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Theorem logtayl 26793

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 12902	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12605	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ∈ ℤ)
3		eqeq1 2773	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
4		oveq2 7421	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (1 / 𝑘) = (1 / 𝑛))
5	3, 4	ifbieq2d 4519	. . . . . . 7 ⊢ (𝑘 = 𝑛 → if(𝑘 = 0, 0, (1 / 𝑘)) = if(𝑛 = 0, 0, (1 / 𝑛)))
6		oveq2 7421	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝐴↑𝑘) = (𝐴↑𝑛))
7	5, 6	oveq12d 7431	. . . . . 6 ⊢ (𝑘 = 𝑛 → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
8		eqid 2769	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘))) = (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))
9		ovex 7446	. . . . . 6 ⊢ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) ∈ V
10	7, 8, 9	fvmpt 6992	. . . . 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 486	. . . 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 11201	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 0 ∈ ℂ)
13		elnn0 12508	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
14	13	bilani 509	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (𝑛 ∈ ℕ ∨ 𝑛 = 0))
15	14	ord 877	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 ∈ ℕ → 𝑛 = 0))
16	15	con1d 146	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 = 0 → 𝑛 ∈ ℕ))
17	16	imp 411	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
18	17	nnrecred 12289	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℝ)
19	18	recnd 11239	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℂ)
20	12, 19	ifclda 4528	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → if(𝑛 = 0, 0, (1 / 𝑛)) ∈ ℂ)
21		expcl 14117	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (𝐴↑𝑛) ∈ ℂ)
22	21	adantlr 727	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (𝐴↑𝑛) ∈ ℂ)
23	20, 22	mulcld 11231	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ₀) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) ∈ ℂ)
24		logtayllem 26792	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ∈ dom ⇝ )
25	1, 2, 11, 23, 24	isumclim2 15811	. . 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)))
26		simpl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ)
27		0cn 11200	. . . . . . . 8 ⊢ 0 ∈ ℂ
28		eqid 2769	. . . . . . . . 9 ⊢ (abs ∘ − ) = (abs ∘ − )
29	28	cnmetdval 24898	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴(abs ∘ − )0) = (abs‘(𝐴 − 0)))
30	26, 27, 29	sylancl 597	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) = (abs‘(𝐴 − 0)))
31		subid1 11480	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
32	31	adantr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 − 0) = 𝐴)
33	32	fveq2d 6888	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴 − 0)) = (abs‘𝐴))
34	30, 33	eqtrd 2804	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) = (abs‘𝐴))
35		simpr 489	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1)
36	34, 35	eqbrtrd 5137	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴(abs ∘ − )0) < 1)
37		cnxmet 24900	. . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
38		1xr 11270	. . . . . . 7 ⊢ 1 ∈ ℝ^*
39		elbl3 24520	. . . . . . 7 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
40	37, 38, 39	mpanl12 714	. . . . . 6 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
41	27, 26, 40	sylancr 598	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝐴(abs ∘ − )0) < 1))
42	36, 41	mpbird 260	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ (0(ball‘(abs ∘ − ))1))
43		tru 1571	. . . . . 6 ⊢ ⊤
44		eqid 2769	. . . . . . . 8 ⊢ (0(ball‘(abs ∘ − ))1) = (0(ball‘(abs ∘ − ))1)
45		0cnd 11201	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℂ)
46	38	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℝ^*)
47		ax-1cn 11160	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
48		blssm 24546	. . . . . . . . . . . . . . 15 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
49	37, 27, 38, 48	mp3an 1487	. . . . . . . . . . . . . 14 ⊢ (0(ball‘(abs ∘ − ))1) ⊆ ℂ
50	49	sseli 3941	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 𝑦 ∈ ℂ)
51		subcl 11458	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − 𝑦) ∈ ℂ)
52	47, 50, 51	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ∈ ℂ)
53	50	abscld 15492	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ ℝ)
54	28	cnmetdval 24898	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑦(abs ∘ − )0) = (abs‘(𝑦 − 0)))
55	50, 27, 54	sylancl 597	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) = (abs‘(𝑦 − 0)))
56	50	subid1d 11560	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦 − 0) = 𝑦)
57	56	fveq2d 6888	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(𝑦 − 0)) = (abs‘𝑦))
58	55, 57	eqtrd 2804	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) = (abs‘𝑦))
59		elbl3 24520	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
60	37, 38, 59	mpanl12 714	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
61	27, 50, 60	sylancr 598	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑦(abs ∘ − )0) < 1))
62	61	ibi 270	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑦(abs ∘ − )0) < 1)
63	58, 62	eqbrtrrd 5139	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < 1)
64	53, 63	gtned 11347	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ≠ (abs‘𝑦))
65		abs1 15350	. . . . . . . . . . . . . . . 16 ⊢ (abs‘1) = 1
66		fveq2 6884	. . . . . . . . . . . . . . . 16 ⊢ (1 = 𝑦 → (abs‘1) = (abs‘𝑦))
67	65, 66	eqtr3id 2818	. . . . . . . . . . . . . . 15 ⊢ (1 = 𝑦 → 1 = (abs‘𝑦))
68	67	necon3i 2996	. . . . . . . . . . . . . 14 ⊢ (1 ≠ (abs‘𝑦) → 1 ≠ 𝑦)
69	64, 68	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ≠ 𝑦)
70		subeq0 11486	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((1 − 𝑦) = 0 ↔ 1 = 𝑦))
71	70	necon3bid 3008	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((1 − 𝑦) ≠ 0 ↔ 1 ≠ 𝑦))
72	47, 50, 71	sylancr 598	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 − 𝑦) ≠ 0 ↔ 1 ≠ 𝑦))
73	69, 72	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ≠ 0)
