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Theorem logtayl2 25244

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 12280	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12012	. . 3 ⊢ (𝐴 ∈ 𝑆 → 1 ∈ ℤ)
3		neg1cn 11750	. . . 4 ⊢ -1 ∈ ℂ
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑆 → -1 ∈ ℂ)
5		ax-1cn 10594	. . . . . 6 ⊢ 1 ∈ ℂ
6		logtayl2.s	. . . . . . . . 9 ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1)
7	6	eleq2i 2904	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ (1(ball‘(abs ∘ − ))1))
8		cnxmet 23380	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
9		1xr 10699	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
10		elbl 22997	. . . . . . . . 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 1457	. . . . . . . 8 ⊢ (𝐴 ∈ (1(ball‘(abs ∘ − ))1) ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1))
12	7, 11	bitri 277	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1))
13	12	simplbi 500	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ ℂ)
14		subcl 10884	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
15	5, 13, 14	sylancr 589	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (1 − 𝐴) ∈ ℂ)
16		eqid 2821	. . . . . . . 8 ⊢ (abs ∘ − ) = (abs ∘ − )
17	16	cnmetdval 23378	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1(abs ∘ − )𝐴) = (abs‘(1 − 𝐴)))
18	5, 13, 17	sylancr 589	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1(abs ∘ − )𝐴) = (abs‘(1 − 𝐴)))
19	12	simprbi 499	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1(abs ∘ − )𝐴) < 1)
20	18, 19	eqbrtrrd 5089	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (abs‘(1 − 𝐴)) < 1)
21		logtayl 25242	. . . . 5 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (abs‘(1 − 𝐴)) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘(1 − (1 − 𝐴))))
22	15, 20, 21	syl2anc 586	. . . 4 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘(1 − (1 − 𝐴))))
23		nncan 10914	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − (1 − 𝐴)) = 𝐴)
24	5, 13, 23	sylancr 589	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1 − (1 − 𝐴)) = 𝐴)
25	24	fveq2d 6673	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (log‘(1 − (1 − 𝐴))) = (log‘𝐴))
26	25	negeqd 10879	. . . 4 ⊢ (𝐴 ∈ 𝑆 → -(log‘(1 − (1 − 𝐴))) = -(log‘𝐴))
27	22, 26	breqtrd 5091	. . 3 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘𝐴))
28		oveq2 7163	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((1 − 𝐴)↑𝑘) = ((1 − 𝐴)↑𝑛))
29		id 22	. . . . . . 7 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
30	28, 29	oveq12d 7173	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((1 − 𝐴)↑𝑘) / 𝑘) = (((1 − 𝐴)↑𝑛) / 𝑛))
31		eqid 2821	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘)) = (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))
32		ovex 7188	. . . . . 6 ⊢ (((1 − 𝐴)↑𝑛) / 𝑛) ∈ V
33	30, 31, 32	fvmpt 6767	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) = (((1 − 𝐴)↑𝑛) / 𝑛))
34	33	adantl 484	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) = (((1 − 𝐴)↑𝑛) / 𝑛))
35		nnnn0 11903	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
36		expcl 13446	. . . . . 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 11645	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
39	38	adantl 484	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
40		nnne0 11670	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
41	40	adantl 484	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
42	37, 39, 41	divcld 11415	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((1 − 𝐴)↑𝑛) / 𝑛) ∈ ℂ)
43	34, 42	eqeltrd 2913	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) ∈ ℂ)
44	37, 39, 41	divnegd 11428	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -(((1 − 𝐴)↑𝑛) / 𝑛) = (-((1 − 𝐴)↑𝑛) / 𝑛))
45	42	mulm1d 11091	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)) = -(((1 − 𝐴)↑𝑛) / 𝑛))
46	35	adantl 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
47		expcl 13446	. . . . . . . . . 10 ⊢ ((-1 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (-1↑𝑛) ∈ ℂ)
48	3, 46, 47	sylancr 589	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑𝑛) ∈ ℂ)
49		subcl 10884	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
50	13, 5, 49	sylancl 588	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → (𝐴 − 1) ∈ ℂ)
51		expcl 13446	. . . . . . . . . 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 11092	. . . . . . . 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 11752	. . . . . . . . . . . 12 ⊢ -1 ≠ 0
56	55	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -1 ≠ 0)
57		nnz 12003	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
58	57	adantl 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
59	54, 56, 58	expm1d 13519	. . . . . . . . . 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 10605	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
62	61	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 1 ≠ 0)
63	48, 60, 62	divneg2d 11429	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((-1↑𝑛) / 1) = ((-1↑𝑛) / -1))
64	48	div1d 11407	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑𝑛) / 1) = (-1↑𝑛))
