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Theorem logtayl2 26162

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 12862	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12590	. . 3 ⊢ (𝐴 ∈ 𝑆 → 1 ∈ ℤ)
3		neg1cn 12323	. . . 4 ⊢ -1 ∈ ℂ
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑆 → -1 ∈ ℂ)
5		ax-1cn 11165	. . . . . 6 ⊢ 1 ∈ ℂ
6		logtayl2.s	. . . . . . . . 9 ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1)
7	6	eleq2i 2826	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ (1(ball‘(abs ∘ − ))1))
8		cnxmet 24281	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
9		1xr 11270	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
10		elbl 23886	. . . . . . . . 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 1462	. . . . . . . 8 ⊢ (𝐴 ∈ (1(ball‘(abs ∘ − ))1) ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1))
12	7, 11	bitri 275	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1))
13	12	simplbi 499	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ ℂ)
14		subcl 11456	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
15	5, 13, 14	sylancr 588	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (1 − 𝐴) ∈ ℂ)
16		eqid 2733	. . . . . . . 8 ⊢ (abs ∘ − ) = (abs ∘ − )
17	16	cnmetdval 24279	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1(abs ∘ − )𝐴) = (abs‘(1 − 𝐴)))
18	5, 13, 17	sylancr 588	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1(abs ∘ − )𝐴) = (abs‘(1 − 𝐴)))
19	12	simprbi 498	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1(abs ∘ − )𝐴) < 1)
20	18, 19	eqbrtrrd 5172	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (abs‘(1 − 𝐴)) < 1)
21		logtayl 26160	. . . . 5 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (abs‘(1 − 𝐴)) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘(1 − (1 − 𝐴))))
22	15, 20, 21	syl2anc 585	. . . 4 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘(1 − (1 − 𝐴))))
23		nncan 11486	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − (1 − 𝐴)) = 𝐴)
24	5, 13, 23	sylancr 588	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1 − (1 − 𝐴)) = 𝐴)
25	24	fveq2d 6893	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (log‘(1 − (1 − 𝐴))) = (log‘𝐴))
26	25	negeqd 11451	. . . 4 ⊢ (𝐴 ∈ 𝑆 → -(log‘(1 − (1 − 𝐴))) = -(log‘𝐴))
27	22, 26	breqtrd 5174	. . 3 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘𝐴))
28		oveq2 7414	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((1 − 𝐴)↑𝑘) = ((1 − 𝐴)↑𝑛))
29		id 22	. . . . . . 7 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
30	28, 29	oveq12d 7424	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((1 − 𝐴)↑𝑘) / 𝑘) = (((1 − 𝐴)↑𝑛) / 𝑛))
31		eqid 2733	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘)) = (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))
32		ovex 7439	. . . . . 6 ⊢ (((1 − 𝐴)↑𝑛) / 𝑛) ∈ V
33	30, 31, 32	fvmpt 6996	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) = (((1 − 𝐴)↑𝑛) / 𝑛))
34	33	adantl 483	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) = (((1 − 𝐴)↑𝑛) / 𝑛))
35		nnnn0 12476	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
36		expcl 14042	. . . . . 6 ⊢ (((1 − 𝐴) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((1 − 𝐴)↑𝑛) ∈ ℂ)
37	15, 35, 36	syl2an 597	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) ∈ ℂ)
38		nncn 12217	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
39	38	adantl 483	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
40		nnne0 12243	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
41	40	adantl 483	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
42	37, 39, 41	divcld 11987	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((1 − 𝐴)↑𝑛) / 𝑛) ∈ ℂ)
43	34, 42	eqeltrd 2834	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) ∈ ℂ)
44	37, 39, 41	divnegd 12000	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -(((1 − 𝐴)↑𝑛) / 𝑛) = (-((1 − 𝐴)↑𝑛) / 𝑛))
45	42	mulm1d 11663	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)) = -(((1 − 𝐴)↑𝑛) / 𝑛))
46	35	adantl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
47		expcl 14042	. . . . . . . . . 10 ⊢ ((-1 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (-1↑𝑛) ∈ ℂ)
48	3, 46, 47	sylancr 588	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑𝑛) ∈ ℂ)
49		subcl 11456	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
50	13, 5, 49	sylancl 587	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → (𝐴 − 1) ∈ ℂ)
51		expcl 14042	. . . . . . . . . 10 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((𝐴 − 1)↑𝑛) ∈ ℂ)
52	50, 35, 51	syl2an 597	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝐴 − 1)↑𝑛) ∈ ℂ)
53	48, 52	mulneg1d 11664	. . . . . . . 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 12325	. . . . . . . . . . . 12 ⊢ -1 ≠ 0
56	55	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -1 ≠ 0)
57		nnz 12576	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
58	57	adantl 483	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
59	54, 56, 58	expm1d 14118	. . . . . . . . . 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 11176	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
62	61	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 1 ≠ 0)
63	48, 60, 62	divneg2d 12001	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((-1↑𝑛) / 1) = ((-1↑𝑛) / -1))
64	48	div1d 11979	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑𝑛) / 1) = (-1↑𝑛))
65	64	negeqd 11451	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((-1↑𝑛) / 1) = -(-1↑𝑛))
66	59, 63, 65	3eqtr2d 2779	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) = -(-1↑𝑛))
67	66	oveq1d 7421	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) = (-(-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
68	50	mulm1d 11663	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑆 → (-1 · (𝐴 − 1)) = -(𝐴 − 1))
69		negsubdi2 11516	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
70	13, 5, 69	sylancl 587	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑆 → -(𝐴 − 1) = (1 − 𝐴))
71	68, 70	eqtr2d 2774	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑆 → (1 − 𝐴) = (-1 · (𝐴 − 1)))
72	71	oveq1d 7421	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑆 → ((1 − 𝐴)↑𝑛) = ((-1 · (𝐴 − 1))↑𝑛))
73	72	adantr 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) = ((-1 · (𝐴 − 1))↑𝑛))
74		mulexp 14064	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (𝐴 − 1) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
75	3, 50, 35, 74	mp3an3an 1468	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
76	73, 75	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
77	76	negeqd 11451	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((1 − 𝐴)↑𝑛) = -((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
78	53, 67, 77	3eqtr4d 2783	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) = -((1 − 𝐴)↑𝑛))
79	78	oveq1d 7421	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛) = (-((1 − 𝐴)↑𝑛) / 𝑛))
80	44, 45, 79	3eqtr4d 2783	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)) = (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛))
81		nnm1nn0 12510	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
82	81	adantl 483	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ₀)
83		expcl 14042	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ (𝑛 − 1) ∈ ℕ₀) → (-1↑(𝑛 − 1)) ∈ ℂ)
84	3, 82, 83	sylancr 588	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) ∈ ℂ)
85	84, 52, 39, 41	div23d 12024	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
86	80, 85	eqtr2d 2774	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)) = (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)))
87		oveq1 7413	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑘 − 1) = (𝑛 − 1))
88	87	oveq2d 7422	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (-1↑(𝑘 − 1)) = (-1↑(𝑛 − 1)))
89	88, 29	oveq12d 7424	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((-1↑(𝑘 − 1)) / 𝑘) = ((-1↑(𝑛 − 1)) / 𝑛))
90		oveq2 7414	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐴 − 1)↑𝑘) = ((𝐴 − 1)↑𝑛))
91	89, 90	oveq12d 7424	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
92		eqid 2733	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘))) = (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))
93		ovex 7439	. . . . . 6 ⊢ (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)) ∈ V
94	91, 92, 93	fvmpt 6996	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))‘𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
95	94	adantl 483	. . . 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 7422	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛)) = (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)))
97	86, 95, 96	3eqtr4d 2783	. . 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 15601	. 2 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))) ⇝ (-1 · -(log‘𝐴)))
99	6	dvlog2lem 26152	. . . . . . . 8 ⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0))
100	99	sseli 3978	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ (ℂ ∖ (-∞(,]0)))
101		eqid 2733	. . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
102	101	logdmn0 26140	. . . . . . 7 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ≠ 0)
103	100, 102	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ≠ 0)
104	13, 103	logcld 26071	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (log‘𝐴) ∈ ℂ)
105	104	negcld 11555	. . . 4 ⊢ (𝐴 ∈ 𝑆 → -(log‘𝐴) ∈ ℂ)
106	105	mulm1d 11663	. . 3 ⊢ (𝐴 ∈ 𝑆 → (-1 · -(log‘𝐴)) = --(log‘𝐴))
107	104	negnegd 11559	. . 3 ⊢ (𝐴 ∈ 𝑆 → --(log‘𝐴) = (log‘𝐴))
108	106, 107	eqtrd 2773	. 2 ⊢ (𝐴 ∈ 𝑆 → (-1 · -(log‘𝐴)) = (log‘𝐴))
109	98, 108	breqtrd 5174	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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3945 class class class wbr 5148 ↦ cmpt 5231 ∘ ccom 5680 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 0cc0 11107 1c1 11108 + caddc 11110 · cmul 11112 -∞cmnf 11243 ℝ^*cxr 11244 < clt 11245 − cmin 11441 -cneg 11442 / cdiv 11868 ℕcn 12209 ℕ₀cn0 12469 ℤcz 12555 (,]cioc 13322 seqcseq 13963 ↑cexp 14024 abscabs 15178 ⇝ cli 15425 ∞Metcxmet 20922 ballcbl 20924 logclog 26055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-tan 16012 df-pi 16013 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-mulg 18946 df-cntz 19176 df-cmn 19645 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-nei 22594 df-lp 22632 df-perf 22633 df-cn 22723 df-cnp 22724 df-haus 22811 df-cmp 22883 df-tx 23058 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-xms 23818 df-ms 23819 df-tms 23820 df-cncf 24386 df-limc 25375 df-dv 25376 df-ulm 25881 df-log 26057

This theorem is referenced by: stirlinglem5 44781

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