loglesqrt - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > loglesqrt		Structured version Visualization version GIF version

Theorem loglesqrt 24499

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 10040	. . . 4 ⊢ 0 ∈ ℝ
2	1	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ ℝ)
3		simpl 473	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
4		elicc2 12238	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥 ∈ (0[,]𝐴) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴)))
5	1, 3, 4	sylancr 695	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴)))
6	5	biimpa 501	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴))
7	6	simp1d 1073	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → 𝑥 ∈ ℝ)
8	6	simp2d 1074	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → 0 ≤ 𝑥)
9	7, 8	ge0p1rpd 11902	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → (𝑥 + 1) ∈ ℝ⁺)
10	9	fvresd 6208	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → ((log ↾ ℝ⁺)‘(𝑥 + 1)) = (log‘(𝑥 + 1)))
11	10	mpteq2dva 4744	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) = (𝑥 ∈ (0[,]𝐴) ↦ (log‘(𝑥 + 1))))
12		eqid 2622	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	cnfldtopon 22586	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
14	7	ex 450	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) → 𝑥 ∈ ℝ))
15	14	ssrdv 3609	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ ℝ)
16		ax-resscn 9993	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
17	15, 16	syl6ss 3615	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ ℂ)
18		resttopon 20965	. . . . . . 7 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (0[,]𝐴) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) ∈ (TopOn‘(0[,]𝐴)))
19	13, 17, 18	sylancr 695	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) ∈ (TopOn‘(0[,]𝐴)))
20		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) = (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1))
21	9, 20	fmptd 6385	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)):(0[,]𝐴)⟶ℝ⁺)
22		rpssre 11843	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ ℝ
23	22, 16	sstri 3612	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℂ
24	12	addcn 22668	. . . . . . . . . . 11 ⊢ + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
25	24	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
26		ssid 3624	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
27		cncfmptid 22715	. . . . . . . . . . 11 ⊢ (((0[,]𝐴) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]𝐴) ↦ 𝑥) ∈ ((0[,]𝐴)–cn→ℂ))
28	17, 26, 27	sylancl 694	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ 𝑥) ∈ ((0[,]𝐴)–cn→ℂ))
29		1cnd 10056	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 1 ∈ ℂ)
30	26	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℂ ⊆ ℂ)
31		cncfmptc 22714	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (0[,]𝐴) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]𝐴) ↦ 1) ∈ ((0[,]𝐴)–cn→ℂ))
32	29, 17, 30, 31	syl3anc 1326	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ 1) ∈ ((0[,]𝐴)–cn→ℂ))
33	12, 25, 28, 32	cncfmpt2f 22717	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℂ))
34		cncffvrn 22701	. . . . . . . . 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⁺))
35	23, 33, 34	sylancr 695	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℝ⁺) ↔ (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)):(0[,]𝐴)⟶ℝ⁺))
36	21, 35	mpbird 247	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ ((0[,]𝐴)–cn→ℝ⁺))
37		eqid 2622	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) = ((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴))
38		eqid 2622	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ⁺) = ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)
39	12, 37, 38	cncfcn 22712	. . . . . . . 8 ⊢ (((0[,]𝐴) ⊆ ℂ ∧ ℝ⁺ ⊆ ℂ) → ((0[,]𝐴)–cn→ℝ⁺) = (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)))
40	17, 23, 39	sylancl 694	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((0[,]𝐴)–cn→ℝ⁺) = (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)))
41	36, 40	eleqtrd 2703	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (𝑥 + 1)) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ⁺)))
42		relogcn 24384	. . . . . . . 8 ⊢ (log ↾ ℝ⁺) ∈ (ℝ⁺–cn→ℝ)
43		eqid 2622	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
44	12, 38, 43	cncfcn 22712	. . . . . . . . 9 ⊢ ((ℝ⁺ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ⁺–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
