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Theorem loglesqrt 27094

Description: An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.) (Proof shortened by AV, 2-Aug-2021.)

**Assertion**
Ref	Expression
loglesqrt	⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ≤ (√‘𝐴))

Proof of Theorem loglesqrt Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0re 11313	. . . 4 ⊢ 0 ∈ ℝ
2	1	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ ℝ)
3		simpl 488	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
4		elicc2 13545	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥 ∈ (0[,]𝐴) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴)))
5	1, 3, 4	sylancr 599	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴)))
6	5	biimpa 482	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴))
7	6	simp1d 1160	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → 𝑥 ∈ ℝ)
8	6	simp2d 1161	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → 0 ≤ 𝑥)
9	7, 8	ge0p1rpd 13197	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → (𝑥 + 1) ∈ ℝ⁺)
10	9	fvresd 6905	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → ((log ↾ ℝ⁺)‘(𝑥 + 1)) = (log‘(𝑥 + 1)))
11	10	mpteq2dva 5197	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) = (𝑥 ∈ (0[,]𝐴) ↦ (log‘(𝑥 + 1))))
12		eqid 2760	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 25106	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	7	ex 418	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) → 𝑥 ∈ ℝ))
15	14	ssrdv 3914	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ ℝ)
16		ax-resscn 11260	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
17	15, 16	sstrdi 3920	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ ℂ)
18		resttopon 23484	. . . . . . 7 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (0[,]𝐴) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) ∈ (TopOn‘(0[,]𝐴)))
19	13, 17, 18	sylancr 599	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) ∈ (TopOn‘(0[,]𝐴)))
20	9	fmpttd 7115	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)):(0[,]𝐴)⟶ℝ⁺)
21		rpssre 13131	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ ℝ
22	21, 16	sstri 3917	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℂ
23	12	addcn 25190	. . . . . . . . . . 11 ⊢ + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
24	23	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
25		ssid 3930	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
26		cncfmptid 25239	. . . . . . . . . . 11 ⊢ (((0[,]𝐴) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]𝐴) ↦ 𝑥) ∈ ((0[,]𝐴)–cn→ℂ))
27	17, 25, 26	sylancl 598	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ 𝑥) ∈ ((0[,]𝐴)–cn→ℂ))
28		1cnd 11305	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 1 ∈ ℂ)
29	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℂ ⊆ ℂ)
30		cncfmptc 25238	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (0[,]𝐴) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]𝐴) ↦ 1) ∈ ((0[,]𝐴)–cn→ℂ))
31	28, 17, 29, 30	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ 1) ∈ ((0[,]𝐴)–cn→ℂ))
32	12, 24, 27, 31	cncfmpt2f 25241	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℂ))
33		cncfcdm 25224	. . . . . . . . 9 ⊢ ((ℝ⁺ ⊆ ℂ ∧ (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℂ)) → ((𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℝ⁺) ↔ (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)):(0[,]𝐴)⟶ℝ⁺))
34	22, 32, 33	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℝ⁺) ↔ (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)):(0[,]𝐴)⟶ℝ⁺))
35	20, 34	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℝ⁺))
36		eqid 2760	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) = ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴))
37		eqid 2760	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ⁺) = ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)
38	12, 36, 37	cncfcn 25236	. . . . . . . 8 ⊢ (((0[,]𝐴) ⊆ ℂ ∧ ℝ⁺ ⊆ ℂ) → ((0[,]𝐴)–cn→ℝ⁺) = (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)))
39	17, 22, 38	sylancl 598	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((0[,]𝐴)–cn→ℝ⁺) = (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)))
40	35, 39	eleqtrd 2862	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)))
41		relogcn 26971	. . . . . . . 8 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
42		eqid 2760	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
43	12, 37, 42	cncfcn 25236	. . . . . . . . 9 ⊢ ((ℝ⁺ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ⁺–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
44	22, 16, 43	mp2an 705	. . . . . . . 8 ⊢ (ℝ⁺–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
45	41, 44	eleqtri 2858	. . . . . . 7 ⊢ (log ↾ ℝ⁺) ∈ (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
