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Theorem loglesqrt 26111

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 11157	. . . 4 ⊢ 0 ∈ ℝ
2	1	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ ℝ)
3		simpl 483	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
4		elicc2 13329	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥 ∈ (0[,]𝐴) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴)))
5	1, 3, 4	sylancr 587	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴)))
6	5	biimpa 477	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴))
7	6	simp1d 1142	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → 𝑥 ∈ ℝ)
8	6	simp2d 1143	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → 0 ≤ 𝑥)
9	7, 8	ge0p1rpd 12987	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → (𝑥 + 1) ∈ ℝ⁺)
10	9	fvresd 6862	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → ((log ↾ ℝ⁺)‘(𝑥 + 1)) = (log‘(𝑥 + 1)))
11	10	mpteq2dva 5205	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) = (𝑥 ∈ (0[,]𝐴) ↦ (log‘(𝑥 + 1))))
12		eqid 2736	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 24146	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	7	ex 413	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) → 𝑥 ∈ ℝ))
15	14	ssrdv 3950	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ ℝ)
16		ax-resscn 11108	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
17	15, 16	sstrdi 3956	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ ℂ)
18		resttopon 22512	. . . . . . 7 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (0[,]𝐴) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) ∈ (TopOn‘(0[,]𝐴)))
19	13, 17, 18	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) ∈ (TopOn‘(0[,]𝐴)))
20	9	fmpttd 7063	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)):(0[,]𝐴)⟶ℝ⁺)
21		rpssre 12922	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ ℝ
22	21, 16	sstri 3953	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℂ
23	12	addcn 24228	. . . . . . . . . . 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 3966	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
26		cncfmptid 24276	. . . . . . . . . . 11 ⊢ (((0[,]𝐴) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]𝐴) ↦ 𝑥) ∈ ((0[,]𝐴)–cn→ℂ))
27	17, 25, 26	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ 𝑥) ∈ ((0[,]𝐴)–cn→ℂ))
28		1cnd 11150	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 1 ∈ ℂ)
29	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℂ ⊆ ℂ)
30		cncfmptc 24275	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (0[,]𝐴) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]𝐴) ↦ 1) ∈ ((0[,]𝐴)–cn→ℂ))
31	28, 17, 29, 30	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ 1) ∈ ((0[,]𝐴)–cn→ℂ))
32	12, 24, 27, 31	cncfmpt2f 24278	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℂ))
33		cncfcdm 24261	. . . . . . . . 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 587	. . . . . . . 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 256	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℝ⁺))
36		eqid 2736	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) = ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴))
37		eqid 2736	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ⁺) = ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)
38	12, 36, 37	cncfcn 24273	. . . . . . . 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 586	. . . . . . 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 2840	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)))
41		relogcn 25993	. . . . . . . 8 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
42		eqid 2736	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
43	12, 37, 42	cncfcn 24273	. . . . . . . . 9 ⊢ ((ℝ⁺ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ⁺–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
44	22, 16, 43	mp2an 690	. . . . . . . 8 ⊢ (ℝ⁺–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
45	41, 44	eleqtri 2836	. . . . . . 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 23015	. . . . 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 24273	. . . . . 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 586	. . . . 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 2841	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) ∈ ((0[,]𝐴)–cn→ℝ))
51	11, 50	eqeltrrd 2839	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (log‘(𝑥 + 1))) ∈ ((0[,]𝐴)–cn→ℝ))
52		reelprrecn 11143	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
53	52	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
54		simpr 485	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
55		1rp 12919	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
56		rpaddcl 12937	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
57	54, 55, 56	sylancl 586	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
58	57	relogcld 25978	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘(𝑥 + 1)) ∈ ℝ)
59	58	recnd 11183	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘(𝑥 + 1)) ∈ ℂ)
60	57	rpreccld 12967	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ∈ ℝ⁺)
61		1cnd 11150	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
62		relogcl 25931	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ → (log‘𝑦) ∈ ℝ)
63	62	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℝ)
