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Metamath Proof Explorer

Theorem logtayl

Description: The Taylor series for -u log ( 1 - A ) . (Contributed by Mario Carneiro, 1-Apr-2015)

Ref Expression
Assertion logtayl ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ⇝ - ( log ‘ ( 1 − 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 nn0uz ⊢ ℕ0 = ( ℤ≥ ‘ 0 )
2 0zd ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 0 ∈ ℤ )
3 eqeq1 ⊢ ( 𝑘 = 𝑛 → ( 𝑘 = 0 ↔ 𝑛 = 0 ) )
4 oveq2 ⊢ ( 𝑘 = 𝑛 → ( 1 / 𝑘 ) = ( 1 / 𝑛 ) )
5 3 4 ifbieq2d ⊢ ( 𝑘 = 𝑛 → if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) )
6 oveq2 ⊢ ( 𝑘 = 𝑛 → ( 𝐴 ↑ 𝑘 ) = ( 𝐴 ↑ 𝑛 ) )
7 5 6 oveq12d ⊢ ( 𝑘 = 𝑛 → ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) )
8 eqid ⊢ ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) = ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) )
9 ovex ⊢ ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ∈ V
10 7 8 9 fvmpt ⊢ ( 𝑛 ∈ ℕ0 → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) )
11 10 adantl ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) )
12 0cnd ⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → 0 ∈ ℂ )
13 elnn0 ⊢ ( 𝑛 ∈ ℕ0 ↔ ( 𝑛 ∈ ℕ ∨ 𝑛 = 0 ) )
14 13 bilani ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 ∈ ℕ ∨ 𝑛 = 0 ) )
15 14 ord ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( ¬ 𝑛 ∈ ℕ → 𝑛 = 0 ) )
16 15 con1d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( ¬ 𝑛 = 0 → 𝑛 ∈ ℕ ) )
17 16 imp ⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → 𝑛 ∈ ℕ )
18 17 nnrecred ⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 1 / 𝑛 ) ∈ ℝ )
19 18 recnd ⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 1 / 𝑛 ) ∈ ℂ )
20 12 19 ifclda ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) ∈ ℂ )
21 expcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑛 ) ∈ ℂ )
22 21 adantlr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑛 ) ∈ ℂ )
23 20 22 mulcld ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ0 ) → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ∈ ℂ )
24 logtayllem ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ∈ dom ⇝ )
25 1 2 11 23 24 isumclim2 ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ⇝ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) )
26 simpl ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 𝐴 ∈ ℂ )
27 0cn ⊢ 0 ∈ ℂ
28 eqid ⊢ ( abs ∘ − ) = ( abs ∘ − )
29 28 cnmetdval ⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ) → ( 𝐴 ( abs ∘ − ) 0 ) = ( abs ‘ ( 𝐴 − 0 ) ) )
30 26 27 29 sylancl ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ( abs ∘ − ) 0 ) = ( abs ‘ ( 𝐴 − 0 ) ) )
31 subid1 ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 − 0 ) = 𝐴 )
32 31 adantr ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 − 0 ) = 𝐴 )
33 32 fveq2d ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ ( 𝐴 − 0 ) ) = ( abs ‘ 𝐴 ) )
34 30 33 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ( abs ∘ − ) 0 ) = ( abs ‘ 𝐴 ) )
35 simpr ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( abs ‘ 𝐴 ) < 1 )
36 34 35 eqbrtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ( abs ∘ − ) 0 ) < 1 )
37 cnxmet ⊢ ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ )
38 1xr ⊢ 1 ∈ ℝ*
39 elbl3 ⊢ ( ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 1 ∈ ℝ* ) ∧ ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ) ) → ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝐴 ( abs ∘ − ) 0 ) < 1 ) )
40 37 38 39 mpanl12 ⊢ ( ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝐴 ( abs ∘ − ) 0 ) < 1 ) )
41 27 26 40 sylancr ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝐴 ( abs ∘ − ) 0 ) < 1 ) )
42 36 41 mpbird ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) )
43 tru ⊢ ⊤
44 eqid ⊢ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) = ( 0 ( ball ‘ ( abs ∘ − ) ) 1 )
45 0cnd ⊢ ( ⊤ → 0 ∈ ℂ )
46 38 a1i ⊢ ( ⊤ → 1 ∈ ℝ* )
47 ax-1cn ⊢ 1 ∈ ℂ
48 blssm ⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ* ) → ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ℂ )
49 37 27 38 48 mp3an ⊢ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ℂ
50 49 sseli ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 𝑦 ∈ ℂ )
51 subcl ⊢ ( ( 1 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 1 − 𝑦 ) ∈ ℂ )
52 47 50 51 sylancr ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 − 𝑦 ) ∈ ℂ )
53 50 abscld ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) ∈ ℝ )
54 28 cnmetdval ⊢ ( ( 𝑦 ∈ ℂ ∧ 0 ∈ ℂ ) → ( 𝑦 ( abs ∘ − ) 0 ) = ( abs ‘ ( 𝑦 − 0 ) ) )
55 50 27 54 sylancl ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑦 ( abs ∘ − ) 0 ) = ( abs ‘ ( 𝑦 − 0 ) ) )
56 50 subid1d ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑦 − 0 ) = 𝑦 )
57 56 fveq2d ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ ( 𝑦 − 0 ) ) = ( abs ‘ 𝑦 ) )
58 55 57 eqtrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑦 ( abs ∘ − ) 0 ) = ( abs ‘ 𝑦 ) )
59 elbl3 ⊢ ( ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 1 ∈ ℝ* ) ∧ ( 0 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝑦 ( abs ∘ − ) 0 ) < 1 ) )
60 37 38 59 mpanl12 ⊢ ( ( 0 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝑦 ( abs ∘ − ) 0 ) < 1 ) )
61 27 50 60 sylancr ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 𝑦 ( abs ∘ − ) 0 ) < 1 ) )
62 61 ibi ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑦 ( abs ∘ − ) 0 ) < 1 )
63 58 62 eqbrtrrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) < 1 )
64 53 63 gtned ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 1 ≠ ( abs ‘ 𝑦 ) )
65 abs1 ⊢ ( abs ‘ 1 ) = 1
66 fveq2 ⊢ ( 1 = 𝑦 → ( abs ‘ 1 ) = ( abs ‘ 𝑦 ) )
67 65 66 eqtr3id ⊢ ( 1 = 𝑦 → 1 = ( abs ‘ 𝑦 ) )
68 67 necon3i ⊢ ( 1 ≠ ( abs ‘ 𝑦 ) → 1 ≠ 𝑦 )
69 64 68 syl ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 1 ≠ 𝑦 )
70 subeq0 ⊢ ( ( 1 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( 1 − 𝑦 ) = 0 ↔ 1 = 𝑦 ) )
71 70 necon3bid ⊢ ( ( 1 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( 1 − 𝑦 ) ≠ 0 ↔ 1 ≠ 𝑦 ) )
72 47 50 71 sylancr ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 1 − 𝑦 ) ≠ 0 ↔ 1 ≠ 𝑦 ) )
73 69 72 mpbird ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 − 𝑦 ) ≠ 0 )
74 52 73 logcld ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( log ‘ ( 1 − 𝑦 ) ) ∈ ℂ )
75 74 negcld ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → - ( log ‘ ( 1 − 𝑦 ) ) ∈ ℂ )
76 75 adantl ⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → - ( log ‘ ( 1 − 𝑦 ) ) ∈ ℂ )
77 76 fmpttd ⊢ ( ⊤ → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) : ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⟶ ℂ )
78 50 absge0d ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 0 ≤ ( abs ‘ 𝑦 ) )
79 53 rexrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) ∈ ℝ* )
80 peano2re ⊢ ( ( abs ‘ 𝑦 ) ∈ ℝ → ( ( abs ‘ 𝑦 ) + 1 ) ∈ ℝ )
81 53 80 syl ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( abs ‘ 𝑦 ) + 1 ) ∈ ℝ )
82 81 rehalfcld ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ∈ ℝ )
83 82 rexrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ∈ ℝ* )
84 iccssxr ⊢ ( 0 [,] +∞ ) ⊆ ℝ*
85 eqeq1 ⊢ ( 𝑚 = 𝑗 → ( 𝑚 = 0 ↔ 𝑗 = 0 ) )
86 oveq2 ⊢ ( 𝑚 = 𝑗 → ( 1 / 𝑚 ) = ( 1 / 𝑗 ) )
87 85 86 ifbieq2d ⊢ ( 𝑚 = 𝑗 → if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) = if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) )
88 eqid ⊢ ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) = ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) )
89 c0ex ⊢ 0 ∈ V
90 ovex ⊢ ( 1 / 𝑗 ) ∈ V
91 89 90 ifex ⊢ if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) ∈ V
92 87 88 91 fvmpt ⊢ ( 𝑗 ∈ ℕ0 → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑗 ) = if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) )
93 92 eqcomd ⊢ ( 𝑗 ∈ ℕ0 → if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) = ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑗 ) )
94 93 oveq1d ⊢ ( 𝑗 ∈ ℕ0 → ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) = ( ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑗 ) · ( 𝑥 ↑ 𝑗 ) ) )
95 94 mpteq2ia ⊢ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑗 ) · ( 𝑥 ↑ 𝑗 ) ) )
96 95 mpteq2i ⊢ ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) = ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑗 ) · ( 𝑥 ↑ 𝑗 ) ) ) )
97 0cnd ⊢ ( ( ( ⊤ ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑚 = 0 ) → 0 ∈ ℂ )
98 nn0cn ⊢ ( 𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ )
99 98 adantl ⊢ ( ( ⊤ ∧ 𝑚 ∈ ℕ0 ) → 𝑚 ∈ ℂ )
100 neqne ⊢ ( ¬ 𝑚 = 0 → 𝑚 ≠ 0 )
101 reccl ⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑚 ≠ 0 ) → ( 1 / 𝑚 ) ∈ ℂ )
102 99 100 101 syl2an ⊢ ( ( ( ⊤ ∧ 𝑚 ∈ ℕ0 ) ∧ ¬ 𝑚 = 0 ) → ( 1 / 𝑚 ) ∈ ℂ )
103 97 102 ifclda ⊢ ( ( ⊤ ∧ 𝑚 ∈ ℕ0 ) → if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ∈ ℂ )
104 103 fmpttd ⊢ ( ⊤ → ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) : ℕ0 ⟶ ℂ )
105 recn ⊢ ( 𝑟 ∈ ℝ → 𝑟 ∈ ℂ )
106 oveq1 ⊢ ( 𝑥 = 𝑟 → ( 𝑥 ↑ 𝑗 ) = ( 𝑟 ↑ 𝑗 ) )
107 106 oveq2d ⊢ ( 𝑥 = 𝑟 → ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) = ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) )
108 107 mpteq2dv ⊢ ( 𝑥 = 𝑟 → ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) )
109 eqid ⊢ ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) = ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) )
110 nn0ex ⊢ ℕ0 ∈ V
111 110 mptex ⊢ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ∈ V
112 108 109 111 fvmpt ⊢ ( 𝑟 ∈ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) )
113 105 112 syl ⊢ ( 𝑟 ∈ ℝ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) )
114 113 eqcomd ⊢ ( 𝑟 ∈ ℝ → ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) = ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) )
115 114 seqeq3d ⊢ ( 𝑟 ∈ ℝ → seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) = seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) ) )
116 115 eleq1d ⊢ ( 𝑟 ∈ ℝ → ( seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ ↔ seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) ) ∈ dom ⇝ ) )
117 116 rabbiia ⊢ { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } = { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) ) ∈ dom ⇝ }
118 117 supeq1i ⊢ sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) = sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑟 ) ) ∈ dom ⇝ } , ℝ* , < )
119 96 104 118 radcnvcl ⊢ ( ⊤ → sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ( 0 [,] +∞ ) )
120 84 119 sselid ⊢ ( ⊤ → sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ* )
121 43 120 mp1i ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ* )
122 1re ⊢ 1 ∈ ℝ
123 avglt1 ⊢ ( ( ( abs ‘ 𝑦 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( abs ‘ 𝑦 ) < 1 ↔ ( abs ‘ 𝑦 ) < ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) )
124 53 122 123 sylancl ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( abs ‘ 𝑦 ) < 1 ↔ ( abs ‘ 𝑦 ) < ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) )
125 63 124 mpbid ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) < ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) )
126 0red ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 0 ∈ ℝ )
127 126 53 82 78 125 lelttrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 0 < ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) )
128 126 82 127 ltled ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 0 ≤ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) )
