Step |
Hyp |
Ref |
Expression |
1 |
|
0re |
|- 0 e. RR |
2 |
1
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> 0 e. RR ) |
3 |
|
simpl |
|- ( ( A e. RR /\ 0 <_ A ) -> A e. RR ) |
4 |
|
elicc2 |
|- ( ( 0 e. RR /\ A e. RR ) -> ( x e. ( 0 [,] A ) <-> ( x e. RR /\ 0 <_ x /\ x <_ A ) ) ) |
5 |
1 3 4
|
sylancr |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) <-> ( x e. RR /\ 0 <_ x /\ x <_ A ) ) ) |
6 |
5
|
biimpa |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. ( 0 [,] A ) ) -> ( x e. RR /\ 0 <_ x /\ x <_ A ) ) |
7 |
6
|
simp1d |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. ( 0 [,] A ) ) -> x e. RR ) |
8 |
6
|
simp2d |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. ( 0 [,] A ) ) -> 0 <_ x ) |
9 |
7 8
|
ge0p1rpd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. ( 0 [,] A ) ) -> ( x + 1 ) e. RR+ ) |
10 |
9
|
fvresd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. ( 0 [,] A ) ) -> ( ( log |` RR+ ) ` ( x + 1 ) ) = ( log ` ( x + 1 ) ) ) |
11 |
10
|
mpteq2dva |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> ( ( log |` RR+ ) ` ( x + 1 ) ) ) = ( x e. ( 0 [,] A ) |-> ( log ` ( x + 1 ) ) ) ) |
12 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
13 |
12
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
14 |
7
|
ex |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) -> x e. RR ) ) |
15 |
14
|
ssrdv |
|- ( ( A e. RR /\ 0 <_ A ) -> ( 0 [,] A ) C_ RR ) |
16 |
|
ax-resscn |
|- RR C_ CC |
17 |
15 16
|
sstrdi |
|- ( ( A e. RR /\ 0 <_ A ) -> ( 0 [,] A ) C_ CC ) |
18 |
|
resttopon |
|- ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ ( 0 [,] A ) C_ CC ) -> ( ( TopOpen ` CCfld ) |`t ( 0 [,] A ) ) e. ( TopOn ` ( 0 [,] A ) ) ) |
19 |
13 17 18
|
sylancr |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( TopOpen ` CCfld ) |`t ( 0 [,] A ) ) e. ( TopOn ` ( 0 [,] A ) ) ) |
20 |
9
|
fmpttd |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> ( x + 1 ) ) : ( 0 [,] A ) --> RR+ ) |
21 |
|
rpssre |
|- RR+ C_ RR |
22 |
21 16
|
sstri |
|- RR+ C_ CC |
23 |
12
|
addcn |
|- + e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) |
24 |
23
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> + e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) ) |
25 |
|
ssid |
|- CC C_ CC |
26 |
|
cncfmptid |
|- ( ( ( 0 [,] A ) C_ CC /\ CC C_ CC ) -> ( x e. ( 0 [,] A ) |-> x ) e. ( ( 0 [,] A ) -cn-> CC ) ) |
27 |
17 25 26
|
sylancl |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> x ) e. ( ( 0 [,] A ) -cn-> CC ) ) |
28 |
|
1cnd |
|- ( ( A e. RR /\ 0 <_ A ) -> 1 e. CC ) |
29 |
25
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> CC C_ CC ) |
30 |
|
cncfmptc |
|- ( ( 1 e. CC /\ ( 0 [,] A ) C_ CC /\ CC C_ CC ) -> ( x e. ( 0 [,] A ) |-> 1 ) e. ( ( 0 [,] A ) -cn-> CC ) ) |
31 |
28 17 29 30
|
syl3anc |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> 1 ) e. ( ( 0 [,] A ) -cn-> CC ) ) |
32 |
12 24 27 31
|
cncfmpt2f |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> ( x + 1 ) ) e. ( ( 0 [,] A ) -cn-> CC ) ) |
33 |
|
cncffvrn |
|- ( ( 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+ ) ) |
