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Theorem loglesqrt 26822

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 11292	. . . 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 13472	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥 ∈ (0[,]𝐴) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴)))
5	1, 3, 4	sylancr 586	. . . . . . . . 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 13129	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → (𝑥 + 1) ∈ ℝ⁺)
10	9	fvresd 6940	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → ((log ↾ ℝ⁺)‘(𝑥 + 1)) = (log‘(𝑥 + 1)))
11	10	mpteq2dva 5266	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) = (𝑥 ∈ (0[,]𝐴) ↦ (log‘(𝑥 + 1))))
12		eqid 2740	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 24824	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	7	ex 412	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) → 𝑥 ∈ ℝ))
15	14	ssrdv 4014	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ ℝ)
16		ax-resscn 11241	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
17	15, 16	sstrdi 4021	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ ℂ)
18		resttopon 23190	. . . . . . 7 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (0[,]𝐴) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) ∈ (TopOn‘(0[,]𝐴)))
19	13, 17, 18	sylancr 586	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) ∈ (TopOn‘(0[,]𝐴)))
20	9	fmpttd 7149	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)):(0[,]𝐴)⟶ℝ⁺)
21		rpssre 13064	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ ℝ
22	21, 16	sstri 4018	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℂ
23	12	addcn 24906	. . . . . . . . . . 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 4031	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
26		cncfmptid 24958	. . . . . . . . . . 11 ⊢ (((0[,]𝐴) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]𝐴) ↦ 𝑥) ∈ ((0[,]𝐴)–cn→ℂ))
27	17, 25, 26	sylancl 585	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ 𝑥) ∈ ((0[,]𝐴)–cn→ℂ))
28		1cnd 11285	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 1 ∈ ℂ)
29	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℂ ⊆ ℂ)
30		cncfmptc 24957	. . . . . . . . . . 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 24960	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℂ))
33		cncfcdm 24943	. . . . . . . . 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 586	. . . . . . . 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 2740	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) = ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴))
37		eqid 2740	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ⁺) = ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)
38	12, 36, 37	cncfcn 24955	. . . . . . . 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 585	. . . . . . 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 2846	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)))
41		relogcn 26698	. . . . . . . 8 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
42		eqid 2740	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
43	12, 37, 42	cncfcn 24955	. . . . . . . . 9 ⊢ ((ℝ⁺ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ⁺–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
44	22, 16, 43	mp2an 691	. . . . . . . 8 ⊢ (ℝ⁺–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
45	41, 44	eleqtri 2842	. . . . . . 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 23693	. . . . 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 24955	. . . . . 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 585	. . . . 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 2847	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) ∈ ((0[,]𝐴)–cn→ℝ))
51	11, 50	eqeltrrd 2845	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (log‘(𝑥 + 1))) ∈ ((0[,]𝐴)–cn→ℝ))
52		reelprrecn 11276	. . . . 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 13061	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
56		rpaddcl 13079	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
57	54, 55, 56	sylancl 585	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
58	57	relogcld 26683	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘(𝑥 + 1)) ∈ ℝ)
59	58	recnd 11318	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘(𝑥 + 1)) ∈ ℂ)
60	57	rpreccld 13109	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ∈ ℝ⁺)
61		1cnd 11285	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
62		relogcl 26635	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ → (log‘𝑦) ∈ ℝ)
63	62	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℝ)
