Theorem logtayllem 24804

Description: Lemma for logtayl 24805. (Contributed by Mario Carneiro, 1-Apr-2015.)

Assertion

Ref	Expression
logtayllem	⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))) ∈ dom ⇝ )

Distinct variable group:   𝐴,𝑛

Proof of Theorem logtayllem

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 12004	. 2 ⊢ ℕ0 = (ℤ≥‘0)
2		1nn0 11636	. . 3 ⊢ 1 ∈ ℕ0
3	2	a1i 11	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℕ0)
4		oveq2 6913	. . . . 5 ⊢ (𝑛 = 𝑘 → ((abs‘𝐴)↑𝑛) = ((abs‘𝐴)↑𝑘))
5		eqid 2825	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))
6		ovex 6937	. . . . 5 ⊢ ((abs‘𝐴)↑𝑘) ∈ V
7	4, 5, 6	fvmpt 6529	. . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))‘𝑘) = ((abs‘𝐴)↑𝑘))
8	7	adantl 475	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))‘𝑘) = ((abs‘𝐴)↑𝑘))
9		abscl 14395	. . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
10	9	adantr 474	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℝ)
11		reexpcl 13171	. . . 4 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((abs‘𝐴)↑𝑘) ∈ ℝ)
12	10, 11	sylan 577	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → ((abs‘𝐴)↑𝑘) ∈ ℝ)
13	8, 12	eqeltrd 2906	. 2 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))‘𝑘) ∈ ℝ)
14		eqeq1 2829	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 = 0 ↔ 𝑘 = 0))
15		oveq2 6913	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
16	14, 15	ifbieq2d 4331	. . . . . 6 ⊢ (𝑛 = 𝑘 → if(𝑛 = 0, 0, (1 / 𝑛)) = if(𝑘 = 0, 0, (1 / 𝑘)))
17		oveq2 6913	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐴↑𝑛) = (𝐴↑𝑘))
18	16, 17	oveq12d 6923	. . . . 5 ⊢ (𝑛 = 𝑘 → (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)) = (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))
19		eqid 2825	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛))) = (𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))
20		ovex 6937	. . . . 5 ⊢ (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) ∈ V
21	18, 19, 20	fvmpt 6529	. . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))‘𝑘) = (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))
22	21	adantl 475	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))‘𝑘) = (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))
23		0cnd 10349	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) ∧ 𝑘 = 0) → 0 ∈ ℂ)
24		nn0cn 11629	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
25	24	adantl 475	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
26		df-ne 3000	. . . . . . 7 ⊢ (𝑘 ≠ 0 ↔ ¬ 𝑘 = 0)
27	26	biimpri 220	. . . . . 6 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
28		reccl 11017	. . . . . 6 ⊢ ((𝑘 ∈ ℂ ∧ 𝑘 ≠ 0) → (1 / 𝑘) ∈ ℂ)
29	25, 27, 28	syl2an 591	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 = 0) → (1 / 𝑘) ∈ ℂ)
30	23, 29	ifclda 4340	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 0, 0, (1 / 𝑘)) ∈ ℂ)
31		expcl 13172	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ)
32	31	adantlr 708	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ)
33	30, 32	mulcld 10377	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) ∈ ℂ)
34	22, 33	eqeltrd 2906	. 2 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))‘𝑘) ∈ ℂ)
35	10	recnd 10385	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℂ)
36		absidm 14440	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘(abs‘𝐴)) = (abs‘𝐴))
37	36	adantr 474	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(abs‘𝐴)) = (abs‘𝐴))
38		simpr 479	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1)
39	37, 38	eqbrtrd 4895	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(abs‘𝐴)) < 1)
40	35, 39, 8	geolim 14975	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))) ⇝ (1 / (1 − (abs‘𝐴))))
41		seqex 13097	. . . 4 ⊢ seq0( + , (𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))) ∈ V
42		ovex 6937	. . . 4 ⊢ (1 / (1 − (abs‘𝐴))) ∈ V
43	41, 42	breldm 5561	. . 3 ⊢ (seq0( + , (𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))) ⇝ (1 / (1 − (abs‘𝐴))) → seq0( + , (𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))) ∈ dom ⇝ )
44	40, 43	syl 17	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))) ∈ dom ⇝ )
45		1red 10357	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℝ)
46		elnnuz 12006	. . 3 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
47		nnrecre 11393	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
48	47	adantl 475	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
49	48	recnd 10385	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℂ)
50		nnnn0 11626	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
51	50, 32	sylan2 588	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (𝐴↑𝑘) ∈ ℂ)
52	49, 51	absmuld 14570	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (abs‘((1 / 𝑘) · (𝐴↑𝑘))) = ((abs‘(1 / 𝑘)) · (abs‘(𝐴↑𝑘))))
53		nnrp 12125	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
54	53	adantl 475	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
55	54	rpreccld 12166	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ+)