34 |
22 32 33
|
sylancr |
|- ( ( 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+ ) ) |
35 |
20 34
|
mpbird |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> ( x + 1 ) ) e. ( ( 0 [,] A ) -cn-> RR+ ) ) |
36 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t ( 0 [,] A ) ) = ( ( TopOpen ` CCfld ) |`t ( 0 [,] A ) ) |
37 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t RR+ ) = ( ( TopOpen ` CCfld ) |`t RR+ ) |
38 |
12 36 37
|
cncfcn |
|- ( ( ( 0 [,] A ) C_ CC /\ RR+ C_ CC ) -> ( ( 0 [,] A ) -cn-> RR+ ) = ( ( ( TopOpen ` CCfld ) |`t ( 0 [,] A ) ) Cn ( ( TopOpen ` CCfld ) |`t RR+ ) ) ) |
39 |
17 22 38
|
sylancl |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( 0 [,] A ) -cn-> RR+ ) = ( ( ( TopOpen ` CCfld ) |`t ( 0 [,] A ) ) Cn ( ( TopOpen ` CCfld ) |`t RR+ ) ) ) |
40 |
35 39
|
eleqtrd |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> ( x + 1 ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( 0 [,] A ) ) Cn ( ( TopOpen ` CCfld ) |`t RR+ ) ) ) |
41 |
|
relogcn |
|- ( log |` RR+ ) e. ( RR+ -cn-> RR ) |
42 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t RR ) = ( ( TopOpen ` CCfld ) |`t RR ) |
43 |
12 37 42
|
cncfcn |
|- ( ( RR+ C_ CC /\ RR C_ CC ) -> ( RR+ -cn-> RR ) = ( ( ( TopOpen ` CCfld ) |`t RR+ ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) |
44 |
22 16 43
|
mp2an |
|- ( RR+ -cn-> RR ) = ( ( ( TopOpen ` CCfld ) |`t RR+ ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) |
45 |
41 44
|
eleqtri |
|- ( log |` RR+ ) e. ( ( ( TopOpen ` CCfld ) |`t RR+ ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) |
46 |
45
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> ( log |` RR+ ) e. ( ( ( TopOpen ` CCfld ) |`t RR+ ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) |
47 |
19 40 46
|
cnmpt11f |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> ( ( log |` RR+ ) ` ( x + 1 ) ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( 0 [,] A ) ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) |
48 |
12 36 42
|
cncfcn |
|- ( ( ( 0 [,] A ) C_ CC /\ RR C_ CC ) -> ( ( 0 [,] A ) -cn-> RR ) = ( ( ( TopOpen ` CCfld ) |`t ( 0 [,] A ) ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) |
49 |
17 16 48
|
sylancl |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( 0 [,] A ) -cn-> RR ) = ( ( ( TopOpen ` CCfld ) |`t ( 0 [,] A ) ) Cn ( ( TopOpen ` CCfld ) |`t RR ) ) ) |
50 |
47 49
|
eleqtrrd |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> ( ( log |` RR+ ) ` ( x + 1 ) ) ) e. ( ( 0 [,] A ) -cn-> RR ) ) |
51 |
11 50
|
eqeltrrd |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> ( log ` ( x + 1 ) ) ) e. ( ( 0 [,] A ) -cn-> RR ) ) |
52 |
|
reelprrecn |
|- RR e. { RR , CC } |
53 |
52
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> RR e. { RR , CC } ) |
54 |
|
simpr |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> x e. RR+ ) |
55 |
|
1rp |
|- 1 e. RR+ |
56 |
|
rpaddcl |
|- ( ( x e. RR+ /\ 1 e. RR+ ) -> ( x + 1 ) e. RR+ ) |
57 |
54 55 56
|
sylancl |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( x + 1 ) e. RR+ ) |