64	63	recnd 11318	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℂ)
65		rpreccl 13083	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → (1 / 𝑦) ∈ ℝ⁺)
66	65	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (1 / 𝑦) ∈ ℝ⁺)
67		peano2re 11463	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
68	67	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
69	68	recnd 11318	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℂ)
70		1cnd 11285	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
71	16	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ ⊆ ℂ)
72	71	sselda 4008	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
73	53	dvmptid 26015	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
74		0cnd 11283	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ)
75	53, 28	dvmptc 26016	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
76	53, 72, 70, 73, 70, 74, 75	dvmptadd 26018	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ ↦ (1 + 0)))
77		1p0e1 12417	. . . . . . . . 9 ⊢ (1 + 0) = 1
78	77	mpteq2i 5271	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ (1 + 0)) = (𝑥 ∈ ℝ ↦ 1)
79	76, 78	eqtrdi 2796	. . . . . . 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 24844	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
82		ioorp 13485	. . . . . . . . 9 ⊢ (0(,)+∞) = ℝ⁺
83		iooretop 24807	. . . . . . . . 9 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
84	82, 83	eqeltrri 2841	. . . . . . . 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 26021	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ⁺ ↦ 1))
87		relogf1o 26626	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
88		f1of 6862	. . . . . . . . . . 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 6990	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)))
91		fvres 6939	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑦) = (log‘𝑦))
92	91	mpteq2ia 5269	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))
93	90, 92	eqtrdi 2796	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦)))
94	93	oveq2d 7464	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))))
95		dvrelog 26697	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦))
96	94, 95	eqtr3di 2795	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦)))
97		fveq2 6920	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → (log‘𝑦) = (log‘(𝑥 + 1)))
98		oveq2 7456	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → (1 / 𝑦) = (1 / (𝑥 + 1)))
99	53, 53, 57, 61, 64, 66, 86, 96, 97, 98	dvmptco 26030	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 / (𝑥 + 1)) · 1)))
100	60	rpcnd 13101	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ∈ ℂ)
101	100	mulridd 11307	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((1 / (𝑥 + 1)) · 1) = (1 / (𝑥 + 1)))
102	101	mpteq2dva 5266	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ ℝ⁺ ↦ ((1 / (𝑥 + 1)) · 1)) = (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥 + 1))))
103	99, 102	eqtrd 2780	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥 + 1))))
104		ioossicc 13493	. . . . . . . . 9 ⊢ (0(,)𝐴) ⊆ (0[,]𝐴)
105	104	sseli 4004	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)𝐴) → 𝑥 ∈ (0[,]𝐴))
106	105, 7	sylan2 592	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 𝑥 ∈ ℝ)
107		eliooord 13466	. . . . . . . . 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 13096	. . . . . 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 4014	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0(,)𝐴) ⊆ ℝ⁺)
113		iooretop 24807	. . . . 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 26021	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ (0(,)𝐴) ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ (0(,)𝐴) ↦ (1 / (𝑥 + 1))))
116		elrege0 13514	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
117	7, 8, 116	sylanbrc 582	. . . . . . . 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 4014	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ (0[,)+∞))
120	119	resabs1d 6037	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) = (√ ↾ (0[,]𝐴)))
121		sqrtf 15412	. . . . . . 7 ⊢ √:ℂ⟶ℂ
122	121	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → √:ℂ⟶ℂ)
123	122, 17	feqresmpt 6991	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√ ↾ (0[,]𝐴)) = (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)))
124	120, 123	eqtrd 2780	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) = (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)))
125		resqrtcn 26810	. . . . 5 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)