74	52, 73	logcld 26703	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (log‘(1 − 𝑦)) ∈ ℂ)
75	74	negcld 11558	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -(log‘(1 − 𝑦)) ∈ ℂ)
76	75	adantl 486	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → -(log‘(1 − 𝑦)) ∈ ℂ)
77	76	fmpttd 7113	. . . . . . . 8 ⊢ (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))):(0(ball‘(abs ∘ − ))1)⟶ℂ)
78	50	absge0d 15500	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ≤ (abs‘𝑦))
79	53	rexrd 11261	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) ∈ ℝ^*)
80		peano2re 11385	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑦) ∈ ℝ → ((abs‘𝑦) + 1) ∈ ℝ)
81	53, 80	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) + 1) ∈ ℝ)
82	81	rehalfcld 12493	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℝ)
83	82	rexrd 11261	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℝ^*)
84		iccssxr 13459	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) ⊆ ℝ^*
85		eqeq1 2773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑗 → (𝑚 = 0 ↔ 𝑗 = 0))
86		oveq2 7421	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑗 → (1 / 𝑚) = (1 / 𝑗))
87	85, 86	ifbieq2d 4519	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑗 → if(𝑚 = 0, 0, (1 / 𝑚)) = if(𝑗 = 0, 0, (1 / 𝑗)))
88		eqid 2769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚))) = (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))
89		c0ex 11202	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
90		ovex 7446	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 / 𝑗) ∈ V
91	89, 90	ifex 4543	. . . . . . . . . . . . . . . . . . . . 21 ⊢ if(𝑗 = 0, 0, (1 / 𝑗)) ∈ V
92	87, 88, 91	fvmpt 6992	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ℕ₀ → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) = if(𝑗 = 0, 0, (1 / 𝑗)))
93	92	eqcomd 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ₀ → if(𝑗 = 0, 0, (1 / 𝑗)) = ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗))
94	93	oveq1d 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℕ₀ → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)) = (((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥↑𝑗)))
95	94	mpteq2ia 5210	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑗) · (𝑥↑𝑗)))
96	95	mpteq2i 5211	. . . . . . . . . . . . . . . 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))))
97		0cnd 11201	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑚 ∈ ℕ₀) ∧ 𝑚 = 0) → 0 ∈ ℂ)
98		nn0cn 12516	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
99	98	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℂ)
100		neqne 2972	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑚 = 0 → 𝑚 ≠ 0)
101		reccl 11881	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℂ ∧ 𝑚 ≠ 0) → (1 / 𝑚) ∈ ℂ)
102	99, 100, 101	syl2an 607	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑚 ∈ ℕ₀) ∧ ¬ 𝑚 = 0) → (1 / 𝑚) ∈ ℂ)
103	97, 102	ifclda 4528	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑚 ∈ ℕ₀) → if(𝑚 = 0, 0, (1 / 𝑚)) ∈ ℂ)
104	103	fmpttd 7113	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚))):ℕ₀⟶ℂ)
105		recn 11192	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑟 ∈ ℝ → 𝑟 ∈ ℂ)
106		oveq1 7420	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑟 → (𝑥↑𝑗) = (𝑟↑𝑗))
107	106	oveq2d 7429	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑟 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))
108	107	mpteq2dv 5209	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑟 → (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗))))
109		eqid 2769	. . . . . . . . . . . . . . . . . . . . . . 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))))
110		nn0ex 12512	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ₀ ∈ V
111	110	mptex 7224	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗))) ∈ V
112	108, 109, 111	fvmpt 6992	. . . . . . . . . . . . . . . . . . . . . 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))))
113	105, 112	syl 18	. . . . . . . . . . . . . . . . . . . . 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))))
114	113	eqcomd 2775	. . . . . . . . . . . . . . . . . . . 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))
115	114	seqeq3d 14047	. . . . . . . . . . . . . . . . . . 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)))
116	115	eleq1d 2854	. . . . . . . . . . . . . . . . . 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 ⇝ ))
117	116	rabbiia 3427	. . . . . . . . . . . . . . . . 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 ⇝ }
118	117	supeq1i 9409	. . . . . . . . . . . . . . . 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^, < )
119	96, 104, 118	radcnvcl 26548	. . . . . . . . . . . . . . 15 ⊢ (⊤ → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ) ∈ (0[,]+∞))
120	84, 119	sselid 3943	. . . . . . . . . . . . . 14 ⊢ (⊤ → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ^)
121	43, 120	mp1i 14	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ^)
122		1re 11210	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
123		avglt1 12484	. . . . . . . . . . . . . . 15 ⊢ (((abs‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (abs‘𝑦) < (((abs‘𝑦) + 1) / 2)))
124	53, 122, 123	sylancl 597	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) < 1 ↔ (abs‘𝑦) < (((abs‘𝑦) + 1) / 2)))
125	63, 124	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < (((abs‘𝑦) + 1) / 2))
126		0red 11213	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ∈ ℝ)
127	126, 53, 82, 78, 125	lelttrd 11370	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 < (((abs‘𝑦) + 1) / 2))
128	126, 82, 127	ltled 11360	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ≤ (((abs‘𝑦) + 1) / 2))
129	82, 128	absidd 15476	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) = (((abs‘𝑦) + 1) / 2))
130	43, 104	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚))):ℕ₀⟶ℂ)
131	82	recnd 11239	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) ∈ ℂ)
132		oveq1 7420	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (((abs‘𝑦) + 1) / 2) → (𝑥↑𝑗) = ((((abs‘𝑦) + 1) / 2)↑𝑗))
133	132	oveq2d 7429	. . . . . . . . . . . . . . . . . . . 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)))
134	133	mpteq2dv 5209	. . . . . . . . . . . . . . . . . . 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))))
135	110	mptex 7224	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗))) ∈ V
136	134, 109, 135	fvmpt 6992	. . . . . . . . . . . . . . . . . 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))))
137	131, 136	syl 18	. . . . . . . . . . . . . . . . 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))))
138	137	seqeq3d 14047	. . . . . . . . . . . . . . . 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)))))
139		avglt2 12485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((abs‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑦) < 1 ↔ (((abs‘𝑦) + 1) / 2) < 1))
140	53, 122, 139	sylancl 597	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((abs‘𝑦) < 1 ↔ (((abs‘𝑦) + 1) / 2) < 1))
141	63, 140	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (((abs‘𝑦) + 1) / 2) < 1)
142	129, 141	eqbrtrd 5137	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(((abs‘𝑦) + 1) / 2)) < 1)
143		logtayllem 26792	. . . . . . . . . . . . . . . . 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 ⇝ )
144	131, 142, 143	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · ((((abs‘𝑦) + 1) / 2)↑𝑗)))) ∈ dom ⇝ )
145	138, 144	eqeltrd 2869	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → seq0( + , ((𝑥 ∈ ℂ ↦ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))))‘(((abs‘𝑦) + 1) / 2))) ∈ dom ⇝ )
146	96, 130, 118, 131, 145	radcnvle 26551	. . . . . . . . . . . . . 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^*, < ))
147	129, 146	eqbrtrrd 5139	. . . . . . . . . . . . 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^*, < ))
148	79, 83, 121, 125, 147	xrltletrd 13188	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘𝑦) < sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))
149		0re 11212	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
150		elico2 13439	. . . . . . . . . . . . 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^, < ))))
151	149, 121, 150	sylancr 598	. . . . . . . . . . . 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^, < ))))
152	53, 78, 148, 151	mpbir3and 1359	. . . . . . . . . . 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^*, < )))
153		absf 15391	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
154		ffn 6708	. . . . . . . . . . . 12 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
155		elpreima 7056	. . . . . . . . . . . 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^, < )))))
156	153, 154, 155	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^, < ))))
157	50, 152, 156	sylanbrc 594	. . . . . . . . . 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^*, < ))))
158		cnvimass 6087	. . . . . . . . . . . . . . . . 17 ⊢ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) ⊆ dom abs
159	153	fdmi 6720	. . . . . . . . . . . . . . . . 17 ⊢ dom abs = ℂ
160	158, 159	sseqtri 3993	. . . . . . . . . . . . . . . 16 ⊢ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) ⊆ ℂ
161	160	sseli 3941	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < ))) → 𝑦 ∈ ℂ)
162		oveq1 7420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → (𝑥↑𝑗) = (𝑦↑𝑗))
163	162	oveq2d 7429	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗)) = (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))
164	163	mpteq2dv 5209	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑥↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗))))
165	110	mptex 7224	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗))) ∈ V
166	164, 109, 165	fvmpt 6992	. . . . . . . . . . . . . . . . . . 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))))
167	166	adantr 485	. . . . . . . . . . . . . . . . . 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))))
168	167	fveq1d 6886	. . . . . . . . . . . . . . . . 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))
169		eqeq1 2773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → (𝑗 = 0 ↔ 𝑛 = 0))
170		oveq2 7421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → (1 / 𝑗) = (1 / 𝑛))
171	169, 170	ifbieq2d 4519	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → if(𝑗 = 0, 0, (1 / 𝑗)) = if(𝑛 = 0, 0, (1 / 𝑛)))
172		oveq2 7421	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (𝑦↑𝑗) = (𝑦↑𝑛))
173	171, 172	oveq12d 7431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑛 → (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
174		eqid 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗))) = (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))
175		ovex 7446	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) ∈ V
176	173, 174, 175	fvmpt 6992	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ₀ → ((𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
177	176	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑦↑𝑗)))‘𝑛) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
178	168, 177	eqtr2d 2805	. . . . . . . . . . . . . . . 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))
179	178	sumeq2dv 15755	. . . . . . . . . . . . . . 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))
180	161, 179	syl 18	. . . . . . . . . . . . . 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))
181	180	mpteq2ia 5210	. . . . . . . . . . . . 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))
182		eqid 2769	. . . . . . . . . . . . 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^, < )))
183		eqid 2769	. . . . . . . . . . . . 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))
184	96, 181, 104, 118, 182, 183	psercn 26557	. . . . . . . . . . . 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))
185		cncff 25023	. . . . . . . . . . . 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)
186	184, 185	syl 18	. . . . . . . . . . 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)
187	186	fvmptelcdm 7111	. . . . . . . . . 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)
