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Theorem logtayl 24206

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 11598	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 11266	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ∈ ℤ)
3		eqeq1 2614	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
4		oveq2 6557	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
5	3, 4	ifbieq2d 4061	. . . . . . 7 ⊢ (𝑘 = 𝑛 → if(𝑘 = 0, 0, (1 / 𝑘)) = if(𝑛 = 0, 0, (1 / 𝑛)))
6		oveq2 6557	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝐴↑𝑘) = (𝐴↑𝑛))
7	5, 6	oveq12d 6567	. . . . . 6 ⊢ (𝑘 = 𝑛 → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
8		eqid 2610	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘))) = (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))
9		ovex 6577	. . . . . 6 ⊢ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) ∈ V
10	7, 8, 9	fvmpt 6191	. . . . 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 481	. . . 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 9912	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 0 ∈ ℂ)
13		simpr 476	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
14		elnn0 11171	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
15	13, 14	sylib 207	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (𝑛 ∈ ℕ ∨ 𝑛 = 0))
16	15	ord 391	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 ∈ ℕ → 𝑛 = 0))
17	16	con1d 138	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 = 0 → 𝑛 ∈ ℕ))
18	17	imp 444	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
19	18	nnrecred 10943	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℝ)
20	19	recnd 9947	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℂ)
21	12, 20	ifclda 4070	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → if(𝑛 = 0, 0, (1 / 𝑛)) ∈ ℂ)
22		expcl 12740	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (𝐴↑𝑛) ∈ ℂ)
23	22	adantlr 747	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (𝐴↑𝑛) ∈ ℂ)
24	21, 23	mulcld 9939	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) ∈ ℂ)
25		logtayllem 24205	. . . 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 14331	. . 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 472	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ)
28		0cn 9911	. . . . . . . 8 ⊢ 0 ∈ ℂ
29		eqid 2610	. . . . . . . . 9 ⊢ (abs ∘ − ) = (abs ∘ − )
30	29	cnmetdval 22384	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴(abs ∘ − )0) = (abs‘(𝐴 − 0)))
31	27, 28, 30	sylancl 693	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) = (abs‘(𝐴 − 0)))
32		subid1 10180	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
33	32	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 − 0) = 𝐴)
34	33	fveq2d 6107	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴 − 0)) = (abs‘𝐴))
35	31, 34	eqtrd 2644	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) = (abs‘𝐴))
36		simpr 476	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1)
37	35, 36	eqbrtrd 4605	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) < 1)
38		cnxmet 22386	. . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
39		1rp 11712	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
40		rpxr 11716	. . . . . . . 8 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
41	39, 40	ax-mp 5	. . . . . . 7 ⊢ 1 ∈ ℝ^*
42		elbl3 22007	. . . . . . 7 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
43	38, 41, 42	mpanl12 714	. . . . . 6 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
44	28, 27, 43	sylancr 694	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
45	37, 44	mpbird 246	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ (0(ball‘(abs ∘ − ))1))
46		tru 1479	. . . . . 6 ⊢ ⊤
47		eqid 2610	. . . . . . . 8 ⊢ (0(ball‘(abs ∘ − ))1) = (0(ball‘(abs ∘ − ))1)
48		0cnd 9912	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℂ)
49	41	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℝ^*)
50		ax-1cn 9873	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
51		blssm 22033	. . . . . . . . . . . . . . 15 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
52	38, 28, 41, 51	mp3an 1416	. . . . . . . . . . . . . 14 ⊢ (0(ball‘(abs ∘ − ))1) ⊆ ℂ
53	52	sseli 3564	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 𝑦 ∈ ℂ)
54		subcl 10159	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − 𝑦) ∈ ℂ)
55	50, 53, 54	sylancr 694	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ∈ ℂ)
56	53	abscld 14023	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ ℝ)
57	29	cnmetdval 22384	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑦(abs ∘ − )0) = (abs‘(𝑦 − 0)))
58	53, 28, 57	sylancl 693	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) = (abs‘(𝑦 − 0)))
59	53	subid1d 10260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦 − 0) = 𝑦)
60	59	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(𝑦 − 0)) = (abs‘𝑦))
61	58, 60	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) = (abs‘𝑦))
62		elbl3 22007	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
63	38, 41, 62	mpanl12 714	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
64	28, 53, 63	sylancr 694	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
65	64	ibi 255	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) < 1)
66	61, 65	eqbrtrrd 4607	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < 1)
67	56, 66	gtned 10051	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ≠ (abs‘𝑦))
68		abs1 13885	. . . . . . . . . . . . . . . 16 ⊢ (abs‘1) = 1
69		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (1 = 𝑦 → (abs‘1) = (abs‘𝑦))
70	68, 69	syl5eqr 2658	. . . . . . . . . . . . . . 15 ⊢ (1 = 𝑦 → 1 = (abs‘𝑦))