56	55	rpge0d 12160	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / 𝑘))
57	48, 56	absidd 14538	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (abs‘(1 / 𝑘)) = (1 / 𝑘))
58		simpl 476	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ)
59		absexp 14421	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘))
60	58, 50, 59	syl2an 591	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (abs‘(𝐴↑𝑘)) = ((abs‘𝐴)↑𝑘))
61	57, 60	oveq12d 6923	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → ((abs‘(1 / 𝑘)) · (abs‘(𝐴↑𝑘))) = ((1 / 𝑘) · ((abs‘𝐴)↑𝑘)))
62	52, 61	eqtrd 2861	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (abs‘((1 / 𝑘) · (𝐴↑𝑘))) = ((1 / 𝑘) · ((abs‘𝐴)↑𝑘)))
63		1red 10357	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
64	50, 12	sylan2 588	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → ((abs‘𝐴)↑𝑘) ∈ ℝ)
65	51	absge0d 14560	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 0 ≤ (abs‘(𝐴↑𝑘)))
66	65, 60	breqtrd 4899	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 0 ≤ ((abs‘𝐴)↑𝑘))
67		nnge1 11380	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 1 ≤ 𝑘)
68	67	adantl 475	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 1 ≤ 𝑘)
69		0lt1 10874	. . . . . . . . . 10 ⊢ 0 < 1
70	69	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 0 < 1)
71		nnre 11358	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
72	71	adantl 475	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
73		nngt0 11383	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
74	73	adantl 475	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
75		lerec 11236	. . . . . . . . 9 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → (1 ≤ 𝑘 ↔ (1 / 𝑘) ≤ (1 / 1)))
76	63, 70, 72, 74, 75	syl22anc 874	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (1 ≤ 𝑘 ↔ (1 / 𝑘) ≤ (1 / 1)))
77	68, 76	mpbid 224	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ≤ (1 / 1))
78		1div1e1 11042	. . . . . . 7 ⊢ (1 / 1) = 1
79	77, 78	syl6breq 4914	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ≤ 1)
80	48, 63, 64, 66, 79	lemul1ad 11293	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) · ((abs‘𝐴)↑𝑘)) ≤ (1 · ((abs‘𝐴)↑𝑘)))
81	62, 80	eqbrtrd 4895	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (abs‘((1 / 𝑘) · (𝐴↑𝑘))) ≤ (1 · ((abs‘𝐴)↑𝑘)))
82	50, 22	sylan2 588	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))‘𝑘) = (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)))
83		nnne0 11386	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
84	83	adantl 475	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0)
85	84	neneqd 3004	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → ¬ 𝑘 = 0)
86	85	iffalsed 4317	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → if(𝑘 = 0, 0, (1 / 𝑘)) = (1 / 𝑘))
87	86	oveq1d 6920	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (if(𝑘 = 0, 0, (1 / 𝑘)) · (𝐴↑𝑘)) = ((1 / 𝑘) · (𝐴↑𝑘)))
88	82, 87	eqtrd 2861	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))‘𝑘) = ((1 / 𝑘) · (𝐴↑𝑘)))
89	88	fveq2d 6437	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (abs‘((𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))‘𝑘)) = (abs‘((1 / 𝑘) · (𝐴↑𝑘))))
90	50, 8	sylan2 588	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))‘𝑘) = ((abs‘𝐴)↑𝑘))
91	90	oveq2d 6921	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (1 · ((𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))‘𝑘)) = (1 · ((abs‘𝐴)↑𝑘)))
92	81, 89, 91	3brtr4d 4905	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (abs‘((𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))‘𝑘)) ≤ (1 · ((𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))‘𝑘)))
93	46, 92	sylan2br 590	. 2 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ (ℤ≥‘1)) → (abs‘((𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))‘𝑘)) ≤ (1 · ((𝑛 ∈ ℕ0 ↦ ((abs‘𝐴)↑𝑛))‘𝑘)))
94	1, 3, 13, 34, 44, 45, 93	cvgcmpce 14924	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴↑𝑛)))) ∈ dom ⇝ )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ≠ wne 2999  ifcif 4306   class class class wbr 4873   ↦ cmpt 4952  dom cdm 5342  ‘cfv 6123  (class class class)co 6905  ℂcc 10250  ℝcr 10251  0cc0 10252  1c1 10253   + caddc 10255   · cmul 10257   < clt 10391   ≤ cle 10392   − cmin 10585   / cdiv 11009  ℕcn 11350  ℕ₀cn0 11618  ℤ_≥cuz 11968  ℝ⁺crp 12112  seqcseq 13095  ↑cexp 13154  abscabs 14351   ⇝ cli 14592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-pre-sup 10330  ax-addf 10331  ax-mulf 10332

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-inf 8618  df-oi 8684  df-card 9078  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-2 11414  df-3 11415  df-n0 11619  df-z 11705  df-uz 11969  df-rp 12113  df-ico 12469  df-fz 12620  df-fzo 12761  df-fl 12888  df-seq 13096  df-exp 13155  df-hash 13411  df-cj 14216  df-re 14217  df-im 14218  df-sqrt 14352  df-abs 14353  df-limsup 14579  df-clim 14596  df-rlim 14597  df-sum 14794

This theorem is referenced by:  logtayl  24805