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Theorem logtayl2 24408

Description: Power series expression for the logarithm. (Contributed by Mario Carneiro, 31-Mar-2015.)

**Hypothesis**
Ref	Expression
logtayl2.s	⊢ 𝑆 = (1(ball‘(abs ∘ − ))1)

**Assertion**
Ref	Expression
logtayl2	⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))) ⇝ (log‘𝐴))

Distinct variable group: 𝐴,𝑘 Allowed substitution hint: 𝑆(𝑘)

Proof of Theorem logtayl2 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 11723	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 11408	. . 3 ⊢ (𝐴 ∈ 𝑆 → 1 ∈ ℤ)
3		neg1cn 11124	. . . 4 ⊢ -1 ∈ ℂ
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑆 → -1 ∈ ℂ)
5		ax-1cn 9994	. . . . . 6 ⊢ 1 ∈ ℂ
6		logtayl2.s	. . . . . . . . 9 ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1)
7	6	eleq2i 2693	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ (1(ball‘(abs ∘ − ))1))
8		cnxmet 22576	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
9		1rp 11836	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
10		rpxr 11840	. . . . . . . . . 10 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
11	9, 10	ax-mp 5	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
12		elbl 22193	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ^*) → (𝐴 ∈ (1(ball‘(abs ∘ − ))1) ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1)))
13	8, 5, 11, 12	mp3an 1424	. . . . . . . 8 ⊢ (𝐴 ∈ (1(ball‘(abs ∘ − ))1) ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1))
14	7, 13	bitri 264	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1))
15	14	simplbi 476	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ ℂ)
16		subcl 10280	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
17	5, 15, 16	sylancr 695	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (1 − 𝐴) ∈ ℂ)
18		eqid 2622	. . . . . . . 8 ⊢ (abs ∘ − ) = (abs ∘ − )
19	18	cnmetdval 22574	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1(abs ∘ − )𝐴) = (abs‘(1 − 𝐴)))
20	5, 15, 19	sylancr 695	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1(abs ∘ − )𝐴) = (abs‘(1 − 𝐴)))
21	14	simprbi 480	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1(abs ∘ − )𝐴) < 1)
22	20, 21	eqbrtrrd 4677	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (abs‘(1 − 𝐴)) < 1)
23		logtayl 24406	. . . . 5 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (abs‘(1 − 𝐴)) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘(1 − (1 − 𝐴))))
24	17, 22, 23	syl2anc 693	. . . 4 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘(1 − (1 − 𝐴))))
25		nncan 10310	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − (1 − 𝐴)) = 𝐴)
26	5, 15, 25	sylancr 695	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1 − (1 − 𝐴)) = 𝐴)
27	26	fveq2d 6195	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (log‘(1 − (1 − 𝐴))) = (log‘𝐴))
28	27	negeqd 10275	. . . 4 ⊢ (𝐴 ∈ 𝑆 → -(log‘(1 − (1 − 𝐴))) = -(log‘𝐴))
29	24, 28	breqtrd 4679	. . 3 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘𝐴))
30		oveq2 6658	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((1 − 𝐴)↑𝑘) = ((1 − 𝐴)↑𝑛))
31		id 22	. . . . . . 7 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
32	30, 31	oveq12d 6668	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((1 − 𝐴)↑𝑘) / 𝑘) = (((1 − 𝐴)↑𝑛) / 𝑛))
33		eqid 2622	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘)) = (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))
34		ovex 6678	. . . . . 6 ⊢ (((1 − 𝐴)↑𝑛) / 𝑛) ∈ V
35	32, 33, 34	fvmpt 6282	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) = (((1 − 𝐴)↑𝑛) / 𝑛))
36	35	adantl 482	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) = (((1 − 𝐴)↑𝑛) / 𝑛))
37		nnnn0 11299	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
38		expcl 12878	. . . . . 6 ⊢ (((1 − 𝐴) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((1 − 𝐴)↑𝑛) ∈ ℂ)
39	17, 37, 38	syl2an 494	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) ∈ ℂ)
40		nncn 11028	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
41	40	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
42		nnne0 11053	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
43	42	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
44	39, 41, 43	divcld 10801	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((1 − 𝐴)↑𝑛) / 𝑛) ∈ ℂ)
45	36, 44	eqeltrd 2701	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) ∈ ℂ)
46	39, 41, 43	divnegd 10814	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -(((1 − 𝐴)↑𝑛) / 𝑛) = (-((1 − 𝐴)↑𝑛) / 𝑛))
47	44	mulm1d 10482	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)) = -(((1 − 𝐴)↑𝑛) / 𝑛))
48	37	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
49		expcl 12878	. . . . . . . . . 10 ⊢ ((-1 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (-1↑𝑛) ∈ ℂ)
50	3, 48, 49	sylancr 695	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑𝑛) ∈ ℂ)
51		subcl 10280	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
52	15, 5, 51	sylancl 694	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → (𝐴 − 1) ∈ ℂ)
53		expcl 12878	. . . . . . . . . 10 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((𝐴 − 1)↑𝑛) ∈ ℂ)
54	52, 37, 53	syl2an 494	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝐴 − 1)↑𝑛) ∈ ℂ)
55	50, 54	mulneg1d 10483	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-(-1↑𝑛) · ((𝐴 − 1)↑𝑛)) = -((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
56	3	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -1 ∈ ℂ)
57		neg1ne0 11126	. . . . . . . . . . . 12 ⊢ -1 ≠ 0
58	57	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -1 ≠ 0)
59		nnz 11399	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