188	157, 187	sylan2 604	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) ∈ ℂ)
189	188	fmpttd 7113	. . . . . . . 8 ⊢ (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))):(0(ball‘(abs ∘ − ))1)⟶ℂ)
190		cnelprrecn 11195	. . . . . . . . . . . . 13 ⊢ ℂ ∈ {ℝ, ℂ}
191	190	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
192	74	adantl 486	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → (log‘(1 − 𝑦)) ∈ ℂ)
193		ovexd 7448	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → ((1 / (1 − 𝑦)) · -1) ∈ V)
194	28	cnmetdval 24898	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℂ ∧ (1 − 𝑦) ∈ ℂ) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘(1 − (1 − 𝑦))))
195	47, 52, 194	sylancr 598	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘(1 − (1 − 𝑦))))
196		nncan 11489	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − (1 − 𝑦)) = 𝑦)
197	47, 50, 196	sylancr 598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − (1 − 𝑦)) = 𝑦)
198	197	fveq2d 6888	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (abs‘(1 − (1 − 𝑦))) = (abs‘𝑦))
199	195, 198	eqtrd 2804	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) = (abs‘𝑦))
200	199, 63	eqbrtrd 5137	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1(abs ∘ − )(1 − 𝑦)) < 1)
201		elbl 24516	. . . . . . . . . . . . . . . 16 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ^*) → ((1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 − 𝑦) ∈ ℂ ∧ (1(abs ∘ − )(1 − 𝑦)) < 1)))
202	37, 47, 38, 201	mp3an 1487	. . . . . . . . . . . . . . 15 ⊢ ((1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1) ↔ ((1 − 𝑦) ∈ ℂ ∧ (1(abs ∘ − )(1 − 𝑦)) < 1))
203	52, 200, 202	sylanbrc 594	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1))
204	203	adantl 486	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → (1 − 𝑦) ∈ (1(ball‘(abs ∘ − ))1))
205		neg1cn 12205	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℂ
206	205	a1i 11	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ (0(ball‘(abs ∘ − ))1)) → -1 ∈ ℂ)
207		eqid 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (1(ball‘(abs ∘ − ))1) = (1(ball‘(abs ∘ − ))1)
208	207	dvlog2lem 26785	. . . . . . . . . . . . . . . . 17 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ (-∞(,]0))
209	208	sseli 3941	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ∈ (ℂ ∖ (-∞(,]0)))
210	209	eldifad 3925	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ∈ ℂ)
211		eqid 2769	. . . . . . . . . . . . . . . . 17 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
212	211	logdmn0 26773	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) → 𝑥 ≠ 0)
213	209, 212	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → 𝑥 ≠ 0)
214	210, 213	logcld 26703	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → (log‘𝑥) ∈ ℂ)
215	214	adantl 486	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(ball‘(abs ∘ − ))1)) → (log‘𝑥) ∈ ℂ)
216		ovexd 7448	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(ball‘(abs ∘ − ))1)) → (1 / 𝑥) ∈ V)
217		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
218	47, 217, 51	sylancr 598	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → (1 − 𝑦) ∈ ℂ)
219	205	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → -1 ∈ ℂ)
220		1cnd 11204	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
221		0cnd 11201	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 0 ∈ ℂ)
222		1cnd 11204	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → 1 ∈ ℂ)
223	191, 222	dvmptc 26088	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 1)) = (𝑦 ∈ ℂ ↦ 0))
224	191	dvmptid 26087	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1))
225	191, 220, 221, 223, 217, 220, 224	dvmptsub 26097	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (1 − 𝑦))) = (𝑦 ∈ ℂ ↦ (0 − 1)))
226		df-neg 11446	. . . . . . . . . . . . . . . 16 ⊢ -1 = (0 − 1)
227	226	mpteq2i 5211	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℂ ↦ -1) = (𝑦 ∈ ℂ ↦ (0 − 1))
228	225, 227	eqtr4di 2822	. . . . . . . . . . . . . 14 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (1 − 𝑦))) = (𝑦 ∈ ℂ ↦ -1))
229	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
230		eqid 2769	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
231	230	cnfldtopon 24910	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
232	231	toponrestid 23049	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
233	230	cnfldtopn 24909	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
234	233	blopn 24628	. . . . . . . . . . . . . . . 16 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂ_fld))
235	37, 27, 38, 234	mp3an 1487	. . . . . . . . . . . . . . 15 ⊢ (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂ_fld)
236	235	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → (0(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂ_fld))
237	191, 218, 219, 228, 229, 232, 230, 236	dvmptres 26093	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -1))
238		logf1o 26697	. . . . . . . . . . . . . . . . . . . 20 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
239		f1of 6823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
240	238, 239	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ log:(ℂ ∖ {0})⟶ran log
241	211	logdmss 26775	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
242	208, 241	sstri 3954	. . . . . . . . . . . . . . . . . . 19 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})
243		fssres 6747	. . . . . . . . . . . . . . . . . . 19 ⊢ ((log:(ℂ ∖ {0})⟶ran log ∧ (1(ball‘(abs ∘ − ))1) ⊆ (ℂ ∖ {0})) → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log)
244	240, 242, 243	mp2an 704	. . . . . . . . . . . . . . . . . 18 ⊢ (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log
245	244	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)):(1(ball‘(abs ∘ − ))1)⟶ran log)
246	245	feqmptd 6952	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥)))
247		fvres 6903	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) → ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥) = (log‘𝑥))
248	247	mpteq2ia 5210	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ ((log ↾ (1(ball‘(abs ∘ − ))1))‘𝑥)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))
249	246, 248	eqtrdi 2820	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (log ↾ (1(ball‘(abs ∘ − ))1)) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥)))
250	249	oveq2d 7429	. . . . . . . . . . . . . 14 ⊢ (⊤ → (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (ℂ D (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))))
251	207	dvlog2 26786	. . . . . . . . . . . . . 14 ⊢ (ℂ D (log ↾ (1(ball‘(abs ∘ − ))1))) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑥))
252	250, 251	eqtr3di 2819	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (log‘𝑥))) = (𝑥 ∈ (1(ball‘(abs ∘ − ))1) ↦ (1 / 𝑥)))
253		fveq2 6884	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1 − 𝑦) → (log‘𝑥) = (log‘(1 − 𝑦)))
254		oveq2 7421	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1 − 𝑦) → (1 / 𝑥) = (1 / (1 − 𝑦)))
255	191, 191, 204, 206, 215, 216, 237, 252, 253, 254	dvmptco 26102	. . . . . . . . . . . 12 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ ((1 / (1 − 𝑦)) · -1)))
256	191, 192, 193, 255	dvmptneg 26096	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -((1 / (1 − 𝑦)) · -1)))
257	52, 73	reccld 11986	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 / (1 − 𝑦)) ∈ ℂ)
258		mulcom 11188	. . . . . . . . . . . . . . . 16 ⊢ (((1 / (1 − 𝑦)) ∈ ℂ ∧ -1 ∈ ℂ) → ((1 / (1 − 𝑦)) · -1) = (-1 · (1 / (1 − 𝑦))))
259	257, 205, 258	sylancl 597	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 / (1 − 𝑦)) · -1) = (-1 · (1 / (1 − 𝑦))))
260	257	mulm1d 11668	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (-1 · (1 / (1 − 𝑦))) = -(1 / (1 − 𝑦)))
261	259, 260	eqtrd 2804	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → ((1 / (1 − 𝑦)) · -1) = -(1 / (1 − 𝑦)))
262	261	negeqd 11453	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -((1 / (1 − 𝑦)) · -1) = --(1 / (1 − 𝑦)))
263	257	negnegd 11562	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → --(1 / (1 − 𝑦)) = (1 / (1 − 𝑦)))
264	262, 263	eqtrd 2804	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → -((1 / (1 − 𝑦)) · -1) = (1 / (1 − 𝑦)))
265	264	mpteq2ia 5210	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -((1 / (1 − 𝑦)) · -1)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦)))
266	256, 265	eqtrdi 2820	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
267	266	dmeqd 5898	. . . . . . . . 9 ⊢ (⊤ → dom (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))))
268		dmmptg 6246	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (0(ball‘(abs ∘ − ))1)(1 / (1 − 𝑦)) ∈ V → dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))) = (0(ball‘(abs ∘ − ))1))
269		ovexd 7448	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → (1 / (1 − 𝑦)) ∈ V)
270	268, 269	mprg 3091	. . . . . . . . 9 ⊢ dom (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ (1 / (1 − 𝑦))) = (0(ball‘(abs ∘ − ))1)
271	267, 270	eqtrdi 2820	. . . . . . . 8 ⊢ (⊤ → dom (ℂ D (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))) = (0(ball‘(abs ∘ − ))1))
272		sumex 15741	. . . . . . . . . . . 12 ⊢ Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) ∈ V
273	272	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)
274		fveq2 6884	. . . . . . . . . . . . . . 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))
275	274	cbvsumv 15749	. . . . . . . . . . . . . 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)
276	180, 275	eqtrdi 2820	. . . . . . . . . . . . 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))
277	276	mpteq2ia 5210	. . . . . . . . . . . 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))
278		eqid 2769	. . . . . . . . . . . 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))
279	96, 277, 104, 118, 182, 183, 278	pserdv2 26561	. . . . . . . . . . 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)))))
280	157	ssriv 3949	. . . . . . . . . . . 12 ⊢ (0(ball‘(abs ∘ − ))1) ⊆ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑗 ∈ ℕ₀ ↦ (if(𝑗 = 0, 0, (1 / 𝑗)) · (𝑟↑𝑗)))) ∈ dom ⇝ }, ℝ^*, < )))
281	280	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^*, < ))))
282	191, 187, 273, 279, 281, 232, 230, 236	dvmptres 26093	. . . . . . . . . 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)))))
283		nnnn0 12513	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
284	283	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
285		eqeq1 2773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (𝑚 = 0 ↔ 𝑛 = 0))
286		oveq2 7421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
287	285, 286	ifbieq2d 4519	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → if(𝑚 = 0, 0, (1 / 𝑚)) = if(𝑛 = 0, 0, (1 / 𝑛)))
288		ovex 7446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 / 𝑛) ∈ V
289	89, 288	ifex 4543	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑛 = 0, 0, (1 / 𝑛)) ∈ V
290	287, 88, 289	fvmpt 6992	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)))
291	284, 290	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)))
292		nnne0 12272	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
293	292	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
294	293	neneqd 2969	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
295	294	iffalsed 4503	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, 0, (1 / 𝑛)) = (1 / 𝑛))
296	291, 295	eqtrd 2804	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛) = (1 / 𝑛))
297	296	oveq2d 7429	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) = (𝑛 · (1 / 𝑛)))
298		nncn 12243	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
299	298	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
300	299, 293	recidd 11988	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · (1 / 𝑛)) = 1)
301	297, 300	eqtrd 2804	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) = 1)
302	301	oveq1d 7428	. . . . . . . . . . . . . 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))))
303		nnm1nn0 12547	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
304		expcl 14117	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℂ ∧ (𝑛 − 1) ∈ ℕ₀) → (𝑦↑(𝑛 − 1)) ∈ ℂ)
305	50, 303, 304	syl2an 607	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (𝑦↑(𝑛 − 1)) ∈ ℂ)
306	305	mullidd 11229	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → (1 · (𝑦↑(𝑛 − 1))) = (𝑦↑(𝑛 − 1)))
307	302, 306	eqtrd 2804	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ∧ 𝑛 ∈ ℕ) → ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (𝑦↑(𝑛 − 1)))
308	307	sumeq2dv 15755	. . . . . . . . . . . 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)))
309		nnuz 12903	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ_≥‘1)
310		1e0p1 12760	. . . . . . . . . . . . . . . 16 ⊢ 1 = (0 + 1)
311	310	fveq2i 6887	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘1) = (ℤ_≥‘(0 + 1))
312	309, 311	eqtri 2792	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘(0 + 1))
313		oveq1 7420	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (1 + 𝑚) → (𝑛 − 1) = ((1 + 𝑚) − 1))
314	313	oveq2d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (1 + 𝑚) → (𝑦↑(𝑛 − 1)) = (𝑦↑((1 + 𝑚) − 1)))
315		1zzd 12627	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 1 ∈ ℤ)
316		0zd 12605	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → 0 ∈ ℤ)
317	1, 312, 314, 315, 316, 305	isumshft 15895	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)) = Σ𝑚 ∈ ℕ₀ (𝑦↑((1 + 𝑚) − 1)))
318		pncan2 11466	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((1 + 𝑚) − 1) = 𝑚)
319	47, 98, 318	sylancr 598	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ₀ → ((1 + 𝑚) − 1) = 𝑚)
320	319	oveq2d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → (𝑦↑((1 + 𝑚) − 1)) = (𝑦↑𝑚))
321	320	sumeq2i 15751	. . . . . . . . . . . . 13 ⊢ Σ𝑚 ∈ ℕ₀ (𝑦↑((1 + 𝑚) − 1)) = Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚)
322	317, 321	eqtrdi 2820	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ (𝑦↑(𝑛 − 1)) = Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚))
323		geoisum 15933	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℂ ∧ (abs‘𝑦) < 1) → Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚) = (1 / (1 − 𝑦)))
324	50, 63, 323	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑚 ∈ ℕ₀ (𝑦↑𝑚) = (1 / (1 − 𝑦)))
325	308, 322, 324	3eqtrd 2808	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) → Σ𝑛 ∈ ℕ ((𝑛 · ((𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 0, (1 / 𝑚)))‘𝑛)) · (𝑦↑(𝑛 − 1))) = (1 / (1 − 𝑦)))
326	325	mpteq2ia 5210	. . . . . . . . . 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)))
327	282, 326	eqtrdi 2820	. . . . . . . . 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))))
328	266, 327	eqtr4d 2807	. . . . . . . 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)))))
329		1rp 13022	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
330		blcntr 24541	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ⁺) → 0 ∈ (0(ball‘(abs ∘ − ))1))
331	37, 27, 329, 330	mp3an 1487	. . . . . . . . 9 ⊢ 0 ∈ (0(ball‘(abs ∘ − ))1)
332	331	a1i 11	. . . . . . . 8 ⊢ (⊤ → 0 ∈ (0(ball‘(abs ∘ − ))1))
333		oveq2 7421	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 0 → (1 − 𝑦) = (1 − 0))
334		1m0e1 12362	. . . . . . . . . . . . . . . 16 ⊢ (1 − 0) = 1
335	333, 334	eqtrdi 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 0 → (1 − 𝑦) = 1)
336	335	fveq2d 6888	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 0 → (log‘(1 − 𝑦)) = (log‘1))
337		log1 26718	. . . . . . . . . . . . . 14 ⊢ (log‘1) = 0
338	336, 337	eqtrdi 2820	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (log‘(1 − 𝑦)) = 0)
339	338	negeqd 11453	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → -(log‘(1 − 𝑦)) = -0)
340		neg0 11506	. . . . . . . . . . . 12 ⊢ -0 = 0
341	339, 340	eqtrdi 2820	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → -(log‘(1 − 𝑦)) = 0)
342		eqid 2769	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))
343	341, 342, 89	fvmpt 6992	. . . . . . . . . 10 ⊢ (0 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = 0)
344	331, 343	mp1i 14	. . . . . . . . 9 ⊢ (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘0) = 0)
345		oveq1 7420	. . . . . . . . . . . . . . 15 ⊢ (0 = if(𝑛 = 0, 0, (1 / 𝑛)) → (0 · (𝑦↑𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
346	345	eqeq1d 2771	. . . . . . . . . . . . . 14 ⊢ (0 = if(𝑛 = 0, 0, (1 / 𝑛)) → ((0 · (𝑦↑𝑛)) = 0 ↔ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0))
347		oveq1 7420	. . . . . . . . . . . . . . 15 ⊢ ((1 / 𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)) → ((1 / 𝑛) · (𝑦↑𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))
348	347	eqeq1d 2771	. . . . . . . . . . . . . 14 ⊢ ((1 / 𝑛) = if(𝑛 = 0, 0, (1 / 𝑛)) → (((1 / 𝑛) · (𝑦↑𝑛)) = 0 ↔ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0))
349		simpll 778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 𝑦 = 0)
350	349, 27	eqeltrdi 2877	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 𝑦 ∈ ℂ)
351		simplr 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → 𝑛 ∈ ℕ₀)
352	350, 351	expcld 14184	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → (𝑦↑𝑛) ∈ ℂ)
353	352	mul02d 11410	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑛 = 0) → (0 · (𝑦↑𝑛)) = 0)
354		simpll 778	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → 𝑦 = 0)
355	354	oveq1d 7428	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (𝑦↑𝑛) = (0↑𝑛))
356	13	bilani 509	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (𝑛 ∈ ℕ ∨ 𝑛 = 0))
357	356	ord 877	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 ∈ ℕ → 𝑛 = 0))
358	357	con1d 146	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (¬ 𝑛 = 0 → 𝑛 ∈ ℕ))
359	358	imp 411	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
360	359	0expd 14177	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (0↑𝑛) = 0)
361	355, 360	eqtrd 2804	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (𝑦↑𝑛) = 0)
362	361	oveq2d 7429	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · (𝑦↑𝑛)) = ((1 / 𝑛) · 0))
363	359	nnrecred 12289	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℝ)
364	363	recnd 11239	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → (1 / 𝑛) ∈ ℂ)
365	364	mul01d 11411	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · 0) = 0)