45	23, 16, 44	mp2an 708	. . . . . . . 8 ⊢ (ℝ⁺–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
46	42, 45	eleqtri 2699	. . . . . . 7 ⊢ (log ↾ ℝ⁺) ∈ (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
47	46	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) ∈ (((TopOpen‘ℂ_fld) ↾_t ℝ⁺) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
48	19, 41, 47	cnmpt11f 21467	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
49	12, 37, 43	cncfcn 22712	. . . . . 6 ⊢ (((0[,]𝐴) ⊆ ℂ ∧ ℝ ⊆ ℂ) → ((0[,]𝐴)–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
50	17, 16, 49	sylancl 694	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((0[,]𝐴)–cn→ℝ) = (((TopOpen‘ℂ_fld) ↾_t (0[,]𝐴)) Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
51	48, 50	eleqtrrd 2704	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ ((log ↾ ℝ⁺)‘(𝑥 + 1))) ∈ ((0[,]𝐴)–cn→ℝ))
52	11, 51	eqeltrrd 2702	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (log‘(𝑥 + 1))) ∈ ((0[,]𝐴)–cn→ℝ))
53		reelprrecn 10028	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
54	53	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ ∈ {ℝ, ℂ})
55		simpr 477	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
56		1rp 11836	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
57		rpaddcl 11854	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
58	55, 56, 57	sylancl 694	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
59	58	relogcld 24369	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘(𝑥 + 1)) ∈ ℝ)
60	59	recnd 10068	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (log‘(𝑥 + 1)) ∈ ℂ)
61	58	rpreccld 11882	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ∈ ℝ⁺)
62		1cnd 10056	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
63		relogcl 24322	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ → (log‘𝑦) ∈ ℝ)
64	63	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℝ)
65	64	recnd 10068	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (log‘𝑦) ∈ ℂ)
66		rpreccl 11857	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → (1 / 𝑦) ∈ ℝ⁺)
67	66	adantl 482	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ⁺) → (1 / 𝑦) ∈ ℝ⁺)
68		peano2re 10209	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
69	68	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
70	69	recnd 10068	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℂ)
71		1cnd 10056	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
72	16	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ ⊆ ℂ)
73	72	sselda 3603	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
74	54	dvmptid 23720	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
75		0cnd 10033	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ)
76	54, 29	dvmptc 23721	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
77	54, 73, 71, 74, 71, 75, 76	dvmptadd 23723	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ ↦ (1 + 0)))
78		1p0e1 11133	. . . . . . . . 9 ⊢ (1 + 0) = 1
79	78	mpteq2i 4741	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ (1 + 0)) = (𝑥 ∈ ℝ ↦ 1)
80	77, 79	syl6eq 2672	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ ↦ 1))
81	22	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ⁺ ⊆ ℝ)
82	12	tgioo2 22606	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
83		ioorp 12251	. . . . . . . . 9 ⊢ (0(,)+∞) = ℝ⁺
84		iooretop 22569	. . . . . . . . 9 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
85	83, 84	eqeltrri 2698	. . . . . . . 8 ⊢ ℝ⁺ ∈ (topGen‘ran (,))
86	85	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ℝ⁺ ∈ (topGen‘ran (,)))
87	54, 70, 71, 80, 81, 82, 12, 86	dvmptres 23726	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 + 1))) = (𝑥 ∈ ℝ⁺ ↦ 1))
88		dvrelog 24383	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦))
89		relogf1o 24313	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
90		f1of 6137	. . . . . . . . . . 11 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
91	89, 90	mp1i 13	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
92	91	feqmptd 6249	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)))
93		fvres 6207	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑦) = (log‘𝑦))
94	93	mpteq2ia 4740	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑦)) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))
95	92, 94	syl6eq 2672	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log ↾ ℝ⁺) = (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦)))