58 |
57
|
relogcld |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( log ` ( x + 1 ) ) e. RR ) |
59 |
58
|
recnd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( log ` ( x + 1 ) ) e. CC ) |
60 |
57
|
rpreccld |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( 1 / ( x + 1 ) ) e. RR+ ) |
61 |
|
1cnd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> 1 e. CC ) |
62 |
|
relogcl |
|- ( y e. RR+ -> ( log ` y ) e. RR ) |
63 |
62
|
adantl |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ y e. RR+ ) -> ( log ` y ) e. RR ) |
64 |
63
|
recnd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ y e. RR+ ) -> ( log ` y ) e. CC ) |
65 |
|
rpreccl |
|- ( y e. RR+ -> ( 1 / y ) e. RR+ ) |
66 |
65
|
adantl |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ y e. RR+ ) -> ( 1 / y ) e. RR+ ) |
67 |
|
peano2re |
|- ( x e. RR -> ( x + 1 ) e. RR ) |
68 |
67
|
adantl |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR ) -> ( x + 1 ) e. RR ) |
69 |
68
|
recnd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR ) -> ( x + 1 ) e. CC ) |
70 |
|
1cnd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR ) -> 1 e. CC ) |
71 |
16
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> RR C_ CC ) |
72 |
71
|
sselda |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR ) -> x e. CC ) |
73 |
53
|
dvmptid |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( x e. RR |-> x ) ) = ( x e. RR |-> 1 ) ) |
74 |
|
0cnd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR ) -> 0 e. CC ) |
75 |
53 28
|
dvmptc |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( x e. RR |-> 1 ) ) = ( x e. RR |-> 0 ) ) |
76 |
53 72 70 73 70 74 75
|
dvmptadd |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( x e. RR |-> ( x + 1 ) ) ) = ( x e. RR |-> ( 1 + 0 ) ) ) |
77 |
|
1p0e1 |
|- ( 1 + 0 ) = 1 |
78 |
77
|
mpteq2i |
|- ( x e. RR |-> ( 1 + 0 ) ) = ( x e. RR |-> 1 ) |
79 |
76 78
|
eqtrdi |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( x e. RR |-> ( x + 1 ) ) ) = ( x e. RR |-> 1 ) ) |
80 |
21
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> RR+ C_ RR ) |
81 |
12
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
82 |
|
ioorp |
|- ( 0 (,) +oo ) = RR+ |
83 |
|
iooretop |
|- ( 0 (,) +oo ) e. ( topGen ` ran (,) ) |
84 |
82 83
|
eqeltrri |
|- RR+ e. ( topGen ` ran (,) ) |
85 |
84
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> RR+ e. ( topGen ` ran (,) ) ) |
86 |
53 69 70 79 80 81 12 85
|
dvmptres |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( x e. RR+ |-> ( x + 1 ) ) ) = ( x e. RR+ |-> 1 ) ) |
87 |
|
relogf1o |
|- ( log |` RR+ ) : RR+ -1-1-onto-> RR |
88 |
|
f1of |
|- ( ( log |` RR+ ) : RR+ -1-1-onto-> RR -> ( log |` RR+ ) : RR+ --> RR ) |
89 |
87 88
|
mp1i |
|- ( ( A e. RR /\ 0 <_ A ) -> ( log |` RR+ ) : RR+ --> RR ) |
90 |
89
|
feqmptd |
|- ( ( A e. RR /\ 0 <_ A ) -> ( log |` RR+ ) = ( y e. RR+ |-> ( ( log |` RR+ ) ` y ) ) ) |
91 |
|
fvres |
|- ( y e. RR+ -> ( ( log |` RR+ ) ` y ) = ( log ` y ) ) |
92 |
91
|
mpteq2ia |
|- ( y e. RR+ |-> ( ( log |` RR+ ) ` y ) ) = ( y e. RR+ |-> ( log ` