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Theorem loglesqrt 26671

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 11176	. . . 4 ⊢ 0 ∈ ℝ
2	1	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ ℝ)
3		simpl 482	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
4		elicc2 13372	. . . . . . . . . 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 476	. . . . . . . 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 13025	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → (𝑥 + 1) ∈ ℝ⁺)
10	9	fvresd 6878	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → ((log ↾ ℝ⁺)‘(𝑥 + 1)) = (log‘(𝑥 + 1)))
11	10	mpteq2dva 5200	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) = (𝑥 ∈ (0[,]𝐴) ↦ (log‘(𝑥 + 1))))
12		eqid 2729	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 24670	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	7	ex 412	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) → 𝑥 ∈ ℝ))
15	14	ssrdv 3952	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ ℝ)
16		ax-resscn 11125	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
17	15, 16	sstrdi 3959	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ ℂ)
18		resttopon 23048	. . . . . . 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 7087	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)):(0[,]𝐴)⟶ℝ⁺)
21		rpssre 12959	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ ℝ
22	21, 16	sstri 3956	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℂ
23	12	addcn 24754	. . . . . . . . . . 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 3969	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
26		cncfmptid 24806	. . . . . . . . . . 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 11169	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 1 ∈ ℂ)
29	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℂ ⊆ ℂ)
30		cncfmptc 24805	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (0[,]𝐴) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]𝐴) ↦ 1) ∈ ((0[,]𝐴)–cn→ℂ))
31	28, 17, 29, 30	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ 1) ∈ ((0[,]𝐴)–cn→ℂ))
32	12, 24, 27, 31	cncfmpt2f 24808	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℂ))
33		cncfcdm 24791	. . . . . . . . 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 257	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℝ⁺))
36		eqid 2729	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) = ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴))
37		eqid 2729	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ⁺) = ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)
38	12, 36, 37	cncfcn 24803	. . . . . . . 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 2830	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)))
41		relogcn 26547	. . . . . . . 8 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
42		eqid 2729	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
43	12, 37, 42	cncfcn 24803	. . . . . . . . 9 ⊢ ((ℝ⁺ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ⁺–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
44	22, 16, 43	mp2an 692	. . . . . . . 8 ⊢ (ℝ⁺–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
45	41, 44	eleqtri 2826	. . . . . . 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 23551	. . . . 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 24803	. . . . . 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 2831	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) ∈ ((0[,]𝐴)–cn→ℝ))
51	11, 50	eqeltrrd 2829	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (log‘(𝑥 + 1))) ∈ ((0[,]𝐴)–cn→ℝ))
52		reelprrecn 11160	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
53	52	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
54		simpr 484	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
55		1rp 12955	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
56		rpaddcl 12975	. . . . . . 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 26532	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘(𝑥 + 1)) ∈ ℝ)
59	58	recnd 11202	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘(𝑥 + 1)) ∈ ℂ)
60	57	rpreccld 13005	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ∈ ℝ⁺)
61		1cnd 11169	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
62		relogcl 26484	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ → (log‘𝑦) ∈ ℝ)
63	62	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℝ)
64	63	recnd 11202	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℂ)
65		rpreccl 12979	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → (1 / 𝑦) ∈ ℝ⁺)
66	65	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (1 / 𝑦) ∈ ℝ⁺)
67		peano2re 11347	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