96	95	oveq2d 6666	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))))
97	88, 96	syl5reqr 2671	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (log‘𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / 𝑦)))
98		fveq2 6191	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → (log‘𝑦) = (log‘(𝑥 + 1)))
99		oveq2 6658	. . . . . 6 ⊢ (𝑦 = (𝑥 + 1) → (1 / 𝑦) = (1 / (𝑥 + 1)))
100	54, 54, 58, 62, 65, 67, 87, 97, 98, 99	dvmptco 23735	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 / (𝑥 + 1)) · 1)))
101	61	rpcnd 11874	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ∈ ℂ)
102	101	mulid1d 10057	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((1 / (𝑥 + 1)) · 1) = (1 / (𝑥 + 1)))
103	102	mpteq2dva 4744	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ ℝ⁺ ↦ ((1 / (𝑥 + 1)) · 1)) = (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥 + 1))))
104	100, 103	eqtrd 2656	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥 + 1))))
105		ioossicc 12259	. . . . . . . . 9 ⊢ (0(,)𝐴) ⊆ (0[,]𝐴)
106	105	sseli 3599	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)𝐴) → 𝑥 ∈ (0[,]𝐴))
107	106, 7	sylan2 491	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 𝑥 ∈ ℝ)
108		eliooord 12233	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)𝐴) → (0 < 𝑥 ∧ 𝑥 < 𝐴))
109	108	simpld 475	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)𝐴) → 0 < 𝑥)
110	109	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 0 < 𝑥)
111	107, 110	elrpd 11869	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → 𝑥 ∈ ℝ⁺)
112	111	ex 450	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0(,)𝐴) → 𝑥 ∈ ℝ⁺))
113	112	ssrdv 3609	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0(,)𝐴) ⊆ ℝ⁺)
114		iooretop 22569	. . . . 5 ⊢ (0(,)𝐴) ∈ (topGen‘ran (,))
115	114	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0(,)𝐴) ∈ (topGen‘ran (,)))
116	54, 60, 61, 104, 113, 82, 12, 115	dvmptres 23726	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ (0(,)𝐴) ↦ (log‘(𝑥 + 1)))) = (𝑥 ∈ (0(,)𝐴) ↦ (1 / (𝑥 + 1))))
117		elrege0 12278	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
118	7, 8, 117	sylanbrc 698	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0[,]𝐴)) → 𝑥 ∈ (0[,)+∞))
119	118	ex 450	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) → 𝑥 ∈ (0[,)+∞)))
120	119	ssrdv 3609	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0[,]𝐴) ⊆ (0[,)+∞))
121	120	resabs1d 5428	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) = (√ ↾ (0[,]𝐴)))
122		sqrtf 14103	. . . . . . 7 ⊢ √:ℂ⟶ℂ
123	122	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → √:ℂ⟶ℂ)
124	123, 17	feqresmpt 6250	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√ ↾ (0[,]𝐴)) = (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)))
125	121, 124	eqtrd 2656	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) = (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)))
126		resqrtcn 24490	. . . . 5 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)
127		rescncf 22700	. . . . 5 ⊢ ((0[,]𝐴) ⊆ (0[,)+∞) → ((√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) ∈ ((0[,]𝐴)–cn→ℝ)))
128	120, 126, 127	mpisyl 21	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√ ↾ (0[,)+∞)) ↾ (0[,]𝐴)) ∈ ((0[,]𝐴)–cn→ℝ))
129	125, 128	eqeltrrd 2702	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑥 ∈ (0[,]𝐴) ↦ (√‘𝑥)) ∈ ((0[,]𝐴)–cn→ℝ))
130		rpcn 11841	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
131	130	adantl 482	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
132	131	sqrtcld 14176	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
133		2rp 11837	. . . . . 6 ⊢ 2 ∈ ℝ⁺
134		rpsqrtcl 14005	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
135	134	adantl 482	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
136		rpmulcl 11855	. . . . . 6 ⊢ ((2 ∈ ℝ⁺ ∧ (√‘𝑥) ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ⁺)
137	133, 135, 136	sylancr 695	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ⁺)
138	137	rpreccld 11882	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (2 · (√‘𝑥))) ∈ ℝ⁺)
139		dvsqrt 24483	. . . . 5 ⊢ (ℝ D (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / (2 · (√‘𝑥))))
140	139	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / (2 · (√‘𝑥)))))
141	54, 132, 138, 140, 113, 82, 12, 115	dvmptres 23726	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (ℝ D (𝑥 ∈ (0(,)𝐴) ↦ (√‘𝑥))) = (𝑥 ∈ (0(,)𝐴) ↦ (1 / (2 · (√‘𝑥)))))
142	135	rpred 11872	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
143		1re 10039	. . . . . . . . 9 ⊢ 1 ∈ ℝ
