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Theorem logtayl2 26686

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 12911	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12639	. . 3 ⊢ (𝐴 ∈ 𝑆 → 1 ∈ ℤ)
3		neg1cn 12372	. . . 4 ⊢ -1 ∈ ℂ
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑆 → -1 ∈ ℂ)
5		ax-1cn 11207	. . . . . 6 ⊢ 1 ∈ ℂ
6		logtayl2.s	. . . . . . . . 9 ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1)
7	6	eleq2i 2818	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ (1(ball‘(abs ∘ − ))1))
8		cnxmet 24777	. . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
9		1xr 11314	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
10		elbl 24382	. . . . . . . . 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 1458	. . . . . . . 8 ⊢ (𝐴 ∈ (1(ball‘(abs ∘ − ))1) ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1))
12	7, 11	bitri 274	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1(abs ∘ − )𝐴) < 1))
13	12	simplbi 496	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ ℂ)
14		subcl 11500	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
15	5, 13, 14	sylancr 585	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (1 − 𝐴) ∈ ℂ)
16		eqid 2726	. . . . . . . 8 ⊢ (abs ∘ − ) = (abs ∘ − )
17	16	cnmetdval 24775	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1(abs ∘ − )𝐴) = (abs‘(1 − 𝐴)))
18	5, 13, 17	sylancr 585	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1(abs ∘ − )𝐴) = (abs‘(1 − 𝐴)))
19	12	simprbi 495	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1(abs ∘ − )𝐴) < 1)
20	18, 19	eqbrtrrd 5169	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (abs‘(1 − 𝐴)) < 1)
21		logtayl 26684	. . . . 5 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (abs‘(1 − 𝐴)) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘(1 − (1 − 𝐴))))
22	15, 20, 21	syl2anc 582	. . . 4 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘(1 − (1 − 𝐴))))
23		nncan 11530	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − (1 − 𝐴)) = 𝐴)
24	5, 13, 23	sylancr 585	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (1 − (1 − 𝐴)) = 𝐴)
25	24	fveq2d 6897	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (log‘(1 − (1 − 𝐴))) = (log‘𝐴))
26	25	negeqd 11495	. . . 4 ⊢ (𝐴 ∈ 𝑆 → -(log‘(1 − (1 − 𝐴))) = -(log‘𝐴))
27	22, 26	breqtrd 5171	. . 3 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))) ⇝ -(log‘𝐴))
28		oveq2 7424	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((1 − 𝐴)↑𝑘) = ((1 − 𝐴)↑𝑛))
29		id 22	. . . . . . 7 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
30	28, 29	oveq12d 7434	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((1 − 𝐴)↑𝑘) / 𝑘) = (((1 − 𝐴)↑𝑛) / 𝑛))
31		eqid 2726	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘)) = (𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))
32		ovex 7449	. . . . . 6 ⊢ (((1 − 𝐴)↑𝑛) / 𝑛) ∈ V
33	30, 31, 32	fvmpt 7001	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) = (((1 − 𝐴)↑𝑛) / 𝑛))
34	33	adantl 480	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) = (((1 − 𝐴)↑𝑛) / 𝑛))
35		nnnn0 12525	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
36		expcl 14093	. . . . . 6 ⊢ (((1 − 𝐴) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((1 − 𝐴)↑𝑛) ∈ ℂ)
37	15, 35, 36	syl2an 594	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) ∈ ℂ)
38		nncn 12266	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
39	38	adantl 480	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
40		nnne0 12292	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
41	40	adantl 480	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
42	37, 39, 41	divcld 12035	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((1 − 𝐴)↑𝑛) / 𝑛) ∈ ℂ)
43	34, 42	eqeltrd 2826	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛) ∈ ℂ)
44	37, 39, 41	divnegd 12048	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -(((1 − 𝐴)↑𝑛) / 𝑛) = (-((1 − 𝐴)↑𝑛) / 𝑛))
45	42	mulm1d 11707	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)) = -(((1 − 𝐴)↑𝑛) / 𝑛))
46	35	adantl 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
47		expcl 14093	. . . . . . . . . 10 ⊢ ((-1 ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (-1↑𝑛) ∈ ℂ)
48	3, 46, 47	sylancr 585	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑𝑛) ∈ ℂ)
49		subcl 11500	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
50	13, 5, 49	sylancl 584	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → (𝐴 − 1) ∈ ℂ)
51		expcl 14093	. . . . . . . . . 10 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((𝐴 − 1)↑𝑛) ∈ ℂ)
52	50, 35, 51	syl2an 594	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((𝐴 − 1)↑𝑛) ∈ ℂ)
53	48, 52	mulneg1d 11708	. . . . . . . 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 12374	. . . . . . . . . . . 12 ⊢ -1 ≠ 0
56	55	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -1 ≠ 0)
57		nnz 12625	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
58	57	adantl 480	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
59	54, 56, 58	expm1d 14169	. . . . . . . . . 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 11218	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
62	61	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → 1 ≠ 0)
63	48, 60, 62	divneg2d 12049	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((-1↑𝑛) / 1) = ((-1↑𝑛) / -1))
64	48	div1d 12027	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑𝑛) / 1) = (-1↑𝑛))
65	64	negeqd 11495	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((-1↑𝑛) / 1) = -(-1↑𝑛))
66	59, 63, 65	3eqtr2d 2772	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) = -(-1↑𝑛))
67	66	oveq1d 7431	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) = (-(-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