71	70	necon3i 2814	. . . . . . . . . . . . . 14 ⊢ (1 ≠ (abs‘𝑦) → 1 ≠ 𝑦)
72	67, 71	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ≠ 𝑦)
73		subeq0 10186	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((1 − 𝑦) = 0 ↔ 1 = 𝑦))
74	73	necon3bid 2826	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((1 − 𝑦) ≠ 0 ↔ 1 ≠ 𝑦))
75	50, 53, 74	sylancr 694	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 − 𝑦) ≠ 0 ↔ 1 ≠ 𝑦))
76	72, 75	mpbird 246	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ≠ 0)
77	55, 76	logcld 24121	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (log‘(1 − 𝑦)) ∈ ℂ)
78	77	negcld 10258	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -(log‘(1 − 𝑦)) ∈ ℂ)
79	78	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → -(log‘(1 − 𝑦)) ∈ ℂ)
80		eqid 2610	. . . . . . . . 9 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))
81	79, 80	fmptd 6292	. . . . . . . 8 ⊢ (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))):(0(ball‘(abs ∘ − ))1)⟶ℂ)
82	53	absge0d 14031	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ≤ (abs‘𝑦))
83	56	rexrd 9968	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ ℝ^*)
84		peano2re 10088	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑦) ∈ ℝ → ((abs‘𝑦) + 1) ∈ ℝ)
85	56, 84	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) + 1) ∈ ℝ)
86	85	rehalfcld 11156	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℝ)
87	86	rexrd 9968	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℝ^*)
88		iccssxr 12127	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) ⊆ ℝ^*
89		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑗 → (𝑚 = 0 ↔ 𝑗 = 0))
90		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑗 → (1 / 𝑚) = (1 / 𝑗))
91	89, 90	ifbieq2d 4061	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑗 → if(𝑚 = 0, 0, (1 / 𝑚)) = if(𝑗 = 0, 0, (1 / 𝑗)))
92		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚))) = (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))
93		c0ex 9913	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
94		ovex 6577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 / 𝑗) ∈ V
95	93, 94	ifex 4106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ if(𝑗 = 0, 0, (1 / 𝑗)) ∈ V
96	91, 92, 95	fvmpt 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ℕ₀ → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) = if(𝑗 = 0, 0, (1 / 𝑗)))
97	96	eqcomd 2616	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ₀ → if(𝑗 = 0, 0, (1 / 𝑗)) = ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗))
98	97	oveq1d 6564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ₀ → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)) = (((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥↑𝑗)))
99	98	mpteq2ia 4668	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥↑𝑗)))
100	99	mpteq2i 4669	. . . . . . . . . . . . . . . 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))))
101		0cnd 9912	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑚 ∈ ℕ₀) ∧ 𝑚 = 0) → 0 ∈ ℂ)
102		nn0cn 11179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
103	102	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℂ)
104		df-ne 2782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ≠ 0 ↔ ¬ 𝑚 = 0)
105	104	biimpri 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑚 = 0 → 𝑚 ≠ 0)
106		reccl 10571	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℂ ∧ 𝑚 ≠ 0) → (1 / 𝑚) ∈ ℂ)
107	103, 105, 106	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑚 ∈ ℕ₀) ∧ ¬ 𝑚 = 0) → (1 / 𝑚) ∈ ℂ)
108	101, 107	ifclda 4070	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ₀) → if(𝑚 = 0, 0, (1 / 𝑚)) ∈ ℂ)
109	108, 92	fmptd 6292	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚))):ℕ₀⟶ℂ)
110		recn 9905	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑟 ∈ ℝ → 𝑟 ∈ ℂ)
111		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑟 → (𝑥↑𝑗) = (𝑟↑𝑗))
112	111	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑟 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))
113	112	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑟 → (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗))))
114		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . 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))))
115		nn0ex 11175	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ₀ ∈ V
116	115	mptex 6390	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗))) ∈ V
117	113, 114, 116	fvmpt 6191	. . . . . . . . . . . . . . . . . . . . . 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))))
118	110, 117	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))))
119	118	eqcomd 2616	. . . . . . . . . . . . . . . . . . . 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))
120	119	seqeq3d 12671	. . . . . . . . . . . . . . . . . . 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)))
121	120	eleq1d 2672	. . . . . . . . . . . . . . . . . 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 ⇝ ))
122	121	rabbiia 3161	. . . . . . . . . . . . . . . . 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 ⇝ }
123	122	supeq1i 8236	. . . . . . . . . . . . . . . 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^, < )
124	100, 109, 123	radcnvcl 23975	. . . . . . . . . . . . . . 15 ⊢ (⊤ → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ) ∈ (0[,]+∞))
125	88, 124	sseldi 3566	. . . . . . . . . . . . . 14 ⊢ (⊤ → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ^)
126	46, 125	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^)
127		1re 9918	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
128		avglt1 11147	. . . . . . . . . . . . . . 15 ⊢ (((abs‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (abs‘𝑦) < (((abs‘𝑦) + 1) / 2)))
129	56, 127, 128	sylancl 693	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) < 1 ↔ (abs‘𝑦) < (((abs‘𝑦) + 1) / 2)))
130	66, 129	mpbid 221	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < (((abs‘𝑦) + 1) / 2))
131		0red 9920	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ∈ ℝ)
132	131, 56, 86, 82, 130	lelttrd 10074	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 < (((abs‘𝑦) + 1) / 2))
133	131, 86, 132	ltled 10064	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ≤ (((abs‘𝑦) + 1) / 2))
134	86, 133	absidd 14009	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) = (((abs‘𝑦) + 1) / 2))
135	46, 109	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚))):ℕ₀⟶ℂ)
136	86	recnd 9947	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℂ)
137		oveq1 6556	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (((abs‘𝑦) + 1) / 2) → (𝑥↑𝑗) = ((((abs‘𝑦) + 1) / 2)↑𝑗))
138	137	oveq2d 6565	. . . . . . . . . . . . . . . . . . . 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)))
139	138	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . 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))))
140	115	mptex 6390	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗))) ∈ V
141	139, 114, 140	fvmpt 6191	. . . . . . . . . . . . . . . . . 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))))
142	136, 141	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))))
143	142	seqeq3d 12671	. . . . . . . . . . . . . . . 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)))))
144		avglt2 11148	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((abs‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (((abs‘𝑦) + 1) / 2) < 1))
145	56, 127, 144	sylancl 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) < 1 ↔ (((abs‘𝑦) + 1) / 2) < 1))
146	66, 145	mpbid 221	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) < 1)
147	134, 146	eqbrtrd 4605	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) < 1)
148		logtayllem 24205	. . . . . . . . . . . . . . . . 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 ⇝ )
149	136, 147, 148	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))) ∈ dom ⇝ )
150	143, 149	eqeltrd 2688	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘(((abs‘𝑦) + 1) / 2))) ∈ dom ⇝ )
151	100, 135, 123, 136, 150	radcnvle 23978	. . . . . . . . . . . . . 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^*, < ))
152	134, 151	eqbrtrrd 4607	. . . . . . . . . . . . 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^*, < ))
153	83, 87, 126, 130, 152	xrltletrd 11868	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))
154		0re 9919	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
155		elico2 12108	. . . . . . . . . . . . 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^, < ))))
156	154, 126, 155	sylancr 694	. . . . . . . . . . . 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^, < ))))
157	56, 82, 153, 156	mpbir3and 1238	. . . . . . . . . . 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^*, < )))
158		absf 13925	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
159		ffn 5958	. . . . . . . . . . . 12 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
160		elpreima 6245	. . . . . . . . . . . 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^, < )))))
161	158, 159, 160	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^, < ))))
162	53, 157, 161	sylanbrc 695	. . . . . . . . . 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^*, < ))))
163		cnvimass 5404	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) ⊆ dom abs
164	158	fdmi 5965	. . . . . . . . . . . . . . . . . 18 ⊢ dom abs = ℂ
165	163, 164	sseqtri 3600	. . . . . . . . . . . . . . . . 17 ⊢ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) ⊆ ℂ
166	165	sseli 3564	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) → 𝑦 ∈ ℂ)
167		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑦 → (𝑥↑𝑗) = (𝑦↑𝑗))
168	167	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))
169	168	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗))))
170	115	mptex 6390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗))) ∈ V
171	169, 114, 170	fvmpt 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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))))
172	171	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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))))
173	172	fveq1d 6105	. . . . . . . . . . . . . . . . . 18 ⊢ ((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))
174		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑛 → (𝑗 = 0 ↔ 𝑛 = 0))
175		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑛 → (1 / 𝑗) = (1 / 𝑛))
176	174, 175	ifbieq2d 4061	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → if(𝑗 = 0, 0, (1 / 𝑗)) = if(𝑛 = 0, 0, (1 / 𝑛)))
177		oveq2 6557	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → (𝑦↑𝑗) = (𝑦↑𝑛))
178	176, 177	oveq12d 6567	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
179		eqid 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))
180		ovex 6577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) ∈ V
181	178, 179, 180	fvmpt 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ → ((𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
182	181	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
183	173, 182	eqtr2d 2645	. . . . . . . . . . . . . . . . 17 ⊢ ((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))
184	183	sumeq2dv 14281	. . . . . . . . . . . . . . . 16 ⊢ (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))
185	166, 184	syl 17	. . . . . . . . . . . . . . 15 ⊢ (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))
