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Theorem logtayl2 26785

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 12892	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12616	. . 3 ⊢ (𝐴 ∈ 𝑆 → 1 ∈ ℤ)
3		neg1cn 12194	. . . 4 ⊢ -1 ∈ ℂ
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑆 → -1 ∈ ℂ)
5		ax-1cn 11146	. . . . . 6 ⊢ 1 ∈ ℂ
6		logtayl2.s	. . . . . . . . 9 ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1)
7	6	eleq2i 2857	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ (1(ball‘(abs ∘ − ))1))
8		cnxmet 24890	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
9		1xr 11256	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
10		elbl 24506	. . . . . . . . 9 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ^*) → (𝐴 ∈ (1(ball‘(abs ∘ − ))1) ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1)))
11	8, 5, 9, 10	mp3an 1485	. . . . . . . 8 ⊢ (𝐴 ∈ (1(ball‘(abs ∘ − ))1) ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1))
12	7, 11	bitri 278	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1))
13	12	simplbi 501	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ ℂ)
14		subcl 11444	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
15	5, 13, 14	sylancr 598	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (1 − 𝐴) ∈ ℂ)
16		eqid 2765	. . . . . . . 8 ⊢ (abs ∘ − ) = (abs ∘ − )
17	16	cnmetdval 24888	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1(abs ∘ − )𝐴) = (abs‘(1 − 𝐴)))
18	5, 13, 17	sylancr 598	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1(abs ∘ − )𝐴) = (abs‘(1 − 𝐴)))
19	12	simprbi 502	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1(abs ∘ − )𝐴) < 1)
20	18, 19	eqbrtrrd 5129	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (abs‘(1 − 𝐴)) < 1)
21		logtayl 26783	. . . . 5 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (abs‘(1 − 𝐴)) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘(1 − (1 − 𝐴))))
22	15, 20, 21	syl2anc 595	. . . 4 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘(1 − (1 − 𝐴))))
23		nncan 11475	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − (1 − 𝐴)) = 𝐴)
24	5, 13, 23	sylancr 598	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1 − (1 − 𝐴)) = 𝐴)
25	24	fveq2d 6875	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (log‘(1 − (1 − 𝐴))) = (log‘𝐴))
26	25	negeqd 11439	. . . 4 ⊢ (𝐴 ∈ 𝑆 → -(log‘(1 − (1 − 𝐴))) = -(log‘𝐴))
27	22, 26	breqtrd 5131	. . 3 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘𝐴))
28		oveq2 7408	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((1 − 𝐴)↑𝑘) = ((1 − 𝐴)↑𝑛))
29		id 23	. . . . . . 7 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
30	28, 29	oveq12d 7418	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((1 − 𝐴)↑𝑘) / 𝑘) = (((1 − 𝐴)↑𝑛) / 𝑛))
31		eqid 2765	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘)) = (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))
32		ovex 7433	. . . . . 6 ⊢ (((1 − 𝐴)↑𝑛) / 𝑛) ∈ V
33	30, 31, 32	fvmpt 6979	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) = (((1 − 𝐴)↑𝑛) / 𝑛))
34	33	adantl 486	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) = (((1 − 𝐴)↑𝑛) / 𝑛))
35		nnnn0 12502	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
36		expcl 14106	. . . . . 6 ⊢ (((1 − 𝐴) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((1 − 𝐴)↑𝑛) ∈ ℂ)
37	15, 35, 36	syl2an 607	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) ∈ ℂ)
38		nncn 12232	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
39	38	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
40		nnne0 12261	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
41	40	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
42	37, 39, 41	divcld 11982	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((1 − 𝐴)↑𝑛) / 𝑛) ∈ ℂ)
43	34, 42	eqeltrd 2865	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) ∈ ℂ)
44	37, 39, 41	divnegd 11995	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -(((1 − 𝐴)↑𝑛) / 𝑛) = (-((1 − 𝐴)↑𝑛) / 𝑛))
45	42	mulm1d 11654	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)) = -(((1 − 𝐴)↑𝑛) / 𝑛))
46	35	adantl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
47		expcl 14106	. . . . . . . . . 10 ⊢ ((-1 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (-1↑𝑛) ∈ ℂ)
48	3, 46, 47	sylancr 598	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑𝑛) ∈ ℂ)
49		subcl 11444	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
50	13, 5, 49	sylancl 597	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → (𝐴 − 1) ∈ ℂ)
51		expcl 14106	. . . . . . . . . 10 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((𝐴 − 1)↑𝑛) ∈ ℂ)
52	50, 35, 51	syl2an 607	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝐴 − 1)↑𝑛) ∈ ℂ)
53	48, 52	mulneg1d 11655	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-(-1↑𝑛) · ((𝐴 − 1)↑𝑛)) = -((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
54	3	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -1 ∈ ℂ)
55		neg1ne0 12196	. . . . . . . . . . . 12 ⊢ -1 ≠ 0
56	55	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -1 ≠ 0)
57		nnz 12603	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
58	57	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
59	54, 56, 58	expm1d 14183	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) = ((-1↑𝑛) / -1))
60	5	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
61		ax-1ne0 11157	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
62	61	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 1 ≠ 0)
63	48, 60, 62	divneg2d 11996	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((-1↑𝑛) / 1) = ((-1↑𝑛) / -1))
64	48	div1d 11974	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑𝑛) / 1) = (-1↑𝑛))
