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

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ 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   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   RR+crp 11832   (,)cioo 12175   [,)cico 12177   [,]cicc 12178   ^cexp 12860   sqrcsqrt 13973   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746  TopOnctopon 20715   Cn ccn 21028   tX 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