46	45	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) ∈ (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
47	19, 40, 46	cnmpt11f 23988	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
48	12, 36, 42	cncfcn 25236	. . . . . 6 ⊢ (((0[,]𝐴) ⊆ ℂ ∧ ℝ ⊆ ℂ) → ((0[,]𝐴)–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
49	17, 16, 48	sylancl 598	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((0[,]𝐴)–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
50	47, 49	eleqtrrd 2863	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) ∈ ((0[,]𝐴)–cn→ℝ))
51	11, 50	eqeltrrd 2861	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (log‘(𝑥 + 1))) ∈ ((0[,]𝐴)–cn→ℝ))
52		reelprrecn 11295	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
53	52	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
54		simpr 490	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
55		1rp 13127	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
56		rpaddcl 13147	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
57	54, 55, 56	sylancl 598	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
58	57	relogcld 26956	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘(𝑥 + 1)) ∈ ℝ)
59	58	recnd 11340	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘(𝑥 + 1)) ∈ ℂ)
60	57	rpreccld 13177	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ∈ ℝ⁺)
61		1cnd 11305	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
62		relogcl 26908	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ → (log‘𝑦) ∈ ℝ)
63	62	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℝ)
64	63	recnd 11340	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℂ)
65		rpreccl 13151	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → (1 / 𝑦) ∈ ℝ⁺)
66	65	adantl 487	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (1 / 𝑦) ∈ ℝ⁺)
67		peano2re 11486	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
68	67	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
69	68	recnd 11340	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℂ)
70		1cnd 11305	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
71	16	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ ⊆ ℂ)
72	71	sselda 3908	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
73	53	dvmptid 26282	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
74		0cnd 11302	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ)
75	53, 28	dvmptc 26283	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
76	53, 72, 70, 73, 70, 74, 75	dvmptadd 26285	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ ↦ (1 + 0)))
77		1p0e1 12468	. . . . . . . . 9 ⊢ (1 + 0) = 1
78	77	mpteq2i 5200	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ (1 + 0)) = (𝑥 ∈ ℝ ↦ 1)
79	76, 78	eqtrdi 2811	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ ↦ 1))
80	21	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ⁺ ⊆ ℝ)
81		tgioo4 25129	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
82		ioorp 13559	. . . . . . . . 9 ⊢ (0(,)+∞) = ℝ⁺
83		iooretop 25089	. . . . . . . . 9 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
84	82, 83	eqeltrri 2857	. . . . . . . 8 ⊢ ℝ⁺ ∈ (topGen‘ran (,))
85	84	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ⁺ ∈ (topGen‘ran (,)))
86	53, 69, 70, 79, 80, 81, 12, 85	dvmptres 26288	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ⁺ ↦ 1))
87		relogf1o 26899	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
88		f1of 6824	. . . . . . . . . . 11 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
89	87, 88	mp1i 14	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
90	89	feqmptd 6953	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)))
91		fvres 6904	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑦) = (log‘𝑦))
92	91	mpteq2ia 5199	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))
93	90, 92	eqtrdi 2811	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦)))
94	93	oveq2d 7436	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))))
95		dvrelog 26970	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦))
96	94, 95	eqtr3di 2810	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦)))
97		fveq2 6885	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → (log‘𝑦) = (log‘(𝑥 + 1)))
98		oveq2 7428	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → (1 / 𝑦) = (1 / (𝑥 + 1)))
99	53, 53, 57, 61, 64, 66, 86, 96, 97, 98	dvmptco 26297	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 / (𝑥 + 1)) · 1)))
100	60	rpcnd 13169	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ∈ ℂ)
101	100	mulridd 11329	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((1 / (𝑥 + 1)) · 1) = (1 / (𝑥 + 1)))