129 82 128 absidd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) = ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) )
130 43 104 mp1i ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) : ℕ0 ⟶ ℂ )
131 82 recnd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ∈ ℂ )
132 oveq1 ⊢ ( 𝑥 = ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) → ( 𝑥 ↑ 𝑗 ) = ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) )
133 132 oveq2d ⊢ ( 𝑥 = ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) → ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) = ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) )
134 133 mpteq2dv ⊢ ( 𝑥 = ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) → ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) )
135 110 mptex ⊢ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ∈ V
136 134 109 135 fvmpt ⊢ ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ∈ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) )
137 131 136 syl ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) )
138 137 seqeq3d ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) ) = seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ) )
139 avglt2 ⊢ ( ( ( abs ‘ 𝑦 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( abs ‘ 𝑦 ) < 1 ↔ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) < 1 ) )
140 53 122 139 sylancl ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( abs ‘ 𝑦 ) < 1 ↔ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) < 1 ) )
141 63 140 mpbid ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) < 1 )
142 129 141 eqbrtrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) < 1 )
143 logtayllem ⊢ ( ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ∈ ℂ ∧ ( abs ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) < 1 ) → seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ) ∈ dom ⇝ )
144 131 142 143 syl2anc ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ↑ 𝑗 ) ) ) ) ∈ dom ⇝ )
145 138 144 eqeltrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → seq 0 ( + , ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) ) ∈ dom ⇝ )
146 96 130 118 131 145 radcnvle ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ) ≤ sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) )
147 129 146 eqbrtrrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( ( abs ‘ 𝑦 ) + 1 ) / 2 ) ≤ sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) )
148 79 83 121 125 147 xrltletrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) < sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) )
149 0re ⊢ 0 ∈ ℝ
150 elico2 ⊢ ( ( 0 ∈ ℝ ∧ sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ* ) → ( ( abs ‘ 𝑦 ) ∈ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ↔ ( ( abs ‘ 𝑦 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝑦 ) ∧ ( abs ‘ 𝑦 ) < sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) )
151 149 121 150 sylancr ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( abs ‘ 𝑦 ) ∈ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ↔ ( ( abs ‘ 𝑦 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝑦 ) ∧ ( abs ‘ 𝑦 ) < sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) )
152 53 78 148 151 mpbir3and ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ 𝑦 ) ∈ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) )
153 absf ⊢ abs : ℂ ⟶ ℝ
154 ffn ⊢ ( abs : ℂ ⟶ ℝ → abs Fn ℂ )
155 elpreima ⊢ ( abs Fn ℂ → ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↔ ( 𝑦 ∈ ℂ ∧ ( abs ‘ 𝑦 ) ∈ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) )
156 153 154 155 mp2b ⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↔ ( 𝑦 ∈ ℂ ∧ ( abs ‘ 𝑦 ) ∈ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) )
157 50 152 156 sylanbrc ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) )
158 cnvimass ⊢ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ⊆ dom abs
159 153 fdmi ⊢ dom abs = ℂ
160 158 159 sseqtri ⊢ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ⊆ ℂ
161 160 sseli ⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) → 𝑦 ∈ ℂ )
162 oveq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ↑ 𝑗 ) = ( 𝑦 ↑ 𝑗 ) )
163 162 oveq2d ⊢ ( 𝑥 = 𝑦 → ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) = ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) )
164 163 mpteq2dv ⊢ ( 𝑥 = 𝑦 → ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) )
165 110 mptex ⊢ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ∈ V
166 164 109 165 fvmpt ⊢ ( 𝑦 ∈ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) )
167 166 adantr ⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) )
168 167 fveq1d ⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) = ( ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ‘ 𝑛 ) )
169 eqeq1 ⊢ ( 𝑗 = 𝑛 → ( 𝑗 = 0 ↔ 𝑛 = 0 ) )
170 oveq2 ⊢ ( 𝑗 = 𝑛 → ( 1 / 𝑗 ) = ( 1 / 𝑛 ) )
171 169 170 ifbieq2d ⊢ ( 𝑗 = 𝑛 → if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) )
172 oveq2 ⊢ ( 𝑗 = 𝑛 → ( 𝑦 ↑ 𝑗 ) = ( 𝑦 ↑ 𝑛 ) )
173 171 172 oveq12d ⊢ ( 𝑗 = 𝑛 → ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) )
174 eqid ⊢ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) )
175 ovex ⊢ ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ∈ V
176 173 174 175 fvmpt ⊢ ( 𝑛 ∈ ℕ0 → ( ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ‘ 𝑛 ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) )
177 176 adantl ⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑦 ↑ 𝑗 ) ) ) ‘ 𝑛 ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) )
178 168 177 eqtr2d ⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) )