60	59	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
61	56, 58, 60	expm1d 13018	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) = ((-1↑𝑛) / -1))
62	5	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
63		ax-1ne0 10005	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
64	63	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 1 ≠ 0)
65	50, 62, 64	divneg2d 10815	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((-1↑𝑛) / 1) = ((-1↑𝑛) / -1))
66	50	div1d 10793	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑𝑛) / 1) = (-1↑𝑛))
67	66	negeqd 10275	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((-1↑𝑛) / 1) = -(-1↑𝑛))
68	61, 65, 67	3eqtr2d 2662	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) = -(-1↑𝑛))
69	68	oveq1d 6665	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) = (-(-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
70	52	mulm1d 10482	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑆 → (-1 · (𝐴 − 1)) = -(𝐴 − 1))
71		negsubdi2 10340	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
72	15, 5, 71	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑆 → -(𝐴 − 1) = (1 − 𝐴))
73	70, 72	eqtr2d 2657	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑆 → (1 − 𝐴) = (-1 · (𝐴 − 1)))
74	73	oveq1d 6665	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑆 → ((1 − 𝐴)↑𝑛) = ((-1 · (𝐴 − 1))↑𝑛))
75	74	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) = ((-1 · (𝐴 − 1))↑𝑛))
76		mulexp 12899	. . . . . . . . . . . 12 ⊢ ((-1 ∈ ℂ ∧ (𝐴 − 1) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
77	3, 76	mp3an1 1411	. . . . . . . . . . 11 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
78	52, 37, 77	syl2an 494	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
79	75, 78	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
80	79	negeqd 10275	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((1 − 𝐴)↑𝑛) = -((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
81	55, 69, 80	3eqtr4d 2666	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) = -((1 − 𝐴)↑𝑛))
82	81	oveq1d 6665	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛) = (-((1 − 𝐴)↑𝑛) / 𝑛))
83	46, 47, 82	3eqtr4d 2666	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)) = (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛))
84		nnm1nn0 11334	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
85	84	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ₀)
86		expcl 12878	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ (𝑛 − 1) ∈ ℕ₀) → (-1↑(𝑛 − 1)) ∈ ℂ)
87	3, 85, 86	sylancr 695	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) ∈ ℂ)
88	87, 54, 41, 43	div23d 10838	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
89	83, 88	eqtr2d 2657	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)) = (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)))
90		oveq1 6657	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑘 − 1) = (𝑛 − 1))
91	90	oveq2d 6666	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (-1↑(𝑘 − 1)) = (-1↑(𝑛 − 1)))
92	91, 31	oveq12d 6668	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((-1↑(𝑘 − 1)) / 𝑘) = ((-1↑(𝑛 − 1)) / 𝑛))
93		oveq2 6658	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐴 − 1)↑𝑘) = ((𝐴 − 1)↑𝑛))
94	92, 93	oveq12d 6668	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
95		eqid 2622	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘))) = (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))
96		ovex 6678	. . . . . 6 ⊢ (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)) ∈ V
97	94, 95, 96	fvmpt 6282	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))‘𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
98	97	adantl 482	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))‘𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
99	36	oveq2d 6666	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛)) = (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)))
100	89, 98, 99	3eqtr4d 2666	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))‘𝑛) = (-1 · ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛)))
101	1, 2, 4, 29, 45, 100	isermulc2 14388	. 2 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))) ⇝ (-1 · -(log‘𝐴)))
102	6	dvlog2lem 24398	. . . . . . . 8 ⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0))
103	102	sseli 3599	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ (ℂ ∖ (-∞(,]0)))
104		eqid 2622	. . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
105	104	logdmn0 24386	. . . . . . 7 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ≠ 0)
106	103, 105	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ≠ 0)
107	15, 106	logcld 24317	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (log‘𝐴) ∈ ℂ)
108	107	negcld 10379	. . . 4 ⊢ (𝐴 ∈ 𝑆 → -(log‘𝐴) ∈ ℂ)
109	108	mulm1d 10482	. . 3 ⊢ (𝐴 ∈ 𝑆 → (-1 · -(log‘𝐴)) = --(log‘𝐴))
110	107	negnegd 10383	. . 3 ⊢ (𝐴 ∈ 𝑆 → --(log‘𝐴) = (log‘𝐴))
111	109, 110	eqtrd 2656	. 2 ⊢ (𝐴 ∈ 𝑆 → (-1 · -(log‘𝐴)) = (log‘𝐴))
112	101, 111	breqtrd 4679	1 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))) ⇝ (log‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 class class class wbr 4653 ↦ cmpt 4729 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 -∞cmnf 10072 ℝ^*cxr 10073 < clt 10074 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℝ⁺crp 11832 (,]cioc 12176 seqcseq 12801 ↑cexp 12860 abscabs 13974 ⇝ cli 14215 ∞Metcxmt 19731 ballcbl 19733 logclog 24301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-ulm 24131 df-log 24303

This theorem is referenced by: stirlinglem5 40295

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