102	101	mpteq2dva 5197	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ ℝ⁺ ↦ ((1 / (𝑥 + 1)) · 1)) = (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥 + 1))))
103	99, 102	eqtrd 2795	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥 + 1))))
104		ioossicc 13567	. . . . . . . . 9 ⊢ (0(,)𝐴) ⊆ (0[,]𝐴)
105	104	sseli 3904	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)𝐴) → 𝑥 ∈ (0[,]𝐴))
106	105, 7	sylan2 605	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 𝑥 ∈ ℝ)
107		eliooord 13539	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)𝐴) → (0 < 𝑥 ∧ 𝑥 < 𝐴))
108	107	simpld 500	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)𝐴) → 0 < 𝑥)
109	108	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 0 < 𝑥)
110	106, 109	elrpd 13164	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 𝑥 ∈ ℝ⁺)
111	110	ex 418	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0(,)𝐴) → 𝑥 ∈ ℝ⁺))
112	111	ssrdv 3914	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0(,)𝐴) ⊆ ℝ⁺)
113		iooretop 25089	. . . . 5 ⊢ (0(,)𝐴) ∈ (topGen‘ran (,))
114	113	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0(,)𝐴) ∈ (topGen‘ran (,)))
115	53, 59, 60, 103, 112, 81, 12, 114	dvmptres 26288	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ (0(,)𝐴) ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ (0(,)𝐴) ↦ (1 / (𝑥 + 1))))
116		elrege0 13588	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
117	7, 8, 116	sylanbrc 595	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → 𝑥 ∈ (0[,)+∞))
118	117	ex 418	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) → 𝑥 ∈ (0[,)+∞)))
119	118	ssrdv 3914	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ (0[,)+∞))
120	119	resabs1d 5998	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) = (√ ↾ (0[,]𝐴)))
121		sqrtf 15536	. . . . . . 7 ⊢ √:ℂ⟶ℂ
122	121	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → √:ℂ⟶ℂ)
123	122, 17	feqresmpt 6954	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√ ↾ (0[,]𝐴)) = (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)))
124	120, 123	eqtrd 2795	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) = (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)))
125		resqrtcn 27082	. . . . 5 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)
126		rescncf 25223	. . . . 5 ⊢ ((0[,]𝐴) ⊆ (0[,)+∞) → ((√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) ∈ ((0[,]𝐴)–cn→ℝ)))
127	119, 125, 126	mpisyl 22	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) ∈ ((0[,]𝐴)–cn→ℝ))
128	124, 127	eqeltrrd 2861	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)) ∈ ((0[,]𝐴)–cn→ℝ))
129		rpcn 13134	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
130	129	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
131	130	sqrtcld 15612	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
132		2rp 13128	. . . . . 6 ⊢ 2 ∈ ℝ⁺
133		rpsqrtcl 15436	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
134	133	adantl 487	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
135		rpmulcl 13148	. . . . . 6 ⊢ ((2 ∈ ℝ⁺ ∧ (√‘𝑥) ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ⁺)
136	132, 134, 135	sylancr 599	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ⁺)
137	136	rpreccld 13177	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (2 · (√‘𝑥))) ∈ ℝ⁺)
138		dvsqrt 27075	. . . . 5 ⊢ (ℝ D (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / (2 · (√‘𝑥))))
139	138	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / (2 · (√‘𝑥)))))
140	53, 131, 137, 139, 112, 81, 12, 114	dvmptres 26288	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ (0(,)𝐴) ↦ (√‘𝑥))) = (𝑥 ∈ (0(,)𝐴) ↦ (1 / (2 · (√‘𝑥)))))
141	134	rpred 13167	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
142		1re 11311	. . . . . . . . 9 ⊢ 1 ∈ ℝ
143		resubcl 11625	. . . . . . . . 9 ⊢ (((√‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ) → ((√‘𝑥) − 1) ∈ ℝ)
144	141, 142, 143	sylancl 598	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥) − 1) ∈ ℝ)
145	144	sqge0d 14283	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (((√‘𝑥) − 1)↑2))
146	130	sqsqrtd 15614	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥)↑2) = 𝑥)
147	146	oveq1d 7435	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥)↑2) − (2 · (√‘𝑥))) = (𝑥 − (2 · (√‘𝑥))))
148	147	oveq1d 7435	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1) = ((𝑥 − (2 · (√‘𝑥))) + 1))
149		binom2sub1 14368	. . . . . . . . 9 ⊢ ((√‘𝑥) ∈ ℂ → (((√‘𝑥) − 1)↑2) = ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1))
150	131, 149	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥) − 1)↑2) = ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1))
151	136	rpcnd 13169	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℂ)
152	130, 61, 151	addsubd 11693	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((𝑥 + 1) − (2 · (√‘𝑥))) = ((𝑥 − (2 · (√‘𝑥))) + 1))