366	362, 365	eqtrd 2804	. . . . . . . . . . . . . 14 ⊢ (((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) ∧ ¬ 𝑛 = 0) → ((1 / 𝑛) · (𝑦↑𝑛)) = 0)
367	346, 348, 353, 366	ifbothda 4531	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 0 ∧ 𝑛 ∈ ℕ₀) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0)
368	367	sumeq2dv 15755	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = Σ𝑛 ∈ ℕ₀ 0)
369	1	eqimssi 4005	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ⊆ (ℤ_≥‘0)
370	369	orci 878	. . . . . . . . . . . . 13 ⊢ (ℕ₀ ⊆ (ℤ_≥‘0) ∨ ℕ₀ ∈ Fin)
371		sumz 15775	. . . . . . . . . . . . 13 ⊢ ((ℕ₀ ⊆ (ℤ_≥‘0) ∨ ℕ₀ ∈ Fin) → Σ𝑛 ∈ ℕ₀ 0 = 0)
372	370, 371	ax-mp 5	. . . . . . . . . . . 12 ⊢ Σ𝑛 ∈ ℕ₀ 0 = 0
373	368, 372	eqtrdi 2820	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = 0)
374		eqid 2769	. . . . . . . . . . 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)))
375	373, 374, 89	fvmpt 6992	. . . . . . . . . 10 ⊢ (0 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))‘0) = 0)
376	331, 375	mp1i 14	. . . . . . . . 9 ⊢ (⊤ → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)))‘0) = 0)
377	344, 376	eqtr4d 2807	. . . . . . . 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))
378	44, 45, 46, 77, 189, 271, 328, 332, 377	dv11cn 26131	. . . . . . 7 ⊢ (⊤ → (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦))) = (𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛))))
379	378	fveq1d 6886	. . . . . 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))
380	43, 379	mp1i 14	. . . . 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))
381		oveq2 7421	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (1 − 𝑦) = (1 − 𝐴))
382	381	fveq2d 6888	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (log‘(1 − 𝑦)) = (log‘(1 − 𝐴)))
383	382	negeqd 11453	. . . . . 6 ⊢ (𝑦 = 𝐴 → -(log‘(1 − 𝑦)) = -(log‘(1 − 𝐴)))
384		negex 11457	. . . . . 6 ⊢ -(log‘(1 − 𝐴)) ∈ V
385	383, 342, 384	fvmpt 6992	. . . . 5 ⊢ (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → ((𝑦 ∈ (0(ball‘(abs ∘ − ))1) ↦ -(log‘(1 − 𝑦)))‘𝐴) = -(log‘(1 − 𝐴)))
386		oveq1 7420	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦↑𝑛) = (𝐴↑𝑛))
387	386	oveq2d 7429	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
388	387	sumeq2sdv 15756	. . . . . 6 ⊢ (𝑦 = 𝐴 → Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝑦↑𝑛)) = Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
389		sumex 15741	. . . . . 6 ⊢ Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) ∈ V
390	388, 374, 389	fvmpt 6992	. . . . 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)))
391	380, 385, 390	3eqtr3d 2812	. . . 4 ⊢ (𝐴 ∈ (0(ball‘(abs ∘ − ))1) → -(log‘(1 − 𝐴)) = Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
392	42, 391	syl 18	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(log‘(1 − 𝐴)) = Σ𝑛 ∈ ℕ₀ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
393	25, 392	breqtrrd 5143	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ⇝ -(log‘(1 − 𝐴)))
394		seqex 14041	. . . 4 ⊢ seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ∈ V
395	394	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) ∈ V)
396		seqex 14041	. . . 4 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))) ∈ V
397	396	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))) ∈ V)
398		1zzd 12627	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℤ)
399		elnnuz 12904	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
400		fvres 6903	. . . . . 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))
401	399, 400	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))
402	401	eqcomd 2775	. . . 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))
403		addlid 11395	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
404	403	adantl 486	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
405		0cnd 11201	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ∈ ℂ)
406		1eluzge0 12906	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
407	406	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ (ℤ_≥‘0))
408		0cnd 11201	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 = 0) → 0 ∈ ℂ)
409		nn0cn 12516	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
410	409	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
411		neqne 2972	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
412		reccl 11881	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℂ ∧ 𝑘 ≠ 0) → (1 / 𝑘) ∈ ℂ)
413	410, 411, 412	syl2an 607	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 = 0) → (1 / 𝑘) ∈ ℂ)
414	408, 413	ifclda 4528	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → if(𝑘 = 0, 0, (1 / 𝑘)) ∈ ℂ)
415		expcl 14117	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℂ)
416	415	adantlr 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℂ)
417	414, 416	mulcld 11231	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ₀) → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) ∈ ℂ)
418	417	fmpttd 7113	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘))):ℕ₀⟶ℂ)
419		1nn0 12522	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
420		ffvelcdm 7079	. . . . . . . 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)
421	418, 419, 420	sylancl 597	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘1) ∈ ℂ)
422		elfz1eq 13565	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...0) → 𝑛 = 0)
423		1m1e0 12315	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
424	423	oveq2i 7424	. . . . . . . . . 10 ⊢ (0...(1 − 1)) = (0...0)
425	422, 424	eleq2s 2887	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(1 − 1)) → 𝑛 = 0)
426	425	fveq2d 6888	. . . . . . . 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))
427		0nn0 12521	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
428		iftrue 4498	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → if(𝑘 = 0, 0, (1 / 𝑘)) = 0)
429		oveq2 7421	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝐴↑𝑘) = (𝐴↑0))
430	428, 429	oveq12d 7431	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) = (0 · (𝐴↑0)))
431		ovex 7446	. . . . . . . . . . 11 ⊢ (0 · (𝐴↑0)) ∈ V
