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Theorem logtayl 25251

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 ↦ ((𝐴↑𝑘) / 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.

Step	Hyp	Ref	Expression
1		nn0uz 12268	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 11981	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ∈ ℤ)
3		eqeq1 2802	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
4		oveq2 7143	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
5	3, 4	ifbieq2d 4450	. . . . . . 7 ⊢ (𝑘 = 𝑛 → if(𝑘 = 0, 0, (1 / 𝑘)) = if(𝑛 = 0, 0, (1 / 𝑛)))
6		oveq2 7143	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝐴↑𝑘) = (𝐴↑𝑛))
7	5, 6	oveq12d 7153	. . . . . 6 ⊢ (𝑘 = 𝑛 → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
8		eqid 2798	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘))) = (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))
9		ovex 7168	. . . . . 6 ⊢ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) ∈ V
10	7, 8, 9	fvmpt 6745	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
11	10	adantl 485	. . . 4 ⊢ (((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 10623	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 0 ∈ ℂ)
13		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
14		elnn0 11887	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
15	13, 14	sylib 221	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (𝑛 ∈ ℕ ∨ 𝑛 = 0))
16	15	ord 861	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 ∈ ℕ → 𝑛 = 0))
17	16	con1d 147	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 = 0 → 𝑛 ∈ ℕ))
18	17	imp 410	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
19	18	nnrecred 11676	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℝ)
20	19	recnd 10658	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℂ)
21	12, 20	ifclda 4459	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → if(𝑛 = 0, 0, (1 / 𝑛)) ∈ ℂ)
22		expcl 13443	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (𝐴↑𝑛) ∈ ℂ)
23	22	adantlr 714	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (𝐴↑𝑛) ∈ ℂ)
24	21, 23	mulcld 10650	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) ∈ ℂ)
25		logtayllem 25250	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ∈ dom ⇝ )
26	1, 2, 11, 24, 25	isumclim2 15105	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ⇝ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
27		simpl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ)
28		0cn 10622	. . . . . . . 8 ⊢ 0 ∈ ℂ
29		eqid 2798	. . . . . . . . 9 ⊢ (abs ∘ − ) = (abs ∘ − )
30	29	cnmetdval 23376	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴(abs ∘ − )0) = (abs‘(𝐴 − 0)))
31	27, 28, 30	sylancl 589	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) = (abs‘(𝐴 − 0)))
32		subid1 10895	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
33	32	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 − 0) = 𝐴)
34	33	fveq2d 6649	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴 − 0)) = (abs‘𝐴))
35	31, 34	eqtrd 2833	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) = (abs‘𝐴))
36		simpr 488	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1)
37	35, 36	eqbrtrd 5052	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) < 1)
38		cnxmet 23378	. . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
39		1xr 10689	. . . . . . 7 ⊢ 1 ∈ ℝ^*
40		elbl3 22999	. . . . . . 7 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
41	38, 39, 40	mpanl12 701	. . . . . 6 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
42	28, 27, 41	sylancr 590	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
43	37, 42	mpbird 260	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ (0(ball‘(abs ∘ − ))1))
44		tru 1542	. . . . . 6 ⊢ ⊤
45		eqid 2798	. . . . . . . 8 ⊢ (0(ball‘(abs ∘ − ))1) = (0(ball‘(abs ∘ − ))1)
46		0cnd 10623	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℂ)
47	39	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℝ^*)
48		ax-1cn 10584	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
49		blssm 23025	. . . . . . . . . . . . . . 15 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
50	38, 28, 39, 49	mp3an 1458	. . . . . . . . . . . . . 14 ⊢ (0(ball‘(abs ∘ − ))1) ⊆ ℂ
51	50	sseli 3911	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 𝑦 ∈ ℂ)
52		subcl 10874	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − 𝑦) ∈ ℂ)
53	48, 51, 52	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ∈ ℂ)
54	51	abscld 14788	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ ℝ)
55	29	cnmetdval 23376	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑦(abs ∘ − )0) = (abs‘(𝑦 − 0)))
56	51, 28, 55	sylancl 589	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) = (abs‘(𝑦 − 0)))
57	51	subid1d 10975	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦 − 0) = 𝑦)
58	57	fveq2d 6649	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(𝑦 − 0)) = (abs‘𝑦))
59	56, 58	eqtrd 2833	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) = (abs‘𝑦))
60		elbl3 22999	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
61	38, 39, 60	mpanl12 701	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
62	28, 51, 61	sylancr 590	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
63	62	ibi 270	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) < 1)
64	59, 63	eqbrtrrd 5054	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < 1)
65	54, 64	gtned 10764	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ≠ (abs‘𝑦))
66		abs1 14649	. . . . . . . . . . . . . . . 16 ⊢ (abs‘1) = 1
67		fveq2 6645	. . . . . . . . . . . . . . . 16 ⊢ (1 = 𝑦 → (abs‘1) = (abs‘𝑦))
68	66, 67	syl5eqr 2847	. . . . . . . . . . . . . . 15 ⊢ (1 = 𝑦 → 1 = (abs‘𝑦))
69	68	necon3i 3019	. . . . . . . . . . . . . 14 ⊢ (1 ≠ (abs‘𝑦) → 1 ≠ 𝑦)
70	65, 69	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ≠ 𝑦)
71		subeq0 10901	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((1 − 𝑦) = 0 ↔ 1 = 𝑦))
72	71	necon3bid 3031	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((1 − 𝑦) ≠ 0 ↔ 1 ≠ 𝑦))
73	48, 51, 72	sylancr 590	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 − 𝑦) ≠ 0 ↔ 1 ≠ 𝑦))
74	70, 73	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ≠ 0)
75	53, 74	logcld 25162	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (log‘(1 − 𝑦)) ∈ ℂ)
76	75	negcld 10973	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -(log‘(1 − 𝑦)) ∈ ℂ)
77	76	adantl 485	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → -(log‘(1 − 𝑦)) ∈ ℂ)
78	77	fmpttd 6856	. . . . . . . 8 ⊢ (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))):(0(ball‘(abs ∘ − ))1)⟶ℂ)
79	51	absge0d 14796	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ≤ (abs‘𝑦))
80	54	rexrd 10680	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ ℝ^*)
81		peano2re 10802	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑦) ∈ ℝ → ((abs‘𝑦) + 1) ∈ ℝ)
82	54, 81	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) + 1) ∈ ℝ)
83	82	rehalfcld 11872	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℝ)
84	83	rexrd 10680	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℝ^*)
85		iccssxr 12808	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) ⊆ ℝ^*
86		eqeq1 2802	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑗 → (𝑚 = 0 ↔ 𝑗 = 0))
87		oveq2 7143	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑗 → (1 / 𝑚) = (1 / 𝑗))
88	86, 87	ifbieq2d 4450	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑗 → if(𝑚 = 0, 0, (1 / 𝑚)) = if(𝑗 = 0, 0, (1 / 𝑗)))
89		eqid 2798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚))) = (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))
90		c0ex 10624	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
91		ovex 7168	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 / 𝑗) ∈ V
92	90, 91	ifex 4473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ if(𝑗 = 0, 0, (1 / 𝑗)) ∈ V
93	88, 89, 92	fvmpt 6745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ℕ₀ → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) = if(𝑗 = 0, 0, (1 / 𝑗)))
94	93	eqcomd 2804	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ₀ → if(𝑗 = 0, 0, (1 / 𝑗)) = ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗))
95	94	oveq1d 7150	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ₀ → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)) = (((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥↑𝑗)))
96	95	mpteq2ia 5121	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥↑𝑗)))
97	96	mpteq2i 5122	. . . . . . . . . . . . . . . 16 ⊢ (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))))
98		0cnd 10623	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑚 ∈ ℕ₀) ∧ 𝑚 = 0) → 0 ∈ ℂ)
99		nn0cn 11895	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
100	99	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℂ)
101		neqne 2995	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑚 = 0 → 𝑚 ≠ 0)
102		reccl 11294	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℂ ∧ 𝑚 ≠ 0) → (1 / 𝑚) ∈ ℂ)
103	100, 101, 102	syl2an 598	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑚 ∈ ℕ₀) ∧ ¬ 𝑚 = 0) → (1 / 𝑚) ∈ ℂ)
104	98, 103	ifclda 4459	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ₀) → if(𝑚 = 0, 0, (1 / 𝑚)) ∈ ℂ)
105	104	fmpttd 6856	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚))):ℕ₀⟶ℂ)
106		recn 10616	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑟 ∈ ℝ → 𝑟 ∈ ℂ)
107		oveq1 7142	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑟 → (𝑥↑𝑗) = (𝑟↑𝑗))
108	107	oveq2d 7151	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑟 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))
109	108	mpteq2dv 5126	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑟 → (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗))))
110		eqid 2798	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)))) = (𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))
111		nn0ex 11891	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ₀ ∈ V
112	111	mptex 6963	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗))) ∈ V
113	109, 110, 112	fvmpt 6745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (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))))
114	106, 113	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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))))
115	114	eqcomd 2804	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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))
116	115	seqeq3d 13372	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ℝ → seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) = seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑟)))
117	116	eleq1d 2874	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ℝ → (seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ ↔ seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑟)) ∈ dom ⇝ ))
118	117	rabbiia 3419	. . . . . . . . . . . . . . . . 17 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ } = {𝑟 ∈ ℝ ∣ seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑟)) ∈ dom ⇝ }
119	118	supeq1i 8895	. . . . . . . . . . . . . . . 16 ⊢ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) = sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑟)) ∈ dom ⇝ }, ℝ^, < )
120	97, 105, 119	radcnvcl 25012	. . . . . . . . . . . . . . 15 ⊢ (⊤ → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ) ∈ (0[,]+∞))
121	85, 120	sseldi 3913	. . . . . . . . . . . . . 14 ⊢ (⊤ → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ^)
122	44, 121	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ^)
123		1re 10630	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
124		avglt1 11863	. . . . . . . . . . . . . . 15 ⊢ (((abs‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (abs‘𝑦) < (((abs‘𝑦) + 1) / 2)))
125	54, 123, 124	sylancl 589	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) < 1 ↔ (abs‘𝑦) < (((abs‘𝑦) + 1) / 2)))
126	64, 125	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < (((abs‘𝑦) + 1) / 2))
127		0red 10633	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ∈ ℝ)