179 178 sumeq2dv ⊢ ( 𝑦 ∈ ℂ → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = Σ 𝑛 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) )
180 161 179 syl ⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = Σ 𝑛 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) )
181 180 mpteq2ia ⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) = ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) )
182 eqid ⊢ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) = ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) )
183 eqid ⊢ if ( sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ , ( ( ( abs ‘ 𝑧 ) + sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) / 2 ) , ( ( abs ‘ 𝑧 ) + 1 ) ) = if ( sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ , ( ( ( abs ‘ 𝑧 ) + sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) / 2 ) , ( ( abs ‘ 𝑧 ) + 1 ) )
184 96 181 104 118 182 183 psercn ⊢ ( ⊤ → ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ∈ ( ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) –cn→ ℂ ) )
185 cncff ⊢ ( ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ∈ ( ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) –cn→ ℂ ) → ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) : ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ⟶ ℂ )
186 184 185 syl ⊢ ( ⊤ → ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) : ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ⟶ ℂ )
187 186 fvmptelcdm ⊢ ( ( ⊤ ∧ 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ∈ ℂ )
188 157 187 sylan2 ⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ∈ ℂ )
189 188 fmpttd ⊢ ( ⊤ → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) : ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⟶ ℂ )
190 cnelprrecn ⊢ ℂ ∈ { ℝ , ℂ }
191 190 a1i ⊢ ( ⊤ → ℂ ∈ { ℝ , ℂ } )
192 74 adantl ⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → ( log ‘ ( 1 − 𝑦 ) ) ∈ ℂ )
193 ovexd ⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) ∈ V )
194 28 cnmetdval ⊢ ( ( 1 ∈ ℂ ∧ ( 1 − 𝑦 ) ∈ ℂ ) → ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) = ( abs ‘ ( 1 − ( 1 − 𝑦 ) ) ) )
195 47 52 194 sylancr ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) = ( abs ‘ ( 1 − ( 1 − 𝑦 ) ) ) )
196 nncan ⊢ ( ( 1 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 1 − ( 1 − 𝑦 ) ) = 𝑦 )
197 47 50 196 sylancr ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 − ( 1 − 𝑦 ) ) = 𝑦 )
198 197 fveq2d ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( abs ‘ ( 1 − ( 1 − 𝑦 ) ) ) = ( abs ‘ 𝑦 ) )
199 195 198 eqtrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) = ( abs ‘ 𝑦 ) )
200 199 63 eqbrtrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) < 1 )
201 elbl ⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ* ) → ( ( 1 − 𝑦 ) ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( ( 1 − 𝑦 ) ∈ ℂ ∧ ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) < 1 ) ) )
202 37 47 38 201 mp3an ⊢ ( ( 1 − 𝑦 ) ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( ( 1 − 𝑦 ) ∈ ℂ ∧ ( 1 ( abs ∘ − ) ( 1 − 𝑦 ) ) < 1 ) )
203 52 200 202 sylanbrc ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 − 𝑦 ) ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) )
204 203 adantl ⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → ( 1 − 𝑦 ) ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) )
205 neg1cn ⊢ - 1 ∈ ℂ
206 205 a1i ⊢ ( ( ⊤ ∧ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → - 1 ∈ ℂ )
207 eqid ⊢ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) = ( 1 ( ball ‘ ( abs ∘ − ) ) 1 )
208 207 dvlog2lem ⊢ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ( ℂ ∖ ( -∞ (,] 0 ) )
209 208 sseli ⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → 𝑥 ∈ ( ℂ ∖ ( -∞ (,] 0 ) ) )
210 209 eldifad ⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → 𝑥 ∈ ℂ )
211 eqid ⊢ ( ℂ ∖ ( -∞ (,] 0 ) ) = ( ℂ ∖ ( -∞ (,] 0 ) )
212 211 logdmn0 ⊢ ( 𝑥 ∈ ( ℂ ∖ ( -∞ (,] 0 ) ) → 𝑥 ≠ 0 )
213 209 212 syl ⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → 𝑥 ≠ 0 )
214 210 213 logcld ⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( log ‘ 𝑥 ) ∈ ℂ )
215 214 adantl ⊢ ( ( ⊤ ∧ 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → ( log ‘ 𝑥 ) ∈ ℂ )
216 ovexd ⊢ ( ( ⊤ ∧ 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) → ( 1 / 𝑥 ) ∈ V )
217 simpr ⊢ ( ( ⊤ ∧ 𝑦 ∈ ℂ ) → 𝑦 ∈ ℂ )
218 47 217 51 sylancr ⊢ ( ( ⊤ ∧ 𝑦 ∈ ℂ ) → ( 1 − 𝑦 ) ∈ ℂ )
219 205 a1i ⊢ ( ( ⊤ ∧ 𝑦 ∈ ℂ ) → - 1 ∈ ℂ )
220 1cnd ⊢ ( ( ⊤ ∧ 𝑦 ∈ ℂ ) → 1 ∈ ℂ )
221 0cnd ⊢ ( ( ⊤ ∧ 𝑦 ∈ ℂ ) → 0 ∈ ℂ )
222 1cnd ⊢ ( ⊤ → 1 ∈ ℂ )
223 191 222 dvmptc ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ℂ ↦ 1 ) ) = ( 𝑦 ∈ ℂ ↦ 0 ) )
224 191 dvmptid ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ℂ ↦ 𝑦 ) ) = ( 𝑦 ∈ ℂ ↦ 1 ) )
225 191 220 221 223 217 220 224 dvmptsub ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( 1 − 𝑦 ) ) ) = ( 𝑦 ∈ ℂ ↦ ( 0 − 1 ) ) )
226 df-neg ⊢ - 1 = ( 0 − 1 )
227 226 mpteq2i ⊢ ( 𝑦 ∈ ℂ ↦ - 1 ) = ( 𝑦 ∈ ℂ ↦ ( 0 − 1 ) )
228 225 227 eqtr4di ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( 1 − 𝑦 ) ) ) = ( 𝑦 ∈ ℂ ↦ - 1 ) )
229 49 a1i ⊢ ( ⊤ → ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ℂ )
230 eqid ⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
231 230 cnfldtopon ⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ )
232 231 toponrestid ⊢ ( TopOpen ‘ ℂfld ) = ( ( TopOpen ‘ ℂfld ) ↾t ℂ )
233 230 cnfldtopn ⊢ ( TopOpen ‘ ℂfld ) = ( MetOpen ‘ ( abs ∘ − ) )
234 233 blopn ⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ* ) → ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∈ ( TopOpen ‘ ℂfld ) )
235 37 27 38 234 mp3an ⊢ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∈ ( TopOpen ‘ ℂfld )
236 235 a1i ⊢ ( ⊤ → ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∈ ( TopOpen ‘ ℂfld ) )