186	185	mpteq2ia 4668	. . . . . . . . . . . . . 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))) = (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))
187		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (^◡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^, < )))
188		eqid 2610	. . . . . . . . . . . . . 14 ⊢ 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))
189	100, 186, 109, 123, 187, 188	psercn 23984	. . . . . . . . . . . . 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))) ∈ ((^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < )))–cn→ℂ))
190		cncff 22504	. . . . . . . . . . . . 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))) ∈ ((^◡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)
191	189, 190	syl 17	. . . . . . . . . . . 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^, < )))⟶ℂ)
192		eqid 2610	. . . . . . . . . . . . 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 (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
193	192	fmpt 6289	. . . . . . . . . . . 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)) ∈ 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)
194	191, 193	sylibr 223	. . . . . . . . . . 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)) ∈ ℂ)
195	194	r19.21bi 2916	. . . . . . . . . 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)
196	162, 195	sylan2 490	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) ∈ ℂ)
197		eqid 2610	. . . . . . . . 9 ⊢ (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)))
198	196, 197	fmptd 6292	. . . . . . . 8 ⊢ (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))):(0(ball‘(abs ∘ − ))1)⟶ℂ)
199		cnelprrecn 9908	. . . . . . . . . . . . 13 ⊢ ℂ ∈ {ℝ, ℂ}
200	199	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
201	77	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → (log‘(1 − 𝑦)) ∈ ℂ)
202		ovex 6577	. . . . . . . . . . . . 13 ⊢ ((1 / (1 − 𝑦)) · -1) ∈ V
203	202	a1i 11	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → ((1 / (1 − 𝑦)) · -1) ∈ V)
204	29	cnmetdval 22384	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℂ ∧ (1 − 𝑦) ∈ ℂ) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘(1 − (1 − 𝑦))))
205	50, 55, 204	sylancr 694	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘(1 − (1 − 𝑦))))
206		nncan 10189	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − (1 − 𝑦)) = 𝑦)
207	50, 53, 206	sylancr 694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − (1 − 𝑦)) = 𝑦)
208	207	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(1 − (1 − 𝑦))) = (abs‘𝑦))
209	205, 208	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘𝑦))
210	209, 66	eqbrtrd 4605	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) < 1)
211		elbl 22003	. . . . . . . . . . . . . . . 16 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ^*) → ((1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 − 𝑦) ∈ ℂ ∧ (1(abs ∘ − )(1 − 𝑦)) < 1)))
212	38, 50, 41, 211	mp3an 1416	. . . . . . . . . . . . . . 15 ⊢ ((1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 − 𝑦) ∈ ℂ ∧ (1(abs ∘ − )(1 − 𝑦)) < 1))
213	55, 210, 212	sylanbrc 695	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1))
214	213	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → (1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1))
215		neg1cn 11001	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℂ
216	215	a1i 11	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → -1 ∈ ℂ)
217		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
218	217	dvlog2lem 24198	. . . . . . . . . . . . . . . . 17 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
219	218	sseli 3564	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
220	219	eldifad 3552	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ∈ ℂ)
221		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
222	221	logdmn0 24186	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) → 𝑥 ≠ 0)
223	219, 222	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ≠ 0)
224	220, 223	logcld 24121	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑥) ∈ ℂ)
225	224	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(ball‘(abs ∘ − ))1)) → (log‘𝑥) ∈ ℂ)
226		ovex 6577	. . . . . . . . . . . . . 14 ⊢ (1 / 𝑥) ∈ V
227	226	a1i 11	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(ball‘(abs ∘ − ))1)) → (1 / 𝑥) ∈ V)
228		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
229	50, 228, 54	sylancr 694	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → (1 − 𝑦) ∈ ℂ)
230	215	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → -1 ∈ ℂ)
231		1cnd 9935	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
232		0cnd 9912	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 0 ∈ ℂ)
233		1cnd 9935	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → 1 ∈ ℂ)
234	200, 233	dvmptc 23527	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 1)) = (𝑦 ∈ ℂ ↦ 0))
235	200	dvmptid 23526	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1))
236	200, 231, 232, 234, 228, 231, 235	dvmptsub 23536	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (1 − 𝑦))) = (𝑦 ∈ ℂ ↦ (0 − 1)))
237		df-neg 10148	. . . . . . . . . . . . . . . 16 ⊢ -1 = (0 − 1)
238	237	mpteq2i 4669	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℂ ↦ -1) = (𝑦 ∈ ℂ ↦ (0 − 1))
239	236, 238	syl6eqr 2662	. . . . . . . . . . . . . 14 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (1 − 𝑦))) = (𝑦 ∈ ℂ ↦ -1))
240	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
241		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
242	241	cnfldtop 22397	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) ∈ Top