68	67	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
69	68	recnd 11202	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℂ)
70		1cnd 11169	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
71	16	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ ⊆ ℂ)
72	71	sselda 3946	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
73	53	dvmptid 25861	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
74		0cnd 11167	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ)
75	53, 28	dvmptc 25862	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
76	53, 72, 70, 73, 70, 74, 75	dvmptadd 25864	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ ↦ (1 + 0)))
77		1p0e1 12305	. . . . . . . . 9 ⊢ (1 + 0) = 1
78	77	mpteq2i 5203	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ (1 + 0)) = (𝑥 ∈ ℝ ↦ 1)
79	76, 78	eqtrdi 2780	. . . . . . 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 24693	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
82		ioorp 13386	. . . . . . . . 9 ⊢ (0(,)+∞) = ℝ⁺
83		iooretop 24653	. . . . . . . . 9 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
84	82, 83	eqeltrri 2825	. . . . . . . 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 25867	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ⁺ ↦ 1))
87		relogf1o 26475	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
88		f1of 6800	. . . . . . . . . . 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 6929	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)))
91		fvres 6877	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑦) = (log‘𝑦))
92	91	mpteq2ia 5202	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))
93	90, 92	eqtrdi 2780	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦)))
94	93	oveq2d 7403	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))))
95		dvrelog 26546	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦))
96	94, 95	eqtr3di 2779	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦)))
97		fveq2 6858	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → (log‘𝑦) = (log‘(𝑥 + 1)))
98		oveq2 7395	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → (1 / 𝑦) = (1 / (𝑥 + 1)))
99	53, 53, 57, 61, 64, 66, 86, 96, 97, 98	dvmptco 25876	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 / (𝑥 + 1)) · 1)))
100	60	rpcnd 12997	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ∈ ℂ)
101	100	mulridd 11191	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((1 / (𝑥 + 1)) · 1) = (1 / (𝑥 + 1)))
102	101	mpteq2dva 5200	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ ℝ⁺ ↦ ((1 / (𝑥 + 1)) · 1)) = (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥 + 1))))
103	99, 102	eqtrd 2764	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥 + 1))))
104		ioossicc 13394	. . . . . . . . 9 ⊢ (0(,)𝐴) ⊆ (0[,]𝐴)
105	104	sseli 3942	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)𝐴) → 𝑥 ∈ (0[,]𝐴))
106	105, 7	sylan2 593	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 𝑥 ∈ ℝ)
107		eliooord 13366	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)𝐴) → (0 < 𝑥 ∧ 𝑥 < 𝐴))
108	107	simpld 494	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)𝐴) → 0 < 𝑥)
109	108	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 0 < 𝑥)
110	106, 109	elrpd 12992	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 𝑥 ∈ ℝ⁺)
111	110	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0(,)𝐴) → 𝑥 ∈ ℝ⁺))
112	111	ssrdv 3952	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0(,)𝐴) ⊆ ℝ⁺)
113		iooretop 24653	. . . . 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 25867	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ (0(,)𝐴) ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ (0(,)𝐴) ↦ (1 / (𝑥 + 1))))
116		elrege0 13415	. . . . . . . . 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 412	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) → 𝑥 ∈ (0[,)+∞)))
119	118	ssrdv 3952	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ (0[,)+∞))
120	119	resabs1d 5979	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) = (√ ↾ (0[,]𝐴)))
121		sqrtf 15330	. . . . . . 7 ⊢ √:ℂ⟶ℂ
122	121	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → √:ℂ⟶ℂ)
123	122, 17	feqresmpt 6930	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√ ↾ (0[,]𝐴)) = (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)))
124	120, 123	eqtrd 2764	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) = (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)))
125		resqrtcn 26659	. . . . 5 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)
126		rescncf 24790	. . . . 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 2829	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)) ∈ ((0[,]𝐴)–cn→ℝ))
129		rpcn 12962	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
130	129	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
131	130	sqrtcld 15406	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
132		2rp 12956	. . . . . 6 ⊢ 2 ∈ ℝ⁺
133		rpsqrtcl 15230	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
134	133	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
135		rpmulcl 12976	. . . . . 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 13005	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (2 · (√‘𝑥))) ∈ ℝ⁺)
138		dvsqrt 26651	. . . . 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 25867	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ (0(,)𝐴) ↦ (√‘𝑥))) = (𝑥 ∈ (0(,)𝐴) ↦ (1 / (2 · (√‘𝑥)))))
141	134	rpred 12995	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
142		1re 11174	. . . . . . . . 9 ⊢ 1 ∈ ℝ
143		resubcl 11486	. . . . . . . . 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 14102	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (((√‘𝑥) − 1)↑2))
146	130	sqsqrtd 15408	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥)↑2) = 𝑥)
147	146	oveq1d 7402	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥)↑2) − (2 · (√‘𝑥))) = (𝑥 − (2 · (√‘𝑥))))
148	147	oveq1d 7402	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1) = ((𝑥 − (2 · (√‘𝑥))) + 1))
149		binom2sub1 14186	. . . . . . . . 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 12997	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℂ)
152	130, 61, 151	addsubd 11554	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((𝑥 + 1) − (2 · (√‘𝑥))) = ((𝑥 − (2 · (√‘𝑥))) + 1))
153	148, 150, 152	3eqtr4d 2774	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥) − 1)↑2) = ((𝑥 + 1) − (2 · (√‘𝑥))))
154	145, 153	breqtrd 5133	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ ((𝑥 + 1) − (2 · (√‘𝑥))))
155	57	rpred 12995	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ)
156	136	rpred 12995	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ)
157	155, 156	subge0d 11768	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (0 ≤ ((𝑥 + 1) − (2 · (√‘𝑥))) ↔ (2 · (√‘𝑥)) ≤ (𝑥 + 1)))
158	154, 157	mpbid 232	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ≤ (𝑥 + 1))
159	136, 57	lerecd 13014	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) ≤ (𝑥 + 1) ↔ (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥)))))
160	158, 159	mpbid 232	. . . 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 11220	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
163		0xr 11221	. . . . 5 ⊢ 0 ∈ ℝ^*
164		lbicc2 13425	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
165	163, 164	mp3an1 1450	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
166	162, 165	sylan 580	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
167		ubicc2 13426	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
168	163, 167	mp3an1 1450	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
169	162, 168	sylan 580	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
170		simpr 484	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ 𝐴)
171		fv0p1e1 12304	. . . 4 ⊢ (𝑥 = 0 → (log‘(𝑥 + 1)) = (log‘1))
172		log1 26494	. . . 4 ⊢ (log‘1) = 0
173	171, 172	eqtrdi 2780	. . 3 ⊢ (𝑥 = 0 → (log‘(𝑥 + 1)) = 0)
174		fveq2 6858	. . . 4 ⊢ (𝑥 = 0 → (√‘𝑥) = (√‘0))
175		sqrt0 15207	. . . 4 ⊢ (√‘0) = 0
176	174, 175	eqtrdi 2780	. . 3 ⊢ (𝑥 = 0 → (√‘𝑥) = 0)
177		fvoveq1 7410	. . 3 ⊢ (𝑥 = 𝐴 → (log‘(𝑥 + 1)) = (log‘(𝐴 + 1)))
178		fveq2 6858	. . 3 ⊢ (𝑥 = 𝐴 → (√‘𝑥) = (√‘𝐴))
179	2, 3, 51, 115, 128, 140, 161, 166, 169, 170, 173, 176, 177, 178	dvle 25912	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((log‘(𝐴 + 1)) − 0) ≤ ((√‘𝐴) − 0))
180		ge0p1rp 12984	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ⁺)
181	180	relogcld 26532	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ∈ ℝ)
182		resqrtcl 15219	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ)
183	181, 182, 2	lesub1d 11785	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((log‘(𝐴 + 1)) ≤ (√‘𝐴) ↔ ((log‘(𝐴 + 1)) − 0) ≤ ((√‘𝐴) − 0)))
184	179, 183	mpbird 257	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ≤ (√‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ⊆ wss 3914 {cpr 4591 class class class wbr 5107 ↦ cmpt 5188 ran crn 5639 ↾ cres 5640 ⟶wf 6507 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 2c2 12241 ℝ⁺crp 12951 (,)cioo 13306 [,)cico 13308 [,]cicc 13309 ↑cexp 14026 √csqrt 15199 ↾_t crest 17383 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21264 TopOnctopon 22797 Cn ccn 23111 ×_t ctx 23447 –cn→ccncf 24769 D cdv 25764 logclog 26463

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-tan 16037 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-cxp 26466

This theorem is referenced by: rplogsumlem1 27395

W3C validator