128	127, 54, 83, 79, 126	lelttrd 10787	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 < (((abs‘𝑦) + 1) / 2))
129	127, 83, 128	ltled 10777	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ≤ (((abs‘𝑦) + 1) / 2))
130	83, 129	absidd 14774	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) = (((abs‘𝑦) + 1) / 2))
131	44, 105	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚))):ℕ₀⟶ℂ)
132	83	recnd 10658	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℂ)
133		oveq1 7142	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (((abs‘𝑦) + 1) / 2) → (𝑥↑𝑗) = ((((abs‘𝑦) + 1) / 2)↑𝑗))
134	133	oveq2d 7151	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (((abs‘𝑦) + 1) / 2) → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))
135	134	mpteq2dv 5126	. . . . . . . . . . . . . . . . . . 19 ⊢ (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))))
136	111	mptex 6963	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗))) ∈ V
137	135, 110, 136	fvmpt 6745	. . . . . . . . . . . . . . . . . 18 ⊢ ((((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))))
138	132, 137	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (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))))
139	138	seqeq3d 13372	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘(((abs‘𝑦) + 1) / 2))) = seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))))
140		avglt2 11864	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((abs‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (((abs‘𝑦) + 1) / 2) < 1))
141	54, 123, 140	sylancl 589	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) < 1 ↔ (((abs‘𝑦) + 1) / 2) < 1))
142	64, 141	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) < 1)
143	130, 142	eqbrtrd 5052	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) < 1)
144		logtayllem 25250	. . . . . . . . . . . . . . . . 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 ⇝ )
145	132, 143, 144	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))) ∈ dom ⇝ )
146	139, 145	eqeltrd 2890	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘(((abs‘𝑦) + 1) / 2))) ∈ dom ⇝ )
147	97, 131, 119, 132, 146	radcnvle 25015	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))
148	130, 147	eqbrtrrd 5054	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))
149	80, 84, 122, 126, 148	xrltletrd 12542	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))
150		0re 10632	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
151		elico2 12789	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ^) → ((abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )) ↔ ((abs‘𝑦) ∈ ℝ ∧ 0 ≤ (abs‘𝑦) ∧ (abs‘𝑦) < sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))))
152	150, 122, 151	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )) ↔ ((abs‘𝑦) ∈ ℝ ∧ 0 ≤ (abs‘𝑦) ∧ (abs‘𝑦) < sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))))
153	54, 79, 149, 152	mpbir3and 1339	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < )))
154		absf 14689	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
155		ffn 6487	. . . . . . . . . . . 12 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
156		elpreima 6805	. . . . . . . . . . . 12 ⊢ (abs Fn ℂ → (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↔ (𝑦 ∈ ℂ ∧ (abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )))))
157	154, 155, 156	mp2b 10	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↔ (𝑦 ∈ ℂ ∧ (abs‘𝑦) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))))
158	51, 153, 157	sylanbrc 586	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))))
159		cnvimass 5916	. . . . . . . . . . . . . . . . 17 ⊢ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) ⊆ dom abs
160	154	fdmi 6498	. . . . . . . . . . . . . . . . 17 ⊢ dom abs = ℂ
161	159, 160	sseqtri 3951	. . . . . . . . . . . . . . . 16 ⊢ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) ⊆ ℂ
162	161	sseli 3911	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) → 𝑦 ∈ ℂ)
163		oveq1 7142	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → (𝑥↑𝑗) = (𝑦↑𝑗))
164	163	oveq2d 7151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))
165	164	mpteq2dv 5126	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗))))
166	111	mptex 6963	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗))) ∈ V
167	165, 110, 166	fvmpt 6745	. . . . . . . . . . . . . . . . . . 19 ⊢ (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))))
168	167	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((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))))
169	168	fveq1d 6647	. . . . . . . . . . . . . . . . 17 ⊢ ((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))
170		eqeq1 2802	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → (𝑗 = 0 ↔ 𝑛 = 0))
171		oveq2 7143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → (1 / 𝑗) = (1 / 𝑛))
172	170, 171	ifbieq2d 4450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → if(𝑗 = 0, 0, (1 / 𝑗)) = if(𝑛 = 0, 0, (1 / 𝑛)))
173		oveq2 7143	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (𝑦↑𝑗) = (𝑦↑𝑛))
174	172, 173	oveq12d 7153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑛 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
175		eqid 2798	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))
176		ovex 7168	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) ∈ V
177	174, 175, 176	fvmpt 6745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ₀ → ((𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
178	177	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
179	169, 178	eqtr2d 2834	. . . . . . . . . . . . . . . 16 ⊢ ((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))
180	179	sumeq2dv 15052	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℂ → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = Σ𝑛 ∈ ℕ₀ (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑦)‘𝑛))
181	162, 180	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = Σ𝑛 ∈ ℕ₀ (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑦)‘𝑛))