243	241	cnfldtopon 22396	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
244	243	toponunii 20547	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
245	244	restid 15917	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
246	242, 245	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
247	246	eqcomi 2619	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
248	241	cnfldtopn 22395	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
249	248	blopn 22115	. . . . . . . . . . . . . . . 16 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂ_fld))
250	38, 28, 41, 249	mp3an 1416	. . . . . . . . . . . . . . 15 ⊢ (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂ_fld)
251	250	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂ_fld))
252	200, 229, 230, 239, 240, 247, 241, 251	dvmptres 23532	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -1))
253	217	dvlog2 24199	. . . . . . . . . . . . . 14 ⊢ (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑥))
254		logf1o 24115	. . . . . . . . . . . . . . . . . . . 20 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
255		f1of 6050	. . . . . . . . . . . . . . . . . . . 20 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
256	254, 255	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ log:(ℂ ∖ {0})⟶ran log
257	221	logdmss 24188	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
258	218, 257	sstri 3577	. . . . . . . . . . . . . . . . . . 19 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
259		fssres 5983	. . . . . . . . . . . . . . . . . . 19 ⊢ ((log:(ℂ ∖ {0})⟶ran log ∧ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})) → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log)
260	256, 258, 259	mp2an 704	. . . . . . . . . . . . . . . . . 18 ⊢ (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log
261	260	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log)
262	261	feqmptd 6159	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥)))
263		fvres 6117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥) = (log‘𝑥))
264	263	mpteq2ia 4668	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))
265	262, 264	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥)))
266	265	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ (⊤ → (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (ℂ D (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))))
267	253, 266	syl5reqr 2659	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑥)))
268		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1 − 𝑦) → (log‘𝑥) = (log‘(1 − 𝑦)))
269		oveq2 6557	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1 − 𝑦) → (1 / 𝑥) = (1 / (1 − 𝑦)))
270	200, 200, 214, 216, 225, 227, 252, 267, 268, 269	dvmptco 23541	. . . . . . . . . . . 12 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ ((1 / (1 − 𝑦)) · -1)))
271	200, 201, 203, 270	dvmptneg 23535	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -((1 / (1 − 𝑦)) · -1)))
272	55, 76	reccld 10673	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 / (1 − 𝑦)) ∈ ℂ)
273		mulcom 9901	. . . . . . . . . . . . . . . 16 ⊢ (((1 / (1 − 𝑦)) ∈ ℂ ∧ -1 ∈ ℂ) → ((1 / (1 − 𝑦)) · -1) = (-1 · (1 / (1 − 𝑦))))
274	272, 215, 273	sylancl 693	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 / (1 − 𝑦)) · -1) = (-1 · (1 / (1 − 𝑦))))
275	272	mulm1d 10361	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (-1 · (1 / (1 − 𝑦))) = -(1 / (1 − 𝑦)))
276	274, 275	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 / (1 − 𝑦)) · -1) = -(1 / (1 − 𝑦)))
277	276	negeqd 10154	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -((1 / (1 − 𝑦)) · -1) = --(1 / (1 − 𝑦)))
278	272	negnegd 10262	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → --(1 / (1 − 𝑦)) = (1 / (1 − 𝑦)))
279	277, 278	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -((1 / (1 − 𝑦)) · -1) = (1 / (1 − 𝑦)))
280	279	mpteq2ia 4668	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -((1 / (1 − 𝑦)) · -1)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦)))
281	271, 280	syl6eq 2660	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
282	281	dmeqd 5248	. . . . . . . . 9 ⊢ (⊤ → dom (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
283		dmmptg 5549	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (0(ball‘(abs ∘ − ))1)(1 / (1 − 𝑦)) ∈ V → dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))) = (0(ball‘(abs ∘ − ))1))
284		ovex 6577	. . . . . . . . . . 11 ⊢ (1 / (1 − 𝑦)) ∈ V
285	284	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 / (1 − 𝑦)) ∈ V)
286	283, 285	mprg 2910	. . . . . . . . 9 ⊢ dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))) = (0(ball‘(abs ∘ − ))1)
287	282, 286	syl6eq 2660	. . . . . . . 8 ⊢ (⊤ → dom (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (0(ball‘(abs ∘ − ))1))
288		sumex 14266	. . . . . . . . . . . 12 ⊢ Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) ∈ V
289	288	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)
290		fveq2 6103	. . . . . . . . . . . . . . 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))
291	290	cbvsumv 14274	. . . . . . . . . . . . . 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)
292	185, 291	syl6eq 2660	. . . . . . . . . . . . 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))
293	292	mpteq2ia 4668	. . . . . . . . . . . 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))
294		eqid 2610	. . . . . . . . . . . 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))
295	100, 293, 109, 123, 187, 188, 294	pserdv2 23988	. . . . . . . . . . 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)))))
296	162	ssriv 3572	. . . . . . . . . . . 12 ⊢ (0(ball‘(abs ∘ − ))1) ⊆ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < )))