65	64	negeqd 11439	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((-1↑𝑛) / 1) = -(-1↑𝑛))
66	59, 63, 65	3eqtr2d 2806	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) = -(-1↑𝑛))
67	66	oveq1d 7415	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) = (-(-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
68	50	mulm1d 11654	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑆 → (-1 · (𝐴 − 1)) = -(𝐴 − 1))
69		negsubdi2 11505	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
70	13, 5, 69	sylancl 597	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑆 → -(𝐴 − 1) = (1 − 𝐴))
71	68, 70	eqtr2d 2801	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑆 → (1 − 𝐴) = (-1 · (𝐴 − 1)))
72	71	oveq1d 7415	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑆 → ((1 − 𝐴)↑𝑛) = ((-1 · (𝐴 − 1))↑𝑛))
73	72	adantr 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) = ((-1 · (𝐴 − 1))↑𝑛))
74		mulexp 14128	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (𝐴 − 1) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
75	3, 50, 35, 74	mp3an3an 1491	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
76	73, 75	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
77	76	negeqd 11439	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((1 − 𝐴)↑𝑛) = -((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
78	53, 67, 77	3eqtr4d 2810	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) = -((1 − 𝐴)↑𝑛))
79	78	oveq1d 7415	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛) = (-((1 − 𝐴)↑𝑛) / 𝑛))
80	44, 45, 79	3eqtr4d 2810	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)) = (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛))
81		nnm1nn0 12536	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
82	81	adantl 486	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ₀)
83		expcl 14106	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ (𝑛 − 1) ∈ ℕ₀) → (-1↑(𝑛 − 1)) ∈ ℂ)
84	3, 82, 83	sylancr 598	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) ∈ ℂ)
85	84, 52, 39, 41	div23d 12019	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
86	80, 85	eqtr2d 2801	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)) = (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)))
87		oveq1 7407	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑘 − 1) = (𝑛 − 1))
88	87	oveq2d 7416	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (-1↑(𝑘 − 1)) = (-1↑(𝑛 − 1)))
89	88, 29	oveq12d 7418	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((-1↑(𝑘 − 1)) / 𝑘) = ((-1↑(𝑛 − 1)) / 𝑛))
90		oveq2 7408	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐴 − 1)↑𝑘) = ((𝐴 − 1)↑𝑛))
91	89, 90	oveq12d 7418	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
92		eqid 2765	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘))) = (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))
93		ovex 7433	. . . . . 6 ⊢ (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)) ∈ V
94	91, 92, 93	fvmpt 6979	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))‘𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
95	94	adantl 486	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))‘𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
96	34	oveq2d 7416	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛)) = (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)))
97	86, 95, 96	3eqtr4d 2810	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))‘𝑛) = (-1 · ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛)))
98	1, 2, 4, 27, 43, 97	isermulc2 15699	. 2 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))) ⇝ (-1 · -(log‘𝐴)))
99	6	dvlog2lem 26775	. . . . . . . 8 ⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0))
100	99	sseli 3935	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ (ℂ ∖ (-∞(,]0)))
101		eqid 2765	. . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
102	101	logdmn0 26763	. . . . . . 7 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ≠ 0)
103	100, 102	syl 18	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ≠ 0)
104	13, 103	logcld 26693	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (log‘𝐴) ∈ ℂ)
105	104	negcld 11544	. . . 4 ⊢ (𝐴 ∈ 𝑆 → -(log‘𝐴) ∈ ℂ)
106	105	mulm1d 11654	. . 3 ⊢ (𝐴 ∈ 𝑆 → (-1 · -(log‘𝐴)) = --(log‘𝐴))
107	104	negnegd 11548	. . 3 ⊢ (𝐴 ∈ 𝑆 → --(log‘𝐴) = (log‘𝐴))
108	106, 107	eqtrd 2800	. 2 ⊢ (𝐴 ∈ 𝑆 → (-1 · -(log‘𝐴)) = (log‘𝐴))
109	98, 108	breqtrd 5131	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 209 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∖ cdif 3904 class class class wbr 5105 ↦ cmpt 5186 ∘ ccom 5656 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 -∞cmnf 11229 ℝ^*cxr 11230 < clt 11231 − cmin 11429 -cneg 11430 / cdiv 11859 ℕcn 12224 ℕ₀cn0 12495 ℤcz 12582 (,]cioc 13364 seqcseq 14028 ↑cexp 14088 abscabs 15275 ⇝ cli 15525 ∞Metcxmet 21467 ballcbl 21469 logclog 26677

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-fac 14301 df-bc 14330 df-hash 14358 df-shft 15094 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-limsup 15512 df-clim 15529 df-rlim 15530 df-sum 15728 df-ef 16111 df-sin 16113 df-cos 16114 df-tan 16115 df-pi 16116 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-mulg 19125 df-cntz 19378 df-cmn 19843 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-fbas 21479 df-fg 21480 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-nei 23216 df-lp 23254 df-perf 23255 df-cn 23345 df-cnp 23346 df-haus 23433 df-cmp 23505 df-tx 23680 df-hmeo 23873 df-fil 23964 df-fm 24056 df-flim 24057 df-flf 24058 df-xms 24438 df-ms 24439 df-tms 24440 df-cncf 24998 df-limc 25986 df-dv 25987 df-ulm 26498 df-log 26679

This theorem is referenced by: stirlinglem5 46650

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