237 191 218 219 228 229 232 230 236 dvmptres ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 − 𝑦 ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - 1 ) )
238 logf1o ⊢ log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log
239 f1of ⊢ ( log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log → log : ( ℂ ∖ { 0 } ) ⟶ ran log )
240 238 239 ax-mp ⊢ log : ( ℂ ∖ { 0 } ) ⟶ ran log
241 211 logdmss ⊢ ( ℂ ∖ ( -∞ (,] 0 ) ) ⊆ ( ℂ ∖ { 0 } )
242 208 241 sstri ⊢ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ( ℂ ∖ { 0 } )
243 fssres ⊢ ( ( log : ( ℂ ∖ { 0 } ) ⟶ ran log ∧ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ( ℂ ∖ { 0 } ) ) → ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) : ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⟶ ran log )
244 240 242 243 mp2an ⊢ ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) : ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⟶ ran log
245 244 a1i ⊢ ( ⊤ → ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) : ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⟶ ran log )
246 245 feqmptd ⊢ ( ⊤ → ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) = ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) ‘ 𝑥 ) ) )
247 fvres ⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) ‘ 𝑥 ) = ( log ‘ 𝑥 ) )
248 247 mpteq2ia ⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) ‘ 𝑥 ) ) = ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( log ‘ 𝑥 ) )
249 246 248 eqtrdi ⊢ ( ⊤ → ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) = ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( log ‘ 𝑥 ) ) )
250 249 oveq2d ⊢ ( ⊤ → ( ℂ D ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) ) = ( ℂ D ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( log ‘ 𝑥 ) ) ) )
251 207 dvlog2 ⊢ ( ℂ D ( log ↾ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) ) = ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / 𝑥 ) )
252 250 251 eqtr3di ⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( log ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / 𝑥 ) ) )
253 fveq2 ⊢ ( 𝑥 = ( 1 − 𝑦 ) → ( log ‘ 𝑥 ) = ( log ‘ ( 1 − 𝑦 ) ) )
254 oveq2 ⊢ ( 𝑥 = ( 1 − 𝑦 ) → ( 1 / 𝑥 ) = ( 1 / ( 1 − 𝑦 ) ) )
255 191 191 204 206 215 216 237 252 253 254 dvmptco ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( log ‘ ( 1 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) ) )
256 191 192 193 255 dvmptneg ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) ) )
257 52 73 reccld ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 / ( 1 − 𝑦 ) ) ∈ ℂ )
258 mulcom ⊢ ( ( ( 1 / ( 1 − 𝑦 ) ) ∈ ℂ ∧ - 1 ∈ ℂ ) → ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) = ( - 1 · ( 1 / ( 1 − 𝑦 ) ) ) )
259 257 205 258 sylancl ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) = ( - 1 · ( 1 / ( 1 − 𝑦 ) ) ) )
260 257 mulm1d ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( - 1 · ( 1 / ( 1 − 𝑦 ) ) ) = - ( 1 / ( 1 − 𝑦 ) ) )
261 259 260 eqtrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) = - ( 1 / ( 1 − 𝑦 ) ) )
262 261 negeqd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → - ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) = - - ( 1 / ( 1 − 𝑦 ) ) )
263 257 negnegd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → - - ( 1 / ( 1 − 𝑦 ) ) = ( 1 / ( 1 − 𝑦 ) ) )
264 262 263 eqtrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → - ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) = ( 1 / ( 1 − 𝑦 ) ) )
265 264 mpteq2ia ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( ( 1 / ( 1 − 𝑦 ) ) · - 1 ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) )
266 256 265 eqtrdi ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) )
267 266 dmeqd ⊢ ( ⊤ → dom ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ) = dom ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) )
268 dmmptg ⊢ ( ∀ 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ( 1 / ( 1 − 𝑦 ) ) ∈ V → dom ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) = ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) )
269 ovexd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( 1 / ( 1 − 𝑦 ) ) ∈ V )
270 268 269 mprg ⊢ dom ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) = ( 0 ( ball ‘ ( abs ∘ − ) ) 1 )
271 267 270 eqtrdi ⊢ ( ⊤ → dom ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ) = ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) )
272 sumex ⊢ Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ∈ V
273 272 a1i ⊢ ( ( ⊤ ∧ 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ) → Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ∈ V )
274 fveq2 ⊢ ( 𝑛 = 𝑘 → ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) = ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑘 ) )
275 274 cbvsumv ⊢ Σ 𝑛 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑛 ) = Σ 𝑘 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑘 )
276 180 275 eqtrdi ⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = Σ 𝑘 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑘 ) )
277 276 mpteq2ia ⊢ ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) = ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑘 ∈ ℕ0 ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑥 ↑ 𝑗 ) ) ) ) ‘ 𝑦 ) ‘ 𝑘 ) )
278 eqid ⊢ ( 0 ( ball ‘ ( abs ∘ − ) ) ( ( ( abs ‘ 𝑧 ) + if ( sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ , ( ( ( abs ‘ 𝑧 ) + sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) / 2 ) , ( ( abs ‘ 𝑧 ) + 1 ) ) ) / 2 ) ) = ( 0 ( ball ‘ ( abs ∘ − ) ) ( ( ( abs ‘ 𝑧 ) + if ( sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ∈ ℝ , ( ( ( abs ‘ 𝑧 ) + sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) / 2 ) , ( ( abs ‘ 𝑧 ) + 1 ) ) ) / 2 ) )