297	296	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^*, < ))))
298	200, 195, 289, 295, 297, 247, 241, 251	dvmptres 23532	. . . . . . . . . 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)))))
299		nnnn0 11176	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
300	299	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
301		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (𝑚 = 0 ↔ 𝑛 = 0))
302		oveq2 6557	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
303	301, 302	ifbieq2d 4061	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → if(𝑚 = 0, 0, (1 / 𝑚)) = if(𝑛 = 0, 0, (1 / 𝑛)))
304		ovex 6577	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 / 𝑛) ∈ V
305	93, 304	ifex 4106	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑛 = 0, 0, (1 / 𝑛)) ∈ V
306	303, 92, 305	fvmpt 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)))
307	300, 306	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)))
308		nnne0 10930	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
309	308	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
310	309	neneqd 2787	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
311	310	iffalsed 4047	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, 0, (1 / 𝑛)) = (1 / 𝑛))
312	307, 311	eqtrd 2644	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = (1 / 𝑛))
313	312	oveq2d 6565	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) = (𝑛 · (1 / 𝑛)))
314		nncn 10905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
315	314	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
316	315, 309	recidd 10675	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · (1 / 𝑛)) = 1)
317	313, 316	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) = 1)
318	317	oveq1d 6564	. . . . . . . . . . . . . 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))))
319		nnm1nn0 11211	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
320		expcl 12740	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℂ ∧ (𝑛 − 1) ∈ ℕ₀) → (𝑦↑(𝑛 − 1)) ∈ ℂ)
321	53, 319, 320	syl2an 493	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑦↑(𝑛 − 1)) ∈ ℂ)
322	321	mulid2d 9937	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (1 · (𝑦↑(𝑛 − 1))) = (𝑦↑(𝑛 − 1)))
323	318, 322	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (𝑦↑(𝑛 − 1)))
324	323	sumeq2dv 14281	. . . . . . . . . . . 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)))
325		nnuz 11599	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ_≥‘1)
326		1e0p1 11428	. . . . . . . . . . . . . . . 16 ⊢ 1 = (0 + 1)
327	326	fveq2i 6106	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘1) = (ℤ_≥‘(0 + 1))
328	325, 327	eqtri 2632	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘(0 + 1))
329		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (1 + 𝑚) → (𝑛 − 1) = ((1 + 𝑚) − 1))
330	329	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1 + 𝑚) → (𝑦↑(𝑛 − 1)) = (𝑦↑((1 + 𝑚) − 1)))
331		1zzd 11285	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ∈ ℤ)
332		0zd 11266	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ∈ ℤ)
333	1, 328, 330, 331, 332, 321	isumshft 14410	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)) = Σ𝑚 ∈ ℕ₀ (𝑦↑((1 + 𝑚) − 1)))
334		pncan2 10167	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((1 + 𝑚) − 1) = 𝑚)
335	50, 102, 334	sylancr 694	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ₀ → ((1 + 𝑚) − 1) = 𝑚)
336	335	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (𝑦↑((1 + 𝑚) − 1)) = (𝑦↑𝑚))
337	336	sumeq2i 14277	. . . . . . . . . . . . 13 ⊢ Σ𝑚 ∈ ℕ₀ (𝑦↑((1 + 𝑚) − 1)) = Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚)
338	333, 337	syl6eq 2660	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)) = Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚))
339		geoisum 14447	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℂ ∧ (abs‘𝑦) < 1) → Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚) = (1 / (1 − 𝑦)))
340	53, 66, 339	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚) = (1 / (1 − 𝑦)))
341	324, 338, 340	3eqtrd 2648	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (1 / (1 − 𝑦)))
342	341	mpteq2ia 4668	. . . . . . . . . 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)))
343	298, 342	syl6eq 2660	. . . . . . . . 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))))
344	281, 343	eqtr4d 2647	. . . . . . . 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)))))
345		blcntr 22028	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ⁺) → 0 ∈ (0(ball‘(abs ∘ − ))1))
346	38, 28, 39, 345	mp3an 1416	. . . . . . . . 9 ⊢ 0 ∈ (0(ball‘(abs ∘ − ))1)
347	346	a1i 11	. . . . . . . 8 ⊢ (⊤ → 0 ∈ (0(ball‘(abs ∘ − ))1))
348		oveq2 6557	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 0 → (1 − 𝑦) = (1 − 0))
349		1m0e1 11008	. . . . . . . . . . . . . . . 16 ⊢ (1 − 0) = 1
350	348, 349	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 0 → (1 − 𝑦) = 1)
351	350	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 0 → (log‘(1 − 𝑦)) = (log‘1))
352		log1 24136	. . . . . . . . . . . . . 14 ⊢ (log‘1) = 0
353	351, 352	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (log‘(1 − 𝑦)) = 0)
354	353	negeqd 10154	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → -(log‘(1 − 𝑦)) = -0)
355		neg0 10206	. . . . . . . . . . . 12 ⊢ -0 = 0
356	354, 355	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → -(log‘(1 − 𝑦)) = 0)
357	356, 80, 93	fvmpt 6191	. . . . . . . . . 10 ⊢ (0 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = 0)
358	346, 357	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = 0)