68	50	mulm1d 11707	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑆 → (-1 · (𝐴 − 1)) = -(𝐴 − 1))
69		negsubdi2 11560	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝐴 − 1) = (1 − 𝐴))
70	13, 5, 69	sylancl 584	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑆 → -(𝐴 − 1) = (1 − 𝐴))
71	68, 70	eqtr2d 2767	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑆 → (1 − 𝐴) = (-1 · (𝐴 − 1)))
72	71	oveq1d 7431	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑆 → ((1 − 𝐴)↑𝑛) = ((-1 · (𝐴 − 1))↑𝑛))
73	72	adantr 479	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) = ((-1 · (𝐴 − 1))↑𝑛))
74		mulexp 14115	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (𝐴 − 1) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
75	3, 50, 35, 74	mp3an3an 1464	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1 · (𝐴 − 1))↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
76	73, 75	eqtrd 2766	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((1 − 𝐴)↑𝑛) = ((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
77	76	negeqd 11495	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → -((1 − 𝐴)↑𝑛) = -((-1↑𝑛) · ((𝐴 − 1)↑𝑛)))
78	53, 67, 77	3eqtr4d 2776	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → ((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) = -((1 − 𝐴)↑𝑛))
79	78	oveq1d 7431	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛) = (-((1 − 𝐴)↑𝑛) / 𝑛))
80	44, 45, 79	3eqtr4d 2776	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)) = (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛))
81		nnm1nn0 12559	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
82	81	adantl 480	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ₀)
83		expcl 14093	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ (𝑛 − 1) ∈ ℕ₀) → (-1↑(𝑛 − 1)) ∈ ℂ)
84	3, 82, 83	sylancr 585	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1↑(𝑛 − 1)) ∈ ℂ)
85	84, 52, 39, 41	div23d 12072	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) · ((𝐴 − 1)↑𝑛)) / 𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
86	80, 85	eqtr2d 2767	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)) = (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)))
87		oveq1 7423	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑘 − 1) = (𝑛 − 1))
88	87	oveq2d 7432	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (-1↑(𝑘 − 1)) = (-1↑(𝑛 − 1)))
89	88, 29	oveq12d 7434	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((-1↑(𝑘 − 1)) / 𝑘) = ((-1↑(𝑛 − 1)) / 𝑛))
90		oveq2 7424	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐴 − 1)↑𝑘) = ((𝐴 − 1)↑𝑛))
91	89, 90	oveq12d 7434	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
92		eqid 2726	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘))) = (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))
93		ovex 7449	. . . . . 6 ⊢ (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)) ∈ V
94	91, 92, 93	fvmpt 7001	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))‘𝑛) = (((-1↑(𝑛 − 1)) / 𝑛) · ((𝐴 − 1)↑𝑛)))
95	94	adantl 480	. . . 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 7432	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑛 ∈ ℕ) → (-1 · ((𝑘 ∈ ℕ ↦ (((1 − 𝐴)↑𝑘) / 𝑘))‘𝑛)) = (-1 · (((1 − 𝐴)↑𝑛) / 𝑛)))
97	86, 95, 96	3eqtr4d 2776	. . 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 15657	. 2 ⊢ (𝐴 ∈ 𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))) ⇝ (-1 · -(log‘𝐴)))
99	6	dvlog2lem 26676	. . . . . . . 8 ⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0))
100	99	sseli 3974	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ (ℂ ∖ (-∞(,]0)))
101		eqid 2726	. . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
102	101	logdmn0 26664	. . . . . . 7 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ≠ 0)
103	100, 102	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ≠ 0)
104	13, 103	logcld 26594	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (log‘𝐴) ∈ ℂ)
105	104	negcld 11599	. . . 4 ⊢ (𝐴 ∈ 𝑆 → -(log‘𝐴) ∈ ℂ)
106	105	mulm1d 11707	. . 3 ⊢ (𝐴 ∈ 𝑆 → (-1 · -(log‘𝐴)) = --(log‘𝐴))
107	104	negnegd 11603	. . 3 ⊢ (𝐴 ∈ 𝑆 → --(log‘𝐴) = (log‘𝐴))
108	106, 107	eqtrd 2766	. 2 ⊢ (𝐴 ∈ 𝑆 → (-1 · -(log‘𝐴)) = (log‘𝐴))
109	98, 108	breqtrd 5171	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 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3943 class class class wbr 5145 ↦ cmpt 5228 ∘ ccom 5678 ‘cfv 6546 (class class class)co 7416 ℂcc 11147 0cc0 11149 1c1 11150 + caddc 11152 · cmul 11154 -∞cmnf 11287 ℝ^*cxr 11288 < clt 11289 − cmin 11485 -cneg 11486 / cdiv 11912 ℕcn 12258 ℕ₀cn0 12518 ℤcz 12604 (,]cioc 13373 seqcseq 14015 ↑cexp 14075 abscabs 15234 ⇝ cli 15481 ∞Metcxmet 21324 ballcbl 21326 logclog 26578

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13376 df-ioc 13377 df-ico 13378 df-icc 13379 df-fz 13533 df-fzo 13676 df-fl 13806 df-mod 13884 df-seq 14016 df-exp 14076 df-fac 14286 df-bc 14315 df-hash 14343 df-shft 15067 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-limsup 15468 df-clim 15485 df-rlim 15486 df-sum 15686 df-ef 16064 df-sin 16066 df-cos 16067 df-tan 16068 df-pi 16069 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19058 df-cntz 19307 df-cmn 19776 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-fbas 21336 df-fg 21337 df-cnfld 21340 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cld 23011 df-ntr 23012 df-cls 23013 df-nei 23090 df-lp 23128 df-perf 23129 df-cn 23219 df-cnp 23220 df-haus 23307 df-cmp 23379 df-tx 23554 df-hmeo 23747 df-fil 23838 df-fm 23930 df-flim 23931 df-flf 23932 df-xms 24314 df-ms 24315 df-tms 24316 df-cncf 24886 df-limc 25883 df-dv 25884 df-ulm 26403 df-log 26580

This theorem is referenced by: stirlinglem5 45735

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