279 96 277 104 118 182 183 278 pserdv2 ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ) = ( 𝑦 ∈ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) ↦ Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ) )
280 157 ssriv ⊢ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) )
281 280 a1i ⊢ ( ⊤ → ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ( ◡ abs “ ( 0 [,) sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( if ( 𝑗 = 0 , 0 , ( 1 / 𝑗 ) ) · ( 𝑟 ↑ 𝑗 ) ) ) ) ∈ dom ⇝ } , ℝ* , < ) ) ) )
282 191 187 273 279 281 232 230 236 dvmptres ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ) )
283 nnnn0 ⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 )
284 283 adantl ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ0 )
285 eqeq1 ⊢ ( 𝑚 = 𝑛 → ( 𝑚 = 0 ↔ 𝑛 = 0 ) )
286 oveq2 ⊢ ( 𝑚 = 𝑛 → ( 1 / 𝑚 ) = ( 1 / 𝑛 ) )
287 285 286 ifbieq2d ⊢ ( 𝑚 = 𝑛 → if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) )
288 ovex ⊢ ( 1 / 𝑛 ) ∈ V
289 89 288 ifex ⊢ if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) ∈ V
290 287 88 289 fvmpt ⊢ ( 𝑛 ∈ ℕ0 → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) )
291 284 290 syl ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) )
292 nnne0 ⊢ ( 𝑛 ∈ ℕ → 𝑛 ≠ 0 )
293 292 adantl ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ≠ 0 )
294 293 neneqd ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ¬ 𝑛 = 0 )
295 294 iffalsed ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) = ( 1 / 𝑛 ) )
296 291 295 eqtrd ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) = ( 1 / 𝑛 ) )
297 296 oveq2d ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) = ( 𝑛 · ( 1 / 𝑛 ) ) )
298 nncn ⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℂ )
299 298 adantl ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℂ )
300 299 293 recidd ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( 𝑛 · ( 1 / 𝑛 ) ) = 1 )
301 297 300 eqtrd ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) = 1 )
302 301 oveq1d ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) = ( 1 · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) )
303 nnm1nn0 ⊢ ( 𝑛 ∈ ℕ → ( 𝑛 − 1 ) ∈ ℕ0 )
304 expcl ⊢ ( ( 𝑦 ∈ ℂ ∧ ( 𝑛 − 1 ) ∈ ℕ0 ) → ( 𝑦 ↑ ( 𝑛 − 1 ) ) ∈ ℂ )
305 50 303 304 syl2an ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( 𝑦 ↑ ( 𝑛 − 1 ) ) ∈ ℂ )
306 305 mullidd ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( 1 · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) = ( 𝑦 ↑ ( 𝑛 − 1 ) ) )
307 302 306 eqtrd ⊢ ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) = ( 𝑦 ↑ ( 𝑛 − 1 ) ) )
308 307 sumeq2dv ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) = Σ 𝑛 ∈ ℕ ( 𝑦 ↑ ( 𝑛 − 1 ) ) )
309 nnuz ⊢ ℕ = ( ℤ≥ ‘ 1 )
310 1e0p1 ⊢ 1 = ( 0 + 1 )
311 310 fveq2i ⊢ ( ℤ≥ ‘ 1 ) = ( ℤ≥ ‘ ( 0 + 1 ) )
312 309 311 eqtri ⊢ ℕ = ( ℤ≥ ‘ ( 0 + 1 ) )
313 oveq1 ⊢ ( 𝑛 = ( 1 + 𝑚 ) → ( 𝑛 − 1 ) = ( ( 1 + 𝑚 ) − 1 ) )
314 313 oveq2d ⊢ ( 𝑛 = ( 1 + 𝑚 ) → ( 𝑦 ↑ ( 𝑛 − 1 ) ) = ( 𝑦 ↑ ( ( 1 + 𝑚 ) − 1 ) ) )
315 1zzd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 1 ∈ ℤ )
316 0zd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → 0 ∈ ℤ )
317 1 312 314 315 316 305 isumshft ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → Σ 𝑛 ∈ ℕ ( 𝑦 ↑ ( 𝑛 − 1 ) ) = Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ ( ( 1 + 𝑚 ) − 1 ) ) )
318 pncan2 ⊢ ( ( 1 ∈ ℂ ∧ 𝑚 ∈ ℂ ) → ( ( 1 + 𝑚 ) − 1 ) = 𝑚 )
319 47 98 318 sylancr ⊢ ( 𝑚 ∈ ℕ0 → ( ( 1 + 𝑚 ) − 1 ) = 𝑚 )
320 319 oveq2d ⊢ ( 𝑚 ∈ ℕ0 → ( 𝑦 ↑ ( ( 1 + 𝑚 ) − 1 ) ) = ( 𝑦 ↑ 𝑚 ) )
321 320 sumeq2i ⊢ Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ ( ( 1 + 𝑚 ) − 1 ) ) = Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ 𝑚 )
322 317 321 eqtrdi ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → Σ 𝑛 ∈ ℕ ( 𝑦 ↑ ( 𝑛 − 1 ) ) = Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ 𝑚 ) )
323 geoisum ⊢ ( ( 𝑦 ∈ ℂ ∧ ( abs ‘ 𝑦 ) < 1 ) → Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ 𝑚 ) = ( 1 / ( 1 − 𝑦 ) ) )
324 50 63 323 syl2anc ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → Σ 𝑚 ∈ ℕ0 ( 𝑦 ↑ 𝑚 ) = ( 1 / ( 1 − 𝑦 ) ) )
325 308 322 324 3eqtrd ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) = ( 1 / ( 1 − 𝑦 ) ) )
326 325 mpteq2ia ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ ( ( 𝑛 · ( ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 0 , ( 1 / 𝑚 ) ) ) ‘ 𝑛 ) ) · ( 𝑦 ↑ ( 𝑛 − 1 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) )
327 282 326 eqtrdi ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ ( 1 / ( 1 − 𝑦 ) ) ) )
328 266 327 eqtr4d ⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ) = ( ℂ D ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ) )
329 1rp ⊢ 1 ∈ ℝ+
330 blcntr ⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ+ ) → 0 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) )
331 37 27 329 330 mp3an ⊢ 0 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 )
332 331 a1i ⊢ ( ⊤ → 0 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) )
333 oveq2 ⊢ ( 𝑦 = 0 → ( 1 − 𝑦 ) = ( 1 − 0 ) )
334 1m0e1 ⊢ ( 1 − 0 ) = 1
335 333 334 eqtrdi ⊢ ( 𝑦 = 0 → ( 1 − 𝑦 ) = 1 )
336 335 fveq2d ⊢ ( 𝑦 = 0 → ( log ‘ ( 1 − 𝑦 ) ) = ( log ‘ 1 ) )
337 log1 ⊢ ( log ‘ 1 ) = 0
338 336 337 eqtrdi ⊢ ( 𝑦 = 0 → ( log ‘ ( 1 − 𝑦 ) ) = 0 )
339 338 negeqd ⊢ ( 𝑦 = 0 → - ( log ‘ ( 1 − 𝑦 ) ) = - 0 )
340 neg0 ⊢ - 0 = 0
341 339 340 eqtrdi ⊢ ( 𝑦 = 0 → - ( log ‘ ( 1 − 𝑦 ) ) = 0 )
342 eqid ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) )
343 341 342 89 fvmpt ⊢ ( 0 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 0 ) = 0 )
344 331 343 mp1i ⊢ ( ⊤ → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 0 ) = 0 )