65	64	negeqd 10879	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((-1↑𝑛) / 1) = -(-1↑𝑛))
66	59, 63, 65	3eqtr2d 2862	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) = -(-1↑𝑛))
67	66	oveq1d 7170	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) = (-(-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
68	50	mulm1d 11091	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑆 → (-1 · (𝐴 − 1)) = -(𝐴 − 1))
69		negsubdi2 10944	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
70	13, 5, 69	sylancl 588	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑆 → -(𝐴 − 1) = (1 − 𝐴))
71	68, 70	eqtr2d 2857	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑆 → (1 − 𝐴) = (-1 · (𝐴 − 1)))
72	71	oveq1d 7170	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑆 → ((1 − 𝐴)↑𝑛) = ((-1 · (𝐴 − 1))↑𝑛))
73	72	adantr 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) = ((-1 · (𝐴 − 1))↑𝑛))
74		mulexp 13467	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (𝐴 − 1) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
75	3, 50, 35, 74	mp3an3an 1463	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
76	73, 75	eqtrd 2856	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
77	76	negeqd 10879	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((1 − 𝐴)↑𝑛) = -((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
78	53, 67, 77	3eqtr4d 2866	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) = -((1 − 𝐴)↑𝑛))
79	78	oveq1d 7170	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛) = (-((1 − 𝐴)↑𝑛) / 𝑛))
80	44, 45, 79	3eqtr4d 2866	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)) = (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛))
81		nnm1nn0 11937	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
82	81	adantl 484	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ₀)
83		expcl 13446	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ (𝑛 − 1) ∈ ℕ₀) → (-1↑(𝑛 − 1)) ∈ ℂ)
84	3, 82, 83	sylancr 589	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) ∈ ℂ)
85	84, 52, 39, 41	div23d 11452	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
86	80, 85	eqtr2d 2857	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)) = (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)))
87		oveq1 7162	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑘 − 1) = (𝑛 − 1))
88	87	oveq2d 7171	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (-1↑(𝑘 − 1)) = (-1↑(𝑛 − 1)))
89	88, 29	oveq12d 7173	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((-1↑(𝑘 − 1)) / 𝑘) = ((-1↑(𝑛 − 1)) / 𝑛))
90		oveq2 7163	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐴 − 1)↑𝑘) = ((𝐴 − 1)↑𝑛))
91	89, 90	oveq12d 7173	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
92		eqid 2821	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘))) = (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))
93		ovex 7188	. . . . . 6 ⊢ (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)) ∈ V
94	91, 92, 93	fvmpt 6767	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))‘𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
95	94	adantl 484	. . . 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 7171	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛)) = (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)))
97	86, 95, 96	3eqtr4d 2866	. . 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 15013	. 2 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))) ⇝ (-1 · -(log‘𝐴)))
99	6	dvlog2lem 25234	. . . . . . . 8 ⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0))
100	99	sseli 3962	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ (ℂ ∖ (-∞(,]0)))
101		eqid 2821	. . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
102	101	logdmn0 25222	. . . . . . 7 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ≠ 0)
103	100, 102	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ≠ 0)
104	13, 103	logcld 25153	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (log‘𝐴) ∈ ℂ)
105	104	negcld 10983	. . . 4 ⊢ (𝐴 ∈ 𝑆 → -(log‘𝐴) ∈ ℂ)
106	105	mulm1d 11091	. . 3 ⊢ (𝐴 ∈ 𝑆 → (-1 · -(log‘𝐴)) = --(log‘𝐴))
107	104	negnegd 10987	. . 3 ⊢ (𝐴 ∈ 𝑆 → --(log‘𝐴) = (log‘𝐴))
108	106, 107	eqtrd 2856	. 2 ⊢ (𝐴 ∈ 𝑆 → (-1 · -(log‘𝐴)) = (log‘𝐴))
109	98, 108	breqtrd 5091	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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3932 class class class wbr 5065 ↦ cmpt 5145 ∘ ccom 5558 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 -∞cmnf 10672 ℝ^*cxr 10673 < clt 10674 − cmin 10869 -cneg 10870 / cdiv 11296 ℕcn 11637 ℕ₀cn0 11896 ℤcz 11980 (,]cioc 12738 seqcseq 13368 ↑cexp 13428 abscabs 14592 ⇝ cli 14840 ∞Metcxmet 20529 ballcbl 20531 logclog 25137

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-shft 14425 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-limsup 14827 df-clim 14844 df-rlim 14845 df-sum 15042 df-ef 15420 df-sin 15422 df-cos 15423 df-tan 15424 df-pi 15425 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-cmp 21994 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485 df-limc 24463 df-dv 24464 df-ulm 24964 df-log 25139

This theorem is referenced by: stirlinglem5 42362

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