64	63	recnd 11183	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℂ)
65		rpreccl 12941	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → (1 / 𝑦) ∈ ℝ⁺)
66	65	adantl 482	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (1 / 𝑦) ∈ ℝ⁺)
67		peano2re 11328	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
68	67	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
69	68	recnd 11183	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℂ)
70		1cnd 11150	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
71	16	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ ⊆ ℂ)
72	71	sselda 3944	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
73	53	dvmptid 25321	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
74		0cnd 11148	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ)
75	53, 28	dvmptc 25322	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
76	53, 72, 70, 73, 70, 74, 75	dvmptadd 25324	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ ↦ (1 + 0)))
77		1p0e1 12277	. . . . . . . . 9 ⊢ (1 + 0) = 1
78	77	mpteq2i 5210	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ (1 + 0)) = (𝑥 ∈ ℝ ↦ 1)
79	76, 78	eqtrdi 2792	. . . . . . 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	12	tgioo2 24166	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
82		ioorp 13342	. . . . . . . . 9 ⊢ (0(,)+∞) = ℝ⁺
83		iooretop 24129	. . . . . . . . 9 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
84	82, 83	eqeltrri 2835	. . . . . . . 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 25327	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ⁺ ↦ 1))
87		relogf1o 25922	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
88		f1of 6784	. . . . . . . . . . 11 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
89	87, 88	mp1i 13	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
90	89	feqmptd 6910	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)))
91		fvres 6861	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑦) = (log‘𝑦))
92	91	mpteq2ia 5208	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))
93	90, 92	eqtrdi 2792	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦)))
94	93	oveq2d 7373	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))))
95		dvrelog 25992	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦))
96	94, 95	eqtr3di 2791	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦)))
97		fveq2 6842	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → (log‘𝑦) = (log‘(𝑥 + 1)))
98		oveq2 7365	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → (1 / 𝑦) = (1 / (𝑥 + 1)))
99	53, 53, 57, 61, 64, 66, 86, 96, 97, 98	dvmptco 25336	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 / (𝑥 + 1)) · 1)))
100	60	rpcnd 12959	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ∈ ℂ)
101	100	mulid1d 11172	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((1 / (𝑥 + 1)) · 1) = (1 / (𝑥 + 1)))
102	101	mpteq2dva 5205	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ ℝ⁺ ↦ ((1 / (𝑥 + 1)) · 1)) = (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥 + 1))))
103	99, 102	eqtrd 2776	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥 + 1))))
104		ioossicc 13350	. . . . . . . . 9 ⊢ (0(,)𝐴) ⊆ (0[,]𝐴)
105	104	sseli 3940	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)𝐴) → 𝑥 ∈ (0[,]𝐴))
106	105, 7	sylan2 593	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 𝑥 ∈ ℝ)
107		eliooord 13323	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)𝐴) → (0 < 𝑥 ∧ 𝑥 < 𝐴))
108	107	simpld 495	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)𝐴) → 0 < 𝑥)
109	108	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 0 < 𝑥)
110	106, 109	elrpd 12954	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 𝑥 ∈ ℝ⁺)
111	110	ex 413	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0(,)𝐴) → 𝑥 ∈ ℝ⁺))
112	111	ssrdv 3950	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0(,)𝐴) ⊆ ℝ⁺)
113		iooretop 24129	. . . . 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 25327	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ (0(,)𝐴) ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ (0(,)𝐴) ↦ (1 / (𝑥 + 1))))
116		elrege0 13371	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
117	7, 8, 116	sylanbrc 583	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → 𝑥 ∈ (0[,)+∞))
118	117	ex 413	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) → 𝑥 ∈ (0[,)+∞)))
119	118	ssrdv 3950	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ (0[,)+∞))
120	119	resabs1d 5968	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) = (√ ↾ (0[,]𝐴)))
121		sqrtf 15248	. . . . . . 7 ⊢ √:ℂ⟶ℂ
122	121	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → √:ℂ⟶ℂ)
123	122, 17	feqresmpt 6911	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√ ↾ (0[,]𝐴)) = (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)))
124	120, 123	eqtrd 2776	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) = (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)))
125		resqrtcn 26102	. . . . 5 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)
126		rescncf 24260	. . . . 5 ⊢ ((0[,]𝐴) ⊆ (0[,)+∞) → ((√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) ∈ ((0[,]𝐴)–cn→ℝ)))
127	119, 125, 126	mpisyl 21	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) ∈ ((0[,]𝐴)–cn→ℝ))
128	124, 127	eqeltrrd 2839	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)) ∈ ((0[,]𝐴)–cn→ℝ))