345 oveq1 ⊢ ( 0 = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) → ( 0 · ( 𝑦 ↑ 𝑛 ) ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) )
346 345 eqeq1d ⊢ ( 0 = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) → ( ( 0 · ( 𝑦 ↑ 𝑛 ) ) = 0 ↔ ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = 0 ) )
347 oveq1 ⊢ ( ( 1 / 𝑛 ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) → ( ( 1 / 𝑛 ) · ( 𝑦 ↑ 𝑛 ) ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) )
348 347 eqeq1d ⊢ ( ( 1 / 𝑛 ) = if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) → ( ( ( 1 / 𝑛 ) · ( 𝑦 ↑ 𝑛 ) ) = 0 ↔ ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = 0 ) )
349 simpll ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → 𝑦 = 0 )
350 349 27 eqeltrdi ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → 𝑦 ∈ ℂ )
351 simplr ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → 𝑛 ∈ ℕ0 )
352 350 351 expcld ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → ( 𝑦 ↑ 𝑛 ) ∈ ℂ )
353 352 mul02d ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → ( 0 · ( 𝑦 ↑ 𝑛 ) ) = 0 )
354 simpll ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → 𝑦 = 0 )
355 354 oveq1d ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 𝑦 ↑ 𝑛 ) = ( 0 ↑ 𝑛 ) )
356 13 bilani ⊢ ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 ∈ ℕ ∨ 𝑛 = 0 ) )
357 356 ord ⊢ ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) → ( ¬ 𝑛 ∈ ℕ → 𝑛 = 0 ) )
358 357 con1d ⊢ ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) → ( ¬ 𝑛 = 0 → 𝑛 ∈ ℕ ) )
359 358 imp ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → 𝑛 ∈ ℕ )
360 359 0expd ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 0 ↑ 𝑛 ) = 0 )
361 355 360 eqtrd ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 𝑦 ↑ 𝑛 ) = 0 )
362 361 oveq2d ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( ( 1 / 𝑛 ) · ( 𝑦 ↑ 𝑛 ) ) = ( ( 1 / 𝑛 ) · 0 ) )
363 359 nnrecred ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 1 / 𝑛 ) ∈ ℝ )
364 363 recnd ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( 1 / 𝑛 ) ∈ ℂ )
365 364 mul01d ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( ( 1 / 𝑛 ) · 0 ) = 0 )
366 362 365 eqtrd ⊢ ( ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ( ( 1 / 𝑛 ) · ( 𝑦 ↑ 𝑛 ) ) = 0 )
367 346 348 353 366 ifbothda ⊢ ( ( 𝑦 = 0 ∧ 𝑛 ∈ ℕ0 ) → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = 0 )
368 367 sumeq2dv ⊢ ( 𝑦 = 0 → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = Σ 𝑛 ∈ ℕ0 0 )
369 1 eqimssi ⊢ ℕ0 ⊆ ( ℤ≥ ‘ 0 )
370 369 orci ⊢ ( ℕ0 ⊆ ( ℤ≥ ‘ 0 ) ∨ ℕ0 ∈ Fin )
371 sumz ⊢ ( ( ℕ0 ⊆ ( ℤ≥ ‘ 0 ) ∨ ℕ0 ∈ Fin ) → Σ 𝑛 ∈ ℕ0 0 = 0 )
372 370 371 ax-mp ⊢ Σ 𝑛 ∈ ℕ0 0 = 0
373 368 372 eqtrdi ⊢ ( 𝑦 = 0 → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = 0 )
374 eqid ⊢ ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) )
375 373 374 89 fvmpt ⊢ ( 0 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 0 ) = 0 )
376 331 375 mp1i ⊢ ( ⊤ → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 0 ) = 0 )
377 344 376 eqtr4d ⊢ ( ⊤ → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 0 ) = ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 0 ) )
378 44 45 46 77 189 271 328 332 377 dv11cn ⊢ ( ⊤ → ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) = ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) )
379 378 fveq1d ⊢ ( ⊤ → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 𝐴 ) = ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 𝐴 ) )
380 43 379 mp1i ⊢ ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 𝐴 ) = ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 𝐴 ) )
381 oveq2 ⊢ ( 𝑦 = 𝐴 → ( 1 − 𝑦 ) = ( 1 − 𝐴 ) )
382 381 fveq2d ⊢ ( 𝑦 = 𝐴 → ( log ‘ ( 1 − 𝑦 ) ) = ( log ‘ ( 1 − 𝐴 ) ) )
383 382 negeqd ⊢ ( 𝑦 = 𝐴 → - ( log ‘ ( 1 − 𝑦 ) ) = - ( log ‘ ( 1 − 𝐴 ) ) )
384 negex ⊢ - ( log ‘ ( 1 − 𝐴 ) ) ∈ V
385 383 342 384 fvmpt ⊢ ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ - ( log ‘ ( 1 − 𝑦 ) ) ) ‘ 𝐴 ) = - ( log ‘ ( 1 − 𝐴 ) ) )
386 oveq1 ⊢ ( 𝑦 = 𝐴 → ( 𝑦 ↑ 𝑛 ) = ( 𝐴 ↑ 𝑛 ) )
387 386 oveq2d ⊢ ( 𝑦 = 𝐴 → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) )
388 387 sumeq2sdv ⊢ ( 𝑦 = 𝐴 → Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) = Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) )
389 sumex ⊢ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) ∈ V
390 388 374 389 fvmpt ⊢ ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → ( ( 𝑦 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) ↦ Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝑦 ↑ 𝑛 ) ) ) ‘ 𝐴 ) = Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) )
391 380 385 390 3eqtr3d ⊢ ( 𝐴 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 1 ) → - ( log ‘ ( 1 − 𝐴 ) ) = Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) )
392 42 391 syl ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → - ( log ‘ ( 1 − 𝐴 ) ) = Σ 𝑛 ∈ ℕ0 ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) )
393 25 392 breqtrrd ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ⇝ - ( log ‘ ( 1 − 𝐴 ) ) )
394 seqex ⊢ seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ∈ V
395 394 a1i ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ∈ V )
396 seqex ⊢ seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ∈ V
397 396 a1i ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ∈ V )
398 1zzd ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 1 ∈ ℤ )
399 elnnuz ⊢ ( 𝑛 ∈ ℕ ↔ 𝑛 ∈ ( ℤ≥ ‘ 1 ) )
400 fvres ⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 1 ) → ( ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) ‘ 𝑛 ) = ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ‘ 𝑛 ) )
401 399 400 sylbi ⊢ ( 𝑛 ∈ ℕ → ( ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) ‘ 𝑛 ) = ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ‘ 𝑛 ) )