153	148, 150, 152	3eqtr4d 2805	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥) − 1)↑2) = ((𝑥 + 1) − (2 · (√‘𝑥))))
154	145, 153	breqtrd 5130	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ ((𝑥 + 1) − (2 · (√‘𝑥))))
155	57	rpred 13167	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ)
156	136	rpred 13167	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ)
157	155, 156	subge0d 11909	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (0 ≤ ((𝑥 + 1) − (2 · (√‘𝑥))) ↔ (2 · (√‘𝑥)) ≤ (𝑥 + 1)))
158	154, 157	mpbid 235	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ≤ (𝑥 + 1))
159	136, 57	lerecd 13186	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) ≤ (𝑥 + 1) ↔ (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥)))))
160	158, 159	mpbid 235	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥))))
161	110, 160	syldan 603	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥))))
162		rexr 11358	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
163		0xr 11359	. . . . 5 ⊢ 0 ∈ ℝ^*
164		lbicc2 13598	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
165	163, 164	mp3an1 1477	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
166	162, 165	sylan 592	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
167		ubicc2 13599	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
168	163, 167	mp3an1 1477	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
169	162, 168	sylan 592	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
170		simpr 490	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ 𝐴)
171		fv0p1e1 12467	. . . 4 ⊢ (𝑥 = 0 → (log‘(𝑥 + 1)) = (log‘1))
172		log1 26918	. . . 4 ⊢ (log‘1) = 0
173	171, 172	eqtrdi 2811	. . 3 ⊢ (𝑥 = 0 → (log‘(𝑥 + 1)) = 0)
174		fveq2 6885	. . . 4 ⊢ (𝑥 = 0 → (√‘𝑥) = (√‘0))
175		sqrt0 15413	. . . 4 ⊢ (√‘0) = 0
176	174, 175	eqtrdi 2811	. . 3 ⊢ (𝑥 = 0 → (√‘𝑥) = 0)
177		fvoveq1 7443	. . 3 ⊢ (𝑥 = 𝐴 → (log‘(𝑥 + 1)) = (log‘(𝐴 + 1)))
178		fveq2 6885	. . 3 ⊢ (𝑥 = 𝐴 → (√‘𝑥) = (√‘𝐴))
179	2, 3, 51, 115, 128, 140, 161, 166, 169, 170, 173, 176, 177, 178	dvle 26332	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((log‘(𝐴 + 1)) − 0) ≤ ((√‘𝐴) − 0))
180		ge0p1rp 13156	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ⁺)
181	180	relogcld 26956	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ∈ ℝ)
182		resqrtcl 15425	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ)
183	181, 182, 2	lesub1d 11926	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((log‘(𝐴 + 1)) ≤ (√‘𝐴) ↔ ((log‘(𝐴 + 1)) − 0) ≤ ((√‘𝐴) − 0)))
184	179, 183	mpbird 260	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ≤ (√‘𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ⊆ wss 3895 {cpr 4586 class class class wbr 5102 ↦ cmpt 5185 ran crn 5651 ↾ cres 5652 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ℂcc 11201 ℝcr 11202 0cc0 11203 1c1 11204 + caddc 11206 · cmul 11208 +∞cpnf 11343 ℝ^*cxr 11345 < clt 11346 ≤ cle 11347 − cmin 11544 / cdiv 11976 2c2 12400 ℝ⁺crp 13123 (,)cioo 13479 [,)cico 13481 [,]cicc 13482 ↑cexp 14207 √csqrt 15405 ↾_t crest 17596 TopOpenctopn 17597 topGenctg 17613 ℂ_fldccnfld 21683 TopOnctopon 23233 Cn ccn 23547 ×_t ctx 23884 –cn→ccncf 25202 D cdv 26188 logclog 26887

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281 ax-addf 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9507 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-q 13079 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-ioo 13483 df-ioc 13484 df-ico 13485 df-icc 13486 df-fz 13643 df-fzo 13792 df-fl 13935 df-mod 14013 df-seq 14148 df-exp 14208 df-fac 14421 df-bc 14450 df-hash 14478 df-shft 15225 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-limsup 15643 df-clim 15660 df-rlim 15661 df-sum 15859 df-ef 16238 df-sin 16240 df-cos 16241 df-tan 16242 df-pi 16243 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-starv 17448 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-unif 17456 df-hom 17457 df-cco 17458 df-rest 17598 df-topn 17599 df-0g 17617 df-gsum 17618 df-topgen 17619 df-pt 17620 df-prds 17623 df-xrs 17679 df-qtop 17684 df-imas 17685 df-xps 17687 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-submnd 18984 df-mulg 19283 df-cntz 19536 df-cmn 20001 df-psmet 21675 df-xmet 21676 df-met 21677 df-bl 21678 df-mopn 21679 df-fbas 21680 df-fg 21681 df-cnfld 21684 df-top 23217 df-topon 23234 df-topsp 23256 df-bases 23269 df-cld 23342 df-ntr 23343 df-cls 23344 df-nei 23421 df-lp 23459 df-perf 23460 df-cn 23550 df-cnp 23551 df-haus 23638 df-cmp 23710 df-tx 23886 df-hmeo 24079 df-fil 24170 df-fm 24262 df-flim 24263 df-flf 24264 df-xms 24644 df-ms 24645 df-tms 24646 df-cncf 25204 df-limc 26191 df-dv 26192 df-log 26889 df-cxp 26890

This theorem is used by: rplogsumlem1 27816

W3C validator