129		rpcn 12925	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
130	129	adantl 482	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
131	130	sqrtcld 15322	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
132		2rp 12920	. . . . . 6 ⊢ 2 ∈ ℝ⁺
133		rpsqrtcl 15149	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
134	133	adantl 482	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
135		rpmulcl 12938	. . . . . 6 ⊢ ((2 ∈ ℝ⁺ ∧ (√‘𝑥) ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ⁺)
136	132, 134, 135	sylancr 587	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ⁺)
137	136	rpreccld 12967	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (2 · (√‘𝑥))) ∈ ℝ⁺)
138		dvsqrt 26095	. . . . 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 25327	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ (0(,)𝐴) ↦ (√‘𝑥))) = (𝑥 ∈ (0(,)𝐴) ↦ (1 / (2 · (√‘𝑥)))))
141	134	rpred 12957	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
142		1re 11155	. . . . . . . . 9 ⊢ 1 ∈ ℝ
143		resubcl 11465	. . . . . . . . 9 ⊢ (((√‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ) → ((√‘𝑥) − 1) ∈ ℝ)
144	141, 142, 143	sylancl 586	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥) − 1) ∈ ℝ)
145	144	sqge0d 14042	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (((√‘𝑥) − 1)↑2))
146	130	sqsqrtd 15324	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥)↑2) = 𝑥)
147	146	oveq1d 7372	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥)↑2) − (2 · (√‘𝑥))) = (𝑥 − (2 · (√‘𝑥))))
148	147	oveq1d 7372	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1) = ((𝑥 − (2 · (√‘𝑥))) + 1))
149		binom2sub1 14124	. . . . . . . . 9 ⊢ ((√‘𝑥) ∈ ℂ → (((√‘𝑥) − 1)↑2) = ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1))
150	131, 149	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥) − 1)↑2) = ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1))
151	136	rpcnd 12959	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℂ)
152	130, 61, 151	addsubd 11533	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((𝑥 + 1) − (2 · (√‘𝑥))) = ((𝑥 − (2 · (√‘𝑥))) + 1))
153	148, 150, 152	3eqtr4d 2786	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥) − 1)↑2) = ((𝑥 + 1) − (2 · (√‘𝑥))))
154	145, 153	breqtrd 5131	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ ((𝑥 + 1) − (2 · (√‘𝑥))))
155	57	rpred 12957	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ)
156	136	rpred 12957	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ)
157	155, 156	subge0d 11745	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (0 ≤ ((𝑥 + 1) − (2 · (√‘𝑥))) ↔ (2 · (√‘𝑥)) ≤ (𝑥 + 1)))
158	154, 157	mpbid 231	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ≤ (𝑥 + 1))
159	136, 57	lerecd 12976	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) ≤ (𝑥 + 1) ↔ (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥)))))
160	158, 159	mpbid 231	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥))))
161	110, 160	syldan 591	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥))))
162		rexr 11201	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
163		0xr 11202	. . . . 5 ⊢ 0 ∈ ℝ^*
164		lbicc2 13381	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
165	163, 164	mp3an1 1448	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
166	162, 165	sylan 580	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
167		ubicc2 13382	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
168	163, 167	mp3an1 1448	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
169	162, 168	sylan 580	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
170		simpr 485	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ 𝐴)
171		fv0p1e1 12276	. . . 4 ⊢ (𝑥 = 0 → (log‘(𝑥 + 1)) = (log‘1))
172		log1 25941	. . . 4 ⊢ (log‘1) = 0
173	171, 172	eqtrdi 2792	. . 3 ⊢ (𝑥 = 0 → (log‘(𝑥 + 1)) = 0)
174		fveq2 6842	. . . 4 ⊢ (𝑥 = 0 → (√‘𝑥) = (√‘0))
175		sqrt0 15126	. . . 4 ⊢ (√‘0) = 0
176	174, 175	eqtrdi 2792	. . 3 ⊢ (𝑥 = 0 → (√‘𝑥) = 0)
177		fvoveq1 7380	. . 3 ⊢ (𝑥 = 𝐴 → (log‘(𝑥 + 1)) = (log‘(𝐴 + 1)))
178		fveq2 6842	. . 3 ⊢ (𝑥 = 𝐴 → (√‘𝑥) = (√‘𝐴))
179	2, 3, 51, 115, 128, 140, 161, 166, 169, 170, 173, 176, 177, 178	dvle 25371	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((log‘(𝐴 + 1)) − 0) ≤ ((√‘𝐴) − 0))
180		ge0p1rp 12946	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ⁺)
181	180	relogcld 25978	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ∈ ℝ)
182		resqrtcl 15138	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ)
183	181, 182, 2	lesub1d 11762	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((log‘(𝐴 + 1)) ≤ (√‘𝐴) ↔ ((log‘(𝐴 + 1)) − 0) ≤ ((√‘𝐴) − 0)))
184	179, 183	mpbird 256	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ≤ (√‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 {cpr 4588 class class class wbr 5105 ↦ cmpt 5188 ran crn 5634 ↾ cres 5635 ⟶wf 6492 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 +∞cpnf 11186 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 2c2 12208 ℝ⁺crp 12915 (,)cioo 13264 [,)cico 13266 [,]cicc 13267 ↑cexp 13967 √csqrt 15118 ↾_t crest 17302 TopOpenctopn 17303 topGenctg 17319 ℂ_fldccnfld 20796 TopOnctopon 22259 Cn ccn 22575 ×_t ctx 22911 –cn→ccncf 24239 D cdv 25227 logclog 25910

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-tan 15954 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-cxp 25913

This theorem is referenced by: rplogsumlem1 26832

W3C validator