432	430, 8, 431	fvmpt 6992	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘0) = (0 · (𝐴↑0)))
433	427, 432	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘0) = (0 · (𝐴↑0))
434		expcl 14117	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℕ₀) → (𝐴↑0) ∈ ℂ)
435	26, 427, 434	sylancl 597	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑0) ∈ ℂ)
436	435	mul02d 11410	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (0 · (𝐴↑0)) = 0)
437	433, 436	eqtrid 2816	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘0) = 0)
438	426, 437	sylan9eqr 2826	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ (0...(1 − 1))) → ((𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))‘𝑛) = 0)
439	404, 405, 407, 421, 438	seqid 14085	. . . . . 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)))))
440	292	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
441	440	neneqd 2969	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
442	441	iffalsed 4503	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, 0, (1 / 𝑛)) = (1 / 𝑛))
443	442	oveq1d 7428	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) = ((1 / 𝑛) · (𝐴↑𝑛)))
444	283, 22	sylan2 604	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ∈ ℂ)
445	298	adantl 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
446	444, 445, 440	divrec2d 11997	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝐴↑𝑛) / 𝑛) = ((1 / 𝑛) · (𝐴↑𝑛)))
447	443, 446	eqtr4d 2807	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) = ((𝐴↑𝑛) / 𝑛))
448	283, 11	sylan2 604	. . . . . . . . 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)))
449		id 23	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
450	6, 449	oveq12d 7431	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝐴↑𝑘) / 𝑘) = ((𝐴↑𝑛) / 𝑛))
451		eqid 2769	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘)) = (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))
452		ovex 7446	. . . . . . . . . . 11 ⊢ ((𝐴↑𝑛) / 𝑛) ∈ V
453	450, 451, 452	fvmpt 6992	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))‘𝑛) = ((𝐴↑𝑛) / 𝑛))
454	453	adantl 486	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))‘𝑛) = ((𝐴↑𝑛) / 𝑛))
455	447, 448, 454	3eqtr4d 2814	. . . . . . . 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))
456	399, 455	sylan2br 606	. . . . . . 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))
457	398, 456	seqfeq 14065	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ₀ ↦ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴↑𝑘) / 𝑘))))
458	439, 457	eqtrd 2804	. . . . 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))))
459	458	fveq1d 6886	. . . 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))
460	402, 459	sylan9eqr 2826	. . 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))
461	309, 395, 397, 398, 460	climeq 15620	. 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))))
462	393, 461	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 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ⊤wtru 1568 ∈ wcel 2149 ≠ wne 2964 {crab 3423 Vcvv 3463 ∖ cdif 3910 ⊆ wss 3913 ifcif 4492 {csn 4594 {cpr 4596 class class class wbr 5113 ↦ cmpt 5196 ^◡ccnv 5663 dom cdm 5664 ran crn 5665 ↾ cres 5666 “ cima 5667 ∘ ccom 5668 Fn wfn 6534 ⟶wf 6535 –1-1-onto→wf1o 6538 ‘cfv 6539 (class class class)co 7413 Fincfn 8945 supcsup 9402 ℂcc 11100 ℝcr 11101 0cc0 11102 1c1 11103 + caddc 11105 · cmul 11107 +∞cpnf 11242 -∞cmnf 11243 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 − cmin 11443 -cneg 11444 / cdiv 11873 ℕcn 12235 2c2 12297 ℕ₀cn0 12506 ℤ_≥cuz 12864 ℝ⁺crp 13018 (,]cioc 13375 [,)cico 13376 [,]cicc 13377 ...cfz 13537 seqcseq 14039 ↑cexp 14099 abscabs 15287 ⇝ cli 15537 Σcsu 15739 TopOpenctopn 17476 ∞Metcxmet 21478 ballcbl 21480 ℂ_fldccnfld 21493 –cn→ccncf 25006 D cdv 25993 logclog 26687

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5273 ax-pow 5339 ax-pr 5407 ax-un 7735 ax-inf2 9612 ax-cnex 11158 ax-resscn 11159 ax-1cn 11160 ax-icn 11161 ax-addcl 11162 ax-addrcl 11163 ax-mulcl 11164 ax-mulrcl 11165 ax-mulcom 11166 ax-addass 11167 ax-mulass 11168 ax-distr 11169 ax-i2m1 11170 ax-1ne0 11171 ax-1rid 11172 ax-rnegex 11173 ax-rrecex 11174 ax-cnre 11175 ax-pre-lttri 11176 ax-pre-lttrn 11177 ax-pre-ltadd 11178 ax-pre-mulgt0 11179 ax-pre-sup 11180 ax-addf 11181

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7677 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9324 df-fi 9373 df-sup 9404 df-inf 9405 df-oi 9474 df-card 9927 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11445 df-neg 11446 df-div 11874 df-nn 12236 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12865 df-q 12975 df-rp 13019 df-xneg 13139 df-xadd 13140 df-xmul 13141 df-ioo 13378 df-ioc 13379 df-ico 13380 df-icc 13381 df-fz 13538 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-shft 15106 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-sum 15740 df-ef 16123 df-sin 16125 df-cos 16126 df-tan 16127 df-pi 16128 df-struct 17209 df-sets 17226 df-slot 17244 df-ndx 17256 df-base 17272 df-ress 17293 df-plusg 17325 df-mulr 17326 df-starv 17327 df-sca 17328 df-vsca 17329 df-ip 17330 df-tset 17331 df-ple 17332 df-ds 17334 df-unif 17335 df-hom 17336 df-cco 17337 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-pt 17499 df-prds 17502 df-xrs 17558 df-qtop 17563 df-imas 17564 df-xps 17566 df-mre 17640 df-mrc 17641 df-acs 17643 df-mgm 18700 df-sgrp 18779 df-mnd 18795 df-submnd 18844 df-mulg 19136 df-cntz 19389 df-cmn 19854 df-psmet 21485 df-xmet 21486 df-met 21487 df-bl 21488 df-mopn 21489 df-fbas 21490 df-fg 21491 df-cnfld 21494 df-top 23022 df-topon 23039 df-topsp 23061 df-bases 23074 df-cld 23147 df-ntr 23148 df-cls 23149 df-nei 23226 df-lp 23264 df-perf 23265 df-cn 23355 df-cnp 23356 df-haus 23443 df-cmp 23515 df-tx 23690 df-hmeo 23883 df-fil 23974 df-fm 24066 df-flim 24067 df-flf 24068 df-xms 24448 df-ms 24449 df-tms 24450 df-cncf 25008 df-limc 25996 df-dv 25997 df-ulm 26508 df-log 26689

This theorem is referenced by: logtaylsum 26794 logtayl2 26795 atantayl 27070 stirlinglem5 46721

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