402 401 eqcomd ⊢ ( 𝑛 ∈ ℕ → ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ‘ 𝑛 ) = ( ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) ‘ 𝑛 ) )
403 addlid ⊢ ( 𝑛 ∈ ℂ → ( 0 + 𝑛 ) = 𝑛 )
404 403 adantl ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℂ ) → ( 0 + 𝑛 ) = 𝑛 )
405 0cnd ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 0 ∈ ℂ )
406 1eluzge0 ⊢ 1 ∈ ( ℤ≥ ‘ 0 )
407 406 a1i ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → 1 ∈ ( ℤ≥ ‘ 0 ) )
408 0cnd ⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑘 = 0 ) → 0 ∈ ℂ )
409 nn0cn ⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ )
410 409 adantl ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℂ )
411 neqne ⊢ ( ¬ 𝑘 = 0 → 𝑘 ≠ 0 )
412 reccl ⊢ ( ( 𝑘 ∈ ℂ ∧ 𝑘 ≠ 0 ) → ( 1 / 𝑘 ) ∈ ℂ )
413 410 411 412 syl2an ⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) ∧ ¬ 𝑘 = 0 ) → ( 1 / 𝑘 ) ∈ ℂ )
414 408 413 ifclda ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) → if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) ∈ ℂ )
415 expcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑘 ) ∈ ℂ )
416 415 adantlr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑘 ) ∈ ℂ )
417 414 416 mulcld ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑘 ∈ ℕ0 ) → ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ∈ ℂ )
418 417 fmpttd ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) : ℕ0 ⟶ ℂ )
419 1nn0 ⊢ 1 ∈ ℕ0
420 ffvelcdm ⊢ ( ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) : ℕ0 ⟶ ℂ ∧ 1 ∈ ℕ0 ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 1 ) ∈ ℂ )
421 418 419 420 sylancl ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 1 ) ∈ ℂ )
422 elfz1eq ⊢ ( 𝑛 ∈ ( 0 ... 0 ) → 𝑛 = 0 )
423 1m1e0 ⊢ ( 1 − 1 ) = 0
424 423 oveq2i ⊢ ( 0 ... ( 1 − 1 ) ) = ( 0 ... 0 )
425 422 424 eleq2s ⊢ ( 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) → 𝑛 = 0 )
426 425 fveq2d ⊢ ( 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 0 ) )
427 0nn0 ⊢ 0 ∈ ℕ0
428 iftrue ⊢ ( 𝑘 = 0 → if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) = 0 )
429 oveq2 ⊢ ( 𝑘 = 0 → ( 𝐴 ↑ 𝑘 ) = ( 𝐴 ↑ 0 ) )
430 428 429 oveq12d ⊢ ( 𝑘 = 0 → ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) = ( 0 · ( 𝐴 ↑ 0 ) ) )
431 ovex ⊢ ( 0 · ( 𝐴 ↑ 0 ) ) ∈ V
432 430 8 431 fvmpt ⊢ ( 0 ∈ ℕ0 → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 0 ) = ( 0 · ( 𝐴 ↑ 0 ) ) )
433 427 432 ax-mp ⊢ ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 0 ) = ( 0 · ( 𝐴 ↑ 0 ) )
434 expcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℕ0 ) → ( 𝐴 ↑ 0 ) ∈ ℂ )
435 26 427 434 sylancl ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 𝐴 ↑ 0 ) ∈ ℂ )
436 435 mul02d ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( 0 · ( 𝐴 ↑ 0 ) ) = 0 )
437 433 436 eqtrid ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 0 ) = 0 )
438 426 437 sylan9eqr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = 0 )
439 404 405 407 421 438 seqid ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) )
440 292 adantl ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ≠ 0 )
441 440 neneqd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ¬ 𝑛 = 0 )
442 441 iffalsed ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) = ( 1 / 𝑛 ) )
443 442 oveq1d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) = ( ( 1 / 𝑛 ) · ( 𝐴 ↑ 𝑛 ) ) )
444 283 22 sylan2 ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( 𝐴 ↑ 𝑛 ) ∈ ℂ )
445 298 adantl ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℂ )
446 444 445 440 divrec2d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) = ( ( 1 / 𝑛 ) · ( 𝐴 ↑ 𝑛 ) ) )
447 443 446 eqtr4d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) = ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) )
448 283 11 sylan2 ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( if ( 𝑛 = 0 , 0 , ( 1 / 𝑛 ) ) · ( 𝐴 ↑ 𝑛 ) ) )
449 id ⊢ ( 𝑘 = 𝑛 → 𝑘 = 𝑛 )
450 6 449 oveq12d ⊢ ( 𝑘 = 𝑛 → ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) = ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) )
451 eqid ⊢ ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) = ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) )
452 ovex ⊢ ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) ∈ V
453 450 451 452 fvmpt ⊢ ( 𝑛 ∈ ℕ → ( ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ‘ 𝑛 ) = ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) )
454 453 adantl ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ‘ 𝑛 ) = ( ( 𝐴 ↑ 𝑛 ) / 𝑛 ) )
455 447 448 454 3eqtr4d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ‘ 𝑛 ) )
456 399 455 sylan2br ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) → ( ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ‘ 𝑛 ) = ( ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ‘ 𝑛 ) )
457 398 456 seqfeq ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) )
458 439 457 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) )
459 458 fveq1d ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ↾ ( ℤ≥ ‘ 1 ) ) ‘ 𝑛 ) = ( seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ‘ 𝑛 ) )
460 402 459 sylan9eqr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) ∧ 𝑛 ∈ ℕ ) → ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ‘ 𝑛 ) = ( seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ‘ 𝑛 ) )
461 309 395 397 398 460 climeq ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → ( seq 0 ( + , ( 𝑘 ∈ ℕ0 ↦ ( if ( 𝑘 = 0 , 0 , ( 1 / 𝑘 ) ) · ( 𝐴 ↑ 𝑘 ) ) ) ) ⇝ - ( log ‘ ( 1 − 𝐴 ) ) ↔ seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ⇝ - ( log ‘ ( 1 − 𝐴 ) ) ) )
462 393 461 mpbid ⊢ ( ( 𝐴 ∈ ℂ ∧ ( abs ‘ 𝐴 ) < 1 ) → seq 1 ( + , ( 𝑘 ∈ ℕ ↦ ( ( 𝐴 ↑ 𝑘 ) / 𝑘 ) ) ) ⇝ - ( log ‘ ( 1 − 𝐴 ) ) )