359		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (0 = if(𝑛 = 0, 0, (1 / 𝑛)) → (0 · (𝑦↑𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
360	359	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (0 = if(𝑛 = 0, 0, (1 / 𝑛)) → ((0 · (𝑦↑𝑛)) = 0 ↔ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0))
361		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ ((1 / 𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)) → ((1 / 𝑛) · (𝑦↑𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
362	361	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ ((1 / 𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)) → (((1 / 𝑛) · (𝑦↑𝑛)) = 0 ↔ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0))
363		simpll 786	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 𝑦 = 0)
364	363, 28	syl6eqel 2696	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 𝑦 ∈ ℂ)
365		simplr 788	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 𝑛 ∈ ℕ₀)
366	364, 365	expcld 12870	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → (𝑦↑𝑛) ∈ ℂ)
367	366	mul02d 10113	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → (0 · (𝑦↑𝑛)) = 0)
368		simpll 786	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → 𝑦 = 0)
369	368	oveq1d 6564	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (𝑦↑𝑛) = (0↑𝑛))
370		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
371	370, 14	sylib 207	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (𝑛 ∈ ℕ ∨ 𝑛 = 0))
372	371	ord 391	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 ∈ ℕ → 𝑛 = 0))
373	372	con1d 138	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 = 0 → 𝑛 ∈ ℕ))
374	373	imp 444	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
375	374	0expd 12886	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (0↑𝑛) = 0)
376	369, 375	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (𝑦↑𝑛) = 0)
377	376	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · (𝑦↑𝑛)) = ((1 / 𝑛) · 0))
378	374	nnrecred 10943	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℝ)
379	378	recnd 9947	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℂ)
380	379	mul01d 10114	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · 0) = 0)
381	377, 380	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · (𝑦↑𝑛)) = 0)
382	360, 362, 367, 381	ifbothda 4073	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0)
383	382	sumeq2dv 14281	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = Σ𝑛 ∈ ℕ₀ 0)
384	1	eqimssi 3622	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ⊆ (ℤ_≥‘0)
385	384	orci 404	. . . . . . . . . . . . 13 ⊢ (ℕ₀ ⊆ (ℤ_≥‘0) ∨ ℕ₀ ∈ Fin)
386		sumz 14300	. . . . . . . . . . . . 13 ⊢ ((ℕ₀ ⊆ (ℤ_≥‘0) ∨ ℕ₀ ∈ Fin) → Σ𝑛 ∈ ℕ₀ 0 = 0)
387	385, 386	ax-mp 5	. . . . . . . . . . . 12 ⊢ Σ𝑛 ∈ ℕ₀ 0 = 0
388	383, 387	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0)
389	388, 197, 93	fvmpt 6191	. . . . . . . . . 10 ⊢ (0 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))‘0) = 0)
390	346, 389	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))‘0) = 0)
391	358, 390	eqtr4d 2647	. . . . . . . 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))
392	47, 48, 49, 81, 198, 287, 344, 347, 391	dv11cn 23568	. . . . . . 7 ⊢ (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))))
393	392	fveq1d 6105	. . . . . 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))
394	46, 393	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))
395		oveq2 6557	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (1 − 𝑦) = (1 − 𝐴))
396	395	fveq2d 6107	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (log‘(1 − 𝑦)) = (log‘(1 − 𝐴)))
397	396	negeqd 10154	. . . . . 6 ⊢ (𝑦 = 𝐴 → -(log‘(1 − 𝑦)) = -(log‘(1 − 𝐴)))
398		negex 10158	. . . . . 6 ⊢ -(log‘(1 − 𝐴)) ∈ V
399	397, 80, 398	fvmpt 6191	. . . . 5 ⊢ (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘𝐴) = -(log‘(1 − 𝐴)))
400		oveq1 6556	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦↑𝑛) = (𝐴↑𝑛))
401	400	oveq2d 6565	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
402	401	sumeq2sdv 14282	. . . . . 6 ⊢ (𝑦 = 𝐴 → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
403		sumex 14266	. . . . . 6 ⊢ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) ∈ V
404	402, 197, 403	fvmpt 6191	. . . . 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)))
405	394, 399, 404	3eqtr3d 2652	. . . 4 ⊢ (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → -(log‘(1 − 𝐴)) = Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
406	45, 405	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(log‘(1 − 𝐴)) = Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
407	26, 406	breqtrrd 4611	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ⇝ -(log‘(1 − 𝐴)))
408		seqex 12665	. . . 4 ⊢ seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ∈ V
409	408	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ∈ V)
410		seqex 12665	. . . 4 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))) ∈ V
411	410	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))) ∈ V)
412		1zzd 11285	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℤ)
413		elnnuz 11600	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
414		fvres 6117	. . . . . 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))
415	413, 414	sylbi 206	. . . . 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))
416	415	eqcomd 2616	. . . 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))
417		addid2 10098	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
418	417	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
419		0cnd 9912	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ∈ ℂ)
420		1eluzge0 11608	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
421	420	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ (ℤ_≥‘0))
422		0cnd 9912	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 = 0) → 0 ∈ ℂ)
423		nn0cn 11179	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
424	423	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
425		df-ne 2782	. . . . . . . . . . . . 13 ⊢ (𝑘 ≠ 0 ↔ ¬ 𝑘 = 0)
426	425	biimpri 217	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
427		reccl 10571	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℂ ∧ 𝑘 ≠ 0) → (1 / 𝑘) ∈ ℂ)
428	424, 426, 427	syl2an 493	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 = 0) → (1 / 𝑘) ∈ ℂ)
429	422, 428	ifclda 4070	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → if(𝑘 = 0, 0, (1 / 𝑘)) ∈ ℂ)
430		expcl 12740	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℂ)
431	430	adantlr 747	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℂ)
432	429, 431	mulcld 9939	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) ∈ ℂ)
433	432, 8	fmptd 6292	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘))):ℕ₀⟶ℂ)
434		1nn0 11185	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
435		ffvelrn 6265	. . . . . . . 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)
436	433, 434, 435	sylancl 693	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘1) ∈ ℂ)
437		elfz1eq 12223	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...0) → 𝑛 = 0)
438		1m1e0 10966	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
439	438	oveq2i 6560	. . . . . . . . . 10 ⊢ (0...(1 − 1)) = (0...0)
440	437, 439	eleq2s 2706	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(1 − 1)) → 𝑛 = 0)
441	440	fveq2d 6107	. . . . . . . 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))
442		0nn0 11184	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
443		iftrue 4042	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → if(𝑘 = 0, 0, (1 / 𝑘)) = 0)
444		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝐴↑𝑘) = (𝐴↑0))
445	443, 444	oveq12d 6567	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) = (0 · (𝐴↑0)))
446		ovex 6577	. . . . . . . . . . 11 ⊢ (0 · (𝐴↑0)) ∈ V
447	445, 8, 446	fvmpt 6191	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘0) = (0 · (𝐴↑0)))
448	442, 447	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘0) = (0 · (𝐴↑0))
449		expcl 12740	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℕ₀) → (𝐴↑0) ∈ ℂ)
450	27, 442, 449	sylancl 693	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑0) ∈ ℂ)
451	450	mul02d 10113	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (0 · (𝐴↑0)) = 0)
452	448, 451	syl5eq 2656	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘0) = 0)
453	441, 452	sylan9eqr 2666	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ (0...(1 − 1))) → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘𝑛) = 0)
454	418, 419, 421, 436, 453	seqid 12708	. . . . . 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)))))
455	308	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
456	455	neneqd 2787	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
457	456	iffalsed 4047	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, 0, (1 / 𝑛)) = (1 / 𝑛))
458	457	oveq1d 6564	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) = ((1 / 𝑛) · (𝐴↑𝑛)))
459	299, 23	sylan2 490	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ∈ ℂ)
460	314	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
461	459, 460, 455	divrec2d 10684	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝐴↑𝑛) / 𝑛) = ((1 / 𝑛) · (𝐴↑𝑛)))
462	458, 461	eqtr4d 2647	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) = ((𝐴↑𝑛) / 𝑛))
463	299, 11	sylan2 490	. . . . . . . . 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)))
464		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
465	6, 464	oveq12d 6567	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝐴↑𝑘) / 𝑘) = ((𝐴↑𝑛) / 𝑛))
466		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘)) = (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))
467		ovex 6577	. . . . . . . . . . 11 ⊢ ((𝐴↑𝑛) / 𝑛) ∈ V
468	465, 466, 467	fvmpt 6191	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))‘𝑛) = ((𝐴↑𝑛) / 𝑛))
469	468	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))‘𝑛) = ((𝐴↑𝑛) / 𝑛))
470	462, 463, 469	3eqtr4d 2654	. . . . . . . 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))
471	413, 470	sylan2br 492	. . . . . . 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))
472	412, 471	seqfeq 12688	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))))
473	454, 472	eqtrd 2644	. . . . 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))))
474	473	fveq1d 6105	. . . 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))
475	416, 474	sylan9eqr 2666	. . 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))
476	325, 409, 411, 412, 475	climeq 14146	. 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))))
477	407, 476	mpbid 221	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 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ifcif 4036 {csn 4125 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 supcsup 8229 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤ_≥cuz 11563 ℝ⁺crp 11708 (,]cioc 12047 [,)cico 12048 [,]cicc 12049 ...cfz 12197 seqcseq 12663 ↑cexp 12722 abscabs 13822 ⇝ cli 14063 Σcsu 14264 ↾_t crest 15904 TopOpenctopn 15905 ∞Metcxmt 19552 ballcbl 19554 ℂ_fldccnfld 19567 Topctop 20517 –cn→ccncf 22487 D cdv 23433 logclog 24105

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-tan 14641 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-ulm 23935 df-log 24107

This theorem is referenced by: logtaylsum 24207 logtayl2 24208 atantayl 24464 stirlinglem5 38971

W3C validator