126		rescncf 24942	. . . . 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 2845	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)) ∈ ((0[,]𝐴)–cn→ℝ))
129		rpcn 13067	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
130	129	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
131	130	sqrtcld 15486	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
132		2rp 13062	. . . . . 6 ⊢ 2 ∈ ℝ⁺
133		rpsqrtcl 15313	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
134	133	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
135		rpmulcl 13080	. . . . . 6 ⊢ ((2 ∈ ℝ⁺ ∧ (√‘𝑥) ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ⁺)
136	132, 134, 135	sylancr 586	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ⁺)
137	136	rpreccld 13109	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (2 · (√‘𝑥))) ∈ ℝ⁺)
138		dvsqrt 26802	. . . . 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 26021	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ (0(,)𝐴) ↦ (√‘𝑥))) = (𝑥 ∈ (0(,)𝐴) ↦ (1 / (2 · (√‘𝑥)))))
141	134	rpred 13099	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
142		1re 11290	. . . . . . . . 9 ⊢ 1 ∈ ℝ
143		resubcl 11600	. . . . . . . . 9 ⊢ (((√‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ) → ((√‘𝑥) − 1) ∈ ℝ)
144	141, 142, 143	sylancl 585	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥) − 1) ∈ ℝ)
145	144	sqge0d 14187	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (((√‘𝑥) − 1)↑2))
146	130	sqsqrtd 15488	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥)↑2) = 𝑥)
147	146	oveq1d 7463	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥)↑2) − (2 · (√‘𝑥))) = (𝑥 − (2 · (√‘𝑥))))
148	147	oveq1d 7463	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1) = ((𝑥 − (2 · (√‘𝑥))) + 1))
149		binom2sub1 14270	. . . . . . . . 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 13101	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℂ)
152	130, 61, 151	addsubd 11668	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((𝑥 + 1) − (2 · (√‘𝑥))) = ((𝑥 − (2 · (√‘𝑥))) + 1))
153	148, 150, 152	3eqtr4d 2790	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥) − 1)↑2) = ((𝑥 + 1) − (2 · (√‘𝑥))))
154	145, 153	breqtrd 5192	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ ((𝑥 + 1) − (2 · (√‘𝑥))))
155	57	rpred 13099	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ)
156	136	rpred 13099	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ)
157	155, 156	subge0d 11880	. . . . . 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 13118	. . . . 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 590	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥))))
162		rexr 11336	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
163		0xr 11337	. . . . 5 ⊢ 0 ∈ ℝ^*
164		lbicc2 13524	. . . . 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 579	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
167		ubicc2 13525	. . . . 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 579	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
170		simpr 484	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ 𝐴)
171		fv0p1e1 12416	. . . 4 ⊢ (𝑥 = 0 → (log‘(𝑥 + 1)) = (log‘1))
172		log1 26645	. . . 4 ⊢ (log‘1) = 0
173	171, 172	eqtrdi 2796	. . 3 ⊢ (𝑥 = 0 → (log‘(𝑥 + 1)) = 0)
174		fveq2 6920	. . . 4 ⊢ (𝑥 = 0 → (√‘𝑥) = (√‘0))
175		sqrt0 15290	. . . 4 ⊢ (√‘0) = 0
176	174, 175	eqtrdi 2796	. . 3 ⊢ (𝑥 = 0 → (√‘𝑥) = 0)
177		fvoveq1 7471	. . 3 ⊢ (𝑥 = 𝐴 → (log‘(𝑥 + 1)) = (log‘(𝐴 + 1)))
178		fveq2 6920	. . 3 ⊢ (𝑥 = 𝐴 → (√‘𝑥) = (√‘𝐴))
179	2, 3, 51, 115, 128, 140, 161, 166, 169, 170, 173, 176, 177, 178	dvle 26066	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((log‘(𝐴 + 1)) − 0) ≤ ((√‘𝐴) − 0))
180		ge0p1rp 13088	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ⁺)
181	180	relogcld 26683	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ∈ ℝ)
182		resqrtcl 15302	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ)
183	181, 182, 2	lesub1d 11897	. 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 1087 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 {cpr 4650 class class class wbr 5166 ↦ cmpt 5249 ran crn 5701 ↾ cres 5702 ⟶wf 6569 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 +∞cpnf 11321 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 2c2 12348 ℝ⁺crp 13057 (,)cioo 13407 [,)cico 13409 [,]cicc 13410 ↑cexp 14112 √csqrt 15282 ↾_t crest 17480 TopOpenctopn 17481 topGenctg 17497 ℂ_fldccnfld 21387 TopOnctopon 22937 Cn ccn 23253 ×_t ctx 23589 –cn→ccncf 24921 D cdv 25918 logclog 26614

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-tan 16119 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-cxp 26617

This theorem is referenced by: rplogsumlem1 27546

W3C validator