144		resubcl 10345	. . . . . . . . 9 ⊢ (((√‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ) → ((√‘𝑥) − 1) ∈ ℝ)
145	142, 143, 144	sylancl 694	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥) − 1) ∈ ℝ)
146	145	sqge0d 13036	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (((√‘𝑥) − 1)↑2))
147	131	sqsqrtd 14178	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥)↑2) = 𝑥)
148	147	oveq1d 6665	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥)↑2) − (2 · (√‘𝑥))) = (𝑥 − (2 · (√‘𝑥))))
149	148	oveq1d 6665	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1) = ((𝑥 − (2 · (√‘𝑥))) + 1))
150		binom2sub1 12982	. . . . . . . . 9 ⊢ ((√‘𝑥) ∈ ℂ → (((√‘𝑥) − 1)↑2) = ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1))
151	132, 150	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥) − 1)↑2) = ((((√‘𝑥)↑2) − (2 · (√‘𝑥))) + 1))
152	137	rpcnd 11874	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℂ)
153	131, 62, 152	addsubd 10413	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((𝑥 + 1) − (2 · (√‘𝑥))) = ((𝑥 − (2 · (√‘𝑥))) + 1))
154	149, 151, 153	3eqtr4d 2666	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (((√‘𝑥) − 1)↑2) = ((𝑥 + 1) − (2 · (√‘𝑥))))
155	146, 154	breqtrd 4679	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ ((𝑥 + 1) − (2 · (√‘𝑥))))
156	58	rpred 11872	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ)
157	137	rpred 11872	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℝ)
158	156, 157	subge0d 10617	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (0 ≤ ((𝑥 + 1) − (2 · (√‘𝑥))) ↔ (2 · (√‘𝑥)) ≤ (𝑥 + 1)))
159	155, 158	mpbid 222	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ≤ (𝑥 + 1))
160	137, 58	lerecd 11891	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) ≤ (𝑥 + 1) ↔ (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥)))))
161	159, 160	mpbid 222	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥))))
162	111, 161	syldan 487	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ (0(,)𝐴)) → (1 / (𝑥 + 1)) ≤ (1 / (2 · (√‘𝑥))))
163		rexr 10085	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
164		0xr 10086	. . . . 5 ⊢ 0 ∈ ℝ^*
165		lbicc2 12288	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
166	164, 165	mp3an1 1411	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
167	163, 166	sylan 488	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ (0[,]𝐴))
168		ubicc2 12289	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
169	164, 168	mp3an1 1411	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
170	163, 169	sylan 488	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
171		simpr 477	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ 𝐴)
172		oveq1 6657	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
173		0p1e1 11132	. . . . . 6 ⊢ (0 + 1) = 1
174	172, 173	syl6eq 2672	. . . . 5 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
175	174	fveq2d 6195	. . . 4 ⊢ (𝑥 = 0 → (log‘(𝑥 + 1)) = (log‘1))
176		log1 24332	. . . 4 ⊢ (log‘1) = 0
177	175, 176	syl6eq 2672	. . 3 ⊢ (𝑥 = 0 → (log‘(𝑥 + 1)) = 0)
178		fveq2 6191	. . . 4 ⊢ (𝑥 = 0 → (√‘𝑥) = (√‘0))
179		sqrt0 13982	. . . 4 ⊢ (√‘0) = 0
180	178, 179	syl6eq 2672	. . 3 ⊢ (𝑥 = 0 → (√‘𝑥) = 0)
181		oveq1 6657	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
182	181	fveq2d 6195	. . 3 ⊢ (𝑥 = 𝐴 → (log‘(𝑥 + 1)) = (log‘(𝐴 + 1)))
183		fveq2 6191	. . 3 ⊢ (𝑥 = 𝐴 → (√‘𝑥) = (√‘𝐴))
184	2, 3, 52, 116, 129, 141, 162, 167, 170, 171, 177, 180, 182, 183	dvle 23770	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((log‘(𝐴 + 1)) − 0) ≤ ((√‘𝐴) − 0))
185		ge0p1rp 11862	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ⁺)
186	185	relogcld 24369	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ∈ ℝ)
187		resqrtcl 13994	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ)
188	186, 187, 2	lesub1d 10634	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((log‘(𝐴 + 1)) ≤ (√‘𝐴) ↔ ((log‘(𝐴 + 1)) − 0) ≤ ((√‘𝐴) − 0)))
189	184, 188	mpbird 247	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ≤ (√‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 ↾ cres 5116 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 2c2 11070 ℝ⁺crp 11832 (,)cioo 12175 [,)cico 12177 [,]cicc 12178 ↑cexp 12860 √csqrt 13973 ↾_t crest 16081 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 TopOnctopon 20715 Cn ccn 21028 ×_t ctx 21363 –cn→ccncf 22679 D cdv 23627 logclog 24301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304

This theorem is referenced by: rplogsumlem1 25173

W3C validator