364  df-3 11365  df-4 11366  df-5 11367  df-6 11368  df-7 11369  df-8 11370  df-9 11371  df-n0 11560  df-z 11644  df-dec 11760  df-uz 11905  df-q 12008  df-rp 12047  df-xneg 12162  df-xadd 12163  df-xmul 12164  df-ioo 12397  df-ioc 12398  df-ico 12399  df-icc 12400  df-fz 12550  df-fzo 12690  df-fl 12817  df-mod 12893  df-seq 13025  df-exp 13084  df-fac 13281  df-bc 13310  df-hash 13338  df-shft 14030  df-cj 14062  df-re 14063  df-im 14064  df-sqrt 14198  df-abs 14199  df-limsup 14425  df-clim 14442  df-rlim 14443  df-sum 14640  df-ef 15018  df-sin 15020  df-cos 15021  df-tan 15022  df-pi 15023  df-struct 16070  df-ndx 16071  df-slot 16072  df-base 16074  df-sets 16075  df-ress 16076  df-plusg 16166  df-mulr 16167  df-starv 16168  df-sca 16169  df-vsca 16170  df-ip 16171  df-tset 16172  df-ple 16173  df-ds 16175  df-unif 16176  df-hom 16177  df-cco 16178  df-rest 16288  df-topn 16289  df-0g 16307  df-gsum 16308  df-topgen 16309  df-pt 16310  df-prds 16313  df-xrs 16367  df-qtop 16372  df-imas 16373  df-xps 16375  df-mre 16451  df-mrc 16452  df-acs 16454  df-mgm 17447  df-sgrp 17489  df-mnd 17500  df-submnd 17541  df-mulg 17746  df-cntz 17951  df-cmn 18396  df-psmet 19946  df-xmet 19947  df-met 19948  df-bl 19949  df-mopn 19950  df-fbas 19951  df-fg 19952  df-cnfld 19955  df-top 20912  df-topon 20929  df-topsp 20951  df-bases 20964  df-cld 21037  df-ntr 21038  df-cls 21039  df-nei 21116  df-lp 21154  df-perf 21155  df-cn 21245  df-cnp 21246  df-haus 21333  df-cmp 21404  df-tx 21579  df-hmeo 21772  df-fil 21863  df-fm 21955  df-flim 21956  df-flf 21957  df-xms 22338  df-ms 22339  df-tms 22340  df-cncf 22894  df-limc 23844  df-dv 23845  df-ulm 24345  df-log 24517
This theorem is referenced by:  logtaylsum  24621  logtayl2  24622  atantayl  24878  stirlinglem5  40774
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