182	181	mpteq2ia 5121	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))) = (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↦ Σ𝑛 ∈ ℕ₀ (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑦)‘𝑛))
183		eqid 2798	. . . . . . . . . . . . 13 ⊢ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) = (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )))
184		eqid 2798	. . . . . . . . . . . . 13 ⊢ if(sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ, (((abs‘𝑧) + sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )) / 2), ((abs‘𝑧) + 1)) = if(sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ, (((abs‘𝑧) + sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )) / 2), ((abs‘𝑧) + 1))
185	97, 182, 105, 119, 183, 184	psercn 25021	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))) ∈ ((^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )))–cn→ℂ))
186		cncff 23498	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))) ∈ ((^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )))–cn→ℂ) → (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))):(^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )))⟶ℂ)
187	185, 186	syl 17	. . . . . . . . . . 11 ⊢ (⊤ → (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))):(^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )))⟶ℂ)
188	187	fvmptelrn 6854	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < )))) → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) ∈ ℂ)
189	158, 188	sylan2 595	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) ∈ ℂ)
190	189	fmpttd 6856	. . . . . . . 8 ⊢ (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))):(0(ball‘(abs ∘ − ))1)⟶ℂ)
191		cnelprrecn 10619	. . . . . . . . . . . . 13 ⊢ ℂ ∈ {ℝ, ℂ}
192	191	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
193	75	adantl 485	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → (log‘(1 − 𝑦)) ∈ ℂ)
194		ovexd 7170	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → ((1 / (1 − 𝑦)) · -1) ∈ V)
195	29	cnmetdval 23376	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℂ ∧ (1 − 𝑦) ∈ ℂ) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘(1 − (1 − 𝑦))))
196	48, 53, 195	sylancr 590	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘(1 − (1 − 𝑦))))
197		nncan 10904	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − (1 − 𝑦)) = 𝑦)
198	48, 51, 197	sylancr 590	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − (1 − 𝑦)) = 𝑦)
199	198	fveq2d 6649	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(1 − (1 − 𝑦))) = (abs‘𝑦))
200	196, 199	eqtrd 2833	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘𝑦))
201	200, 64	eqbrtrd 5052	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) < 1)
202		elbl 22995	. . . . . . . . . . . . . . . 16 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ^*) → ((1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 − 𝑦) ∈ ℂ ∧ (1(abs ∘ − )(1 − 𝑦)) < 1)))
203	38, 48, 39, 202	mp3an 1458	. . . . . . . . . . . . . . 15 ⊢ ((1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 − 𝑦) ∈ ℂ ∧ (1(abs ∘ − )(1 − 𝑦)) < 1))
204	53, 201, 203	sylanbrc 586	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1))
205	204	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → (1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1))
206		neg1cn 11739	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℂ
207	206	a1i 11	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → -1 ∈ ℂ)
208		eqid 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
209	208	dvlog2lem 25243	. . . . . . . . . . . . . . . . 17 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
210	209	sseli 3911	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
211	210	eldifad 3893	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ∈ ℂ)
212		eqid 2798	. . . . . . . . . . . . . . . . 17 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
213	212	logdmn0 25231	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) → 𝑥 ≠ 0)
214	210, 213	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ≠ 0)
215	211, 214	logcld 25162	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑥) ∈ ℂ)
216	215	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(ball‘(abs ∘ − ))1)) → (log‘𝑥) ∈ ℂ)
217		ovexd 7170	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(ball‘(abs ∘ − ))1)) → (1 / 𝑥) ∈ V)
218		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
219	48, 218, 52	sylancr 590	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → (1 − 𝑦) ∈ ℂ)
220	206	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → -1 ∈ ℂ)
221		1cnd 10625	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
222		0cnd 10623	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 0 ∈ ℂ)
223		1cnd 10625	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → 1 ∈ ℂ)
224	192, 223	dvmptc 24561	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 1)) = (𝑦 ∈ ℂ ↦ 0))
225	192	dvmptid 24560	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1))
226	192, 221, 222, 224, 218, 221, 225	dvmptsub 24570	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (1 − 𝑦))) = (𝑦 ∈ ℂ ↦ (0 − 1)))
227		df-neg 10862	. . . . . . . . . . . . . . . 16 ⊢ -1 = (0 − 1)
228	227	mpteq2i 5122	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℂ ↦ -1) = (𝑦 ∈ ℂ ↦ (0 − 1))
229	226, 228	eqtr4di 2851	. . . . . . . . . . . . . 14 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (1 − 𝑦))) = (𝑦 ∈ ℂ ↦ -1))
230	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
231		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
232	231	cnfldtopon 23388	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
233	232	toponrestid 21526	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
234	231	cnfldtopn 23387	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
235	234	blopn 23107	. . . . . . . . . . . . . . . 16 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂ_fld))
236	38, 28, 39, 235	mp3an 1458	. . . . . . . . . . . . . . 15 ⊢ (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂ_fld)
237	236	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂ_fld))
238	192, 219, 220, 229, 230, 233, 231, 237	dvmptres 24566	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -1))
239	208	dvlog2 25244	. . . . . . . . . . . . . 14 ⊢ (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑥))
240		logf1o 25156	. . . . . . . . . . . . . . . . . . . 20 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
241		f1of 6590	. . . . . . . . . . . . . . . . . . . 20 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
242	240, 241	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ log:(ℂ ∖ {0})⟶ran log
243	212	logdmss 25233	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
244	209, 243	sstri 3924	. . . . . . . . . . . . . . . . . . 19 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
245		fssres 6518	. . . . . . . . . . . . . . . . . . 19 ⊢ ((log:(ℂ ∖ {0})⟶ran log ∧ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})) → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log)
246	242, 244, 245	mp2an 691	. . . . . . . . . . . . . . . . . 18 ⊢ (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log
247	246	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log)
248	247	feqmptd 6708	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥)))
249		fvres 6664	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥) = (log‘𝑥))
250	249	mpteq2ia 5121	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))
251	248, 250	eqtrdi 2849	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥)))
252	251	oveq2d 7151	. . . . . . . . . . . . . 14 ⊢ (⊤ → (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (ℂ D (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))))
253	239, 252	syl5reqr 2848	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑥)))
254		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1 − 𝑦) → (log‘𝑥) = (log‘(1 − 𝑦)))
255		oveq2 7143	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1 − 𝑦) → (1 / 𝑥) = (1 / (1 − 𝑦)))
256	192, 192, 205, 207, 216, 217, 238, 253, 254, 255	dvmptco 24575	. . . . . . . . . . . 12 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ ((1 / (1 − 𝑦)) · -1)))
257	192, 193, 194, 256	dvmptneg 24569	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -((1 / (1 − 𝑦)) · -1)))
258	53, 74	reccld 11398	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 / (1 − 𝑦)) ∈ ℂ)
259		mulcom 10612	. . . . . . . . . . . . . . . 16 ⊢ (((1 / (1 − 𝑦)) ∈ ℂ ∧ -1 ∈ ℂ) → ((1 / (1 − 𝑦)) · -1) = (-1 · (1 / (1 − 𝑦))))
260	258, 206, 259	sylancl 589	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 / (1 − 𝑦)) · -1) = (-1 · (1 / (1 − 𝑦))))
261	258	mulm1d 11081	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (-1 · (1 / (1 − 𝑦))) = -(1 / (1 − 𝑦)))
262	260, 261	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 / (1 − 𝑦)) · -1) = -(1 / (1 − 𝑦)))
263	262	negeqd 10869	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -((1 / (1 − 𝑦)) · -1) = --(1 / (1 − 𝑦)))
264	258	negnegd 10977	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → --(1 / (1 − 𝑦)) = (1 / (1 − 𝑦)))
265	263, 264	eqtrd 2833	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -((1 / (1 − 𝑦)) · -1) = (1 / (1 − 𝑦)))
266	265	mpteq2ia 5121	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -((1 / (1 − 𝑦)) · -1)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦)))
267	257, 266	eqtrdi 2849	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
268	267	dmeqd 5738	. . . . . . . . 9 ⊢ (⊤ → dom (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
269		dmmptg 6063	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (0(ball‘(abs ∘ − ))1)(1 / (1 − 𝑦)) ∈ V → dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))) = (0(ball‘(abs ∘ − ))1))
270		ovexd 7170	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 / (1 − 𝑦)) ∈ V)
271	269, 270	mprg 3120	. . . . . . . . 9 ⊢ dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))) = (0(ball‘(abs ∘ − ))1)
272	268, 271	eqtrdi 2849	. . . . . . . 8 ⊢ (⊤ → dom (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (0(ball‘(abs ∘ − ))1))
273		sumex 15036	. . . . . . . . . . . 12 ⊢ Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) ∈ V
274	273	a1i 11	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < )))) → Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) ∈ V)
275		fveq2 6645	. . . . . . . . . . . . . . 15 ⊢ (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))
276	275	cbvsumv 15045	. . . . . . . . . . . . . 14 ⊢ Σ𝑛 ∈ ℕ₀ (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑦)‘𝑛) = Σ𝑘 ∈ ℕ₀ (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑦)‘𝑘)
277	181, 276	eqtrdi 2849	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = Σ𝑘 ∈ ℕ₀ (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑦)‘𝑘))
278	277	mpteq2ia 5121	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))) = (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↦ Σ𝑘 ∈ ℕ₀ (((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘𝑦)‘𝑘))
279		eqid 2798	. . . . . . . . . . . 12 ⊢ (0(ball‘(abs ∘ − ))(((abs‘𝑧) + if(sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ, (((abs‘𝑧) + sup({𝑟 ∈ ℝ ∣ seq0( + , (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 ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ, (((abs‘𝑧) + sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )) / 2), ((abs‘𝑧) + 1))) / 2))
280	97, 278, 105, 119, 183, 184, 279	pserdv2 25025	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))) = (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ))) ↦ Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1)))))
281	158	ssriv 3919	. . . . . . . . . . . 12 ⊢ (0(ball‘(abs ∘ − ))1) ⊆ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < )))
282	281	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(ball‘(abs ∘ − ))1) ⊆ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))))
283	192, 188, 274, 280, 282, 233, 231, 237	dvmptres 24566	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1)))))
284		nnnn0 11892	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
285	284	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
286		eqeq1 2802	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (𝑚 = 0 ↔ 𝑛 = 0))
287		oveq2 7143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
288	286, 287	ifbieq2d 4450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → if(𝑚 = 0, 0, (1 / 𝑚)) = if(𝑛 = 0, 0, (1 / 𝑛)))
289		ovex 7168	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 / 𝑛) ∈ V
290	90, 289	ifex 4473	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑛 = 0, 0, (1 / 𝑛)) ∈ V
291	288, 89, 290	fvmpt 6745	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)))
292	285, 291	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)))
293		nnne0 11659	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
294	293	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
295	294	neneqd 2992	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
296	295	iffalsed 4436	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, 0, (1 / 𝑛)) = (1 / 𝑛))
297	292, 296	eqtrd 2833	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = (1 / 𝑛))
298	297	oveq2d 7151	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) = (𝑛 · (1 / 𝑛)))
299		nncn 11633	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
300	299	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
301	300, 294	recidd 11400	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · (1 / 𝑛)) = 1)
302	298, 301	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) = 1)
303	302	oveq1d 7150	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (1 · (𝑦↑(𝑛 − 1))))
304		nnm1nn0 11926	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
305		expcl 13443	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℂ ∧ (𝑛 − 1) ∈ ℕ₀) → (𝑦↑(𝑛 − 1)) ∈ ℂ)
306	51, 304, 305	syl2an 598	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑦↑(𝑛 − 1)) ∈ ℂ)
307	306	mulid2d 10648	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (1 · (𝑦↑(𝑛 − 1))) = (𝑦↑(𝑛 − 1)))
308	303, 307	eqtrd 2833	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (𝑦↑(𝑛 − 1)))
309	308	sumeq2dv 15052	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)))
310		nnuz 12269	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ_≥‘1)
311		1e0p1 12128	. . . . . . . . . . . . . . . 16 ⊢ 1 = (0 + 1)
312	311	fveq2i 6648	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘1) = (ℤ_≥‘(0 + 1))
313	310, 312	eqtri 2821	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘(0 + 1))
314		oveq1 7142	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (1 + 𝑚) → (𝑛 − 1) = ((1 + 𝑚) − 1))
315	314	oveq2d 7151	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1 + 𝑚) → (𝑦↑(𝑛 − 1)) = (𝑦↑((1 + 𝑚) − 1)))
316		1zzd 12001	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ∈ ℤ)
317		0zd 11981	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ∈ ℤ)
318	1, 313, 315, 316, 317, 306	isumshft 15186	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)) = Σ𝑚 ∈ ℕ₀ (𝑦↑((1 + 𝑚) − 1)))
319		pncan2 10882	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((1 + 𝑚) − 1) = 𝑚)
320	48, 99, 319	sylancr 590	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ₀ → ((1 + 𝑚) − 1) = 𝑚)
321	320	oveq2d 7151	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (𝑦↑((1 + 𝑚) − 1)) = (𝑦↑𝑚))
322	321	sumeq2i 15048	. . . . . . . . . . . . 13 ⊢ Σ𝑚 ∈ ℕ₀ (𝑦↑((1 + 𝑚) − 1)) = Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚)
323	318, 322	eqtrdi 2849	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)) = Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚))
324		geoisum 15225	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℂ ∧ (abs‘𝑦) < 1) → Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚) = (1 / (1 − 𝑦)))
325	51, 64, 324	syl2anc 587	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚) = (1 / (1 − 𝑦)))
326	309, 323, 325	3eqtrd 2837	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (1 / (1 − 𝑦)))
327	326	mpteq2ia 5121	. . . . . . . . . 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)))
328	283, 327	eqtrdi 2849	. . . . . . . . 9 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
329	267, 328	eqtr4d 2836	. . . . . . . 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)) · (𝑦↑𝑛)))))
330		1rp 12381	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
331		blcntr 23020	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ⁺) → 0 ∈ (0(ball‘(abs ∘ − ))1))
332	38, 28, 330, 331	mp3an 1458	. . . . . . . . 9 ⊢ 0 ∈ (0(ball‘(abs ∘ − ))1)
333	332	a1i 11	. . . . . . . 8 ⊢ (⊤ → 0 ∈ (0(ball‘(abs ∘ − ))1))
334		oveq2 7143	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 0 → (1 − 𝑦) = (1 − 0))
335		1m0e1 11746	. . . . . . . . . . . . . . . 16 ⊢ (1 − 0) = 1
336	334, 335	eqtrdi 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 0 → (1 − 𝑦) = 1)
337	336	fveq2d 6649	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 0 → (log‘(1 − 𝑦)) = (log‘1))
338		log1 25177	. . . . . . . . . . . . . 14 ⊢ (log‘1) = 0
339	337, 338	eqtrdi 2849	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (log‘(1 − 𝑦)) = 0)
340	339	negeqd 10869	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → -(log‘(1 − 𝑦)) = -0)
341		neg0 10921	. . . . . . . . . . . 12 ⊢ -0 = 0
342	340, 341	eqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → -(log‘(1 − 𝑦)) = 0)
343		eqid 2798	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))
344	342, 343, 90	fvmpt 6745	. . . . . . . . . 10 ⊢ (0 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = 0)
345	332, 344	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = 0)
346		oveq1 7142	. . . . . . . . . . . . . . 15 ⊢ (0 = if(𝑛 = 0, 0, (1 / 𝑛)) → (0 · (𝑦↑𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
347	346	eqeq1d 2800	. . . . . . . . . . . . . 14 ⊢ (0 = if(𝑛 = 0, 0, (1 / 𝑛)) → ((0 · (𝑦↑𝑛)) = 0 ↔ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0))
348		oveq1 7142	. . . . . . . . . . . . . . 15 ⊢ ((1 / 𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)) → ((1 / 𝑛) · (𝑦↑𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
349	348	eqeq1d 2800	. . . . . . . . . . . . . 14 ⊢ ((1 / 𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)) → (((1 / 𝑛) · (𝑦↑𝑛)) = 0 ↔ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0))
350		simpll 766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 𝑦 = 0)
351	350, 28	eqeltrdi 2898	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 𝑦 ∈ ℂ)
352		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 𝑛 ∈ ℕ₀)
353	351, 352	expcld 13506	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → (𝑦↑𝑛) ∈ ℂ)
354	353	mul02d 10827	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → (0 · (𝑦↑𝑛)) = 0)
355		simpll 766	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → 𝑦 = 0)
356	355	oveq1d 7150	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (𝑦↑𝑛) = (0↑𝑛))
357		simpr 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
358	357, 14	sylib 221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (𝑛 ∈ ℕ ∨ 𝑛 = 0))
359	358	ord 861	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 ∈ ℕ → 𝑛 = 0))
360	359	con1d 147	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 = 0 → 𝑛 ∈ ℕ))
361	360	imp 410	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
362	361	0expd 13499	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (0↑𝑛) = 0)
363	356, 362	eqtrd 2833	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (𝑦↑𝑛) = 0)
364	363	oveq2d 7151	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · (𝑦↑𝑛)) = ((1 / 𝑛) · 0))
365	361	nnrecred 11676	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℝ)
366	365	recnd 10658	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℂ)
367	366	mul01d 10828	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · 0) = 0)
368	364, 367	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · (𝑦↑𝑛)) = 0)
369	347, 349, 354, 368	ifbothda 4462	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0)
370	369	sumeq2dv 15052	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = Σ𝑛 ∈ ℕ₀ 0)
371	1	eqimssi 3973	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ⊆ (ℤ_≥‘0)
372	371	orci 862	. . . . . . . . . . . . 13 ⊢ (ℕ₀ ⊆ (ℤ_≥‘0) ∨ ℕ₀ ∈ Fin)
373		sumz 15071	. . . . . . . . . . . . 13 ⊢ ((ℕ₀ ⊆ (ℤ_≥‘0) ∨ ℕ₀ ∈ Fin) → Σ𝑛 ∈ ℕ₀ 0 = 0)
374	372, 373	ax-mp 5	. . . . . . . . . . . 12 ⊢ Σ𝑛 ∈ ℕ₀ 0 = 0
375	370, 374	eqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0)
376		eqid 2798	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
377	375, 376, 90	fvmpt 6745	. . . . . . . . . 10 ⊢ (0 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))‘0) = 0)
378	332, 377	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))‘0) = 0)
379	345, 378	eqtr4d 2836	. . . . . . . 8 ⊢ (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))‘0))
380	45, 46, 47, 78, 190, 272, 329, 333, 379	dv11cn 24604	. . . . . . 7 ⊢ (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))))
381	380	fveq1d 6647	. . . . . 6 ⊢ (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘𝐴) = ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))‘𝐴))
382	44, 381	mp1i 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)) · (𝑦↑𝑛)))‘𝐴))
383		oveq2 7143	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (1 − 𝑦) = (1 − 𝐴))
384	383	fveq2d 6649	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (log‘(1 − 𝑦)) = (log‘(1 − 𝐴)))
385	384	negeqd 10869	. . . . . 6 ⊢ (𝑦 = 𝐴 → -(log‘(1 − 𝑦)) = -(log‘(1 − 𝐴)))
386		negex 10873	. . . . . 6 ⊢ -(log‘(1 − 𝐴)) ∈ V
387	385, 343, 386	fvmpt 6745	. . . . 5 ⊢ (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘𝐴) = -(log‘(1 − 𝐴)))
388		oveq1 7142	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦↑𝑛) = (𝐴↑𝑛))
389	388	oveq2d 7151	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
390	389	sumeq2sdv 15053	. . . . . 6 ⊢ (𝑦 = 𝐴 → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
391		sumex 15036	. . . . . 6 ⊢ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) ∈ V
392	390, 376, 391	fvmpt 6745	. . . . 5 ⊢ (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))‘𝐴) = Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
393	382, 387, 392	3eqtr3d 2841	. . . 4 ⊢ (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → -(log‘(1 − 𝐴)) = Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
394	43, 393	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(log‘(1 − 𝐴)) = Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
395	26, 394	breqtrrd 5058	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ⇝ -(log‘(1 − 𝐴)))
396		seqex 13366	. . . 4 ⊢ seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ∈ V
397	396	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ∈ V)
398		seqex 13366	. . . 4 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))) ∈ V
399	398	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))) ∈ V)
400		1zzd 12001	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℤ)
401		elnnuz 12270	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
402		fvres 6664	. . . . . 6 ⊢ (𝑛 ∈ (ℤ_≥‘1) → ((seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ↾ (ℤ_≥‘1))‘𝑛) = (seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘))))‘𝑛))
403	401, 402	sylbi 220	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ↾ (ℤ_≥‘1))‘𝑛) = (seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘))))‘𝑛))
404	403	eqcomd 2804	. . . 4 ⊢ (𝑛 ∈ ℕ → (seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘))))‘𝑛) = ((seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ↾ (ℤ_≥‘1))‘𝑛))
405		addid2 10812	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
406	405	adantl 485	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
407		0cnd 10623	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ∈ ℂ)
408		1eluzge0 12280	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
409	408	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ (ℤ_≥‘0))
410		0cnd 10623	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 = 0) → 0 ∈ ℂ)
411		nn0cn 11895	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
412	411	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
413		neqne 2995	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
414		reccl 11294	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℂ ∧ 𝑘 ≠ 0) → (1 / 𝑘) ∈ ℂ)
415	412, 413, 414	syl2an 598	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 = 0) → (1 / 𝑘) ∈ ℂ)
416	410, 415	ifclda 4459	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → if(𝑘 = 0, 0, (1 / 𝑘)) ∈ ℂ)
417		expcl 13443	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℂ)
418	417	adantlr 714	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℂ)
419	416, 418	mulcld 10650	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) ∈ ℂ)
420	419	fmpttd 6856	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘))):ℕ₀⟶ℂ)
421		1nn0 11901	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
422		ffvelrn 6826	. . . . . . . 8 ⊢ (((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)
423	420, 421, 422	sylancl 589	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘1) ∈ ℂ)
424		elfz1eq 12913	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...0) → 𝑛 = 0)
425		1m1e0 11697	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
426	425	oveq2i 7146	. . . . . . . . . 10 ⊢ (0...(1 − 1)) = (0...0)
427	424, 426	eleq2s 2908	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(1 − 1)) → 𝑛 = 0)
428	427	fveq2d 6649	. . . . . . . 8 ⊢ (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))
429		0nn0 11900	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
430		iftrue 4431	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → if(𝑘 = 0, 0, (1 / 𝑘)) = 0)
431		oveq2 7143	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝐴↑𝑘) = (𝐴↑0))
432	430, 431	oveq12d 7153	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) = (0 · (𝐴↑0)))
433		ovex 7168	. . . . . . . . . . 11 ⊢ (0 · (𝐴↑0)) ∈ V
434	432, 8, 433	fvmpt 6745	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘0) = (0 · (𝐴↑0)))
435	429, 434	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘0) = (0 · (𝐴↑0))
436		expcl 13443	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℕ₀) → (𝐴↑0) ∈ ℂ)
437	27, 429, 436	sylancl 589	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑0) ∈ ℂ)
438	437	mul02d 10827	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (0 · (𝐴↑0)) = 0)
439	435, 438	syl5eq 2845	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘0) = 0)
440	428, 439	sylan9eqr 2855	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ (0...(1 − 1))) → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘𝑛) = 0)
441	406, 407, 409, 423, 440	seqid 13411	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ↾ (ℤ_≥‘1)) = seq1( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))))
442	293	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
443	442	neneqd 2992	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
444	443	iffalsed 4436	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, 0, (1 / 𝑛)) = (1 / 𝑛))
445	444	oveq1d 7150	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) = ((1 / 𝑛) · (𝐴↑𝑛)))
446	284, 23	sylan2 595	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ∈ ℂ)
447	299	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
448	446, 447, 442	divrec2d 11409	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝐴↑𝑛) / 𝑛) = ((1 / 𝑛) · (𝐴↑𝑛)))
449	445, 448	eqtr4d 2836	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) = ((𝐴↑𝑛) / 𝑛))
450	284, 11	sylan2 595	. . . . . . . . 9 ⊢ (((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)))
451		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
452	6, 451	oveq12d 7153	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝐴↑𝑘) / 𝑘) = ((𝐴↑𝑛) / 𝑛))
453		eqid 2798	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘)) = (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))
454		ovex 7168	. . . . . . . . . . 11 ⊢ ((𝐴↑𝑛) / 𝑛) ∈ V
455	452, 453, 454	fvmpt 6745	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))‘𝑛) = ((𝐴↑𝑛) / 𝑛))
456	455	adantl 485	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))‘𝑛) = ((𝐴↑𝑛) / 𝑛))
457	449, 450, 456	3eqtr4d 2843	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘𝑛) = ((𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))‘𝑛))
458	401, 457	sylan2br 597	. . . . . . 7 ⊢ (((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))
459	400, 458	seqfeq 13391	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))))
460	441, 459	eqtrd 2833	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ↾ (ℤ_≥‘1)) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))))
461	460	fveq1d 6647	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ↾ (ℤ_≥‘1))‘𝑛) = (seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘)))‘𝑛))
462	404, 461	sylan9eqr 2855	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘))))‘𝑛) = (seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘)))‘𝑛))
463	310, 397, 399, 400, 462	climeq 14916	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ⇝ -(log‘(1 − 𝐴)) ↔ seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))) ⇝ -(log‘(1 − 𝐴))))
464	395, 463	mpbid 235	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))) ⇝ -(log‘(1 − 𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ≠ wne 2987 {crab 3110 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ifcif 4425 {csn 4525 {cpr 4527 class class class wbr 5030 ↦ cmpt 5110 ^◡ccnv 5518 dom cdm 5519 ran crn 5520 ↾ cres 5521 “ cima 5522 ∘ ccom 5523 Fn wfn 6319 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 supcsup 8888 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 +∞cpnf 10661 -∞cmnf 10662 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤ_≥cuz 12231 ℝ⁺crp 12377 (,]cioc 12727 [,)cico 12728 [,]cicc 12729 ...cfz 12885 seqcseq 13364 ↑cexp 13425 abscabs 14585 ⇝ cli 14833 Σcsu 15034 TopOpenctopn 16687 ∞Metcxmet 20076 ballcbl 20078 ℂ_fldccnfld 20091 –cn→ccncf 23481 D cdv 24466 logclog 25146

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-tan 15417 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-ulm 24972 df-log 25148

This theorem is referenced by: logtaylsum 25252 logtayl2 25253 atantayl 25523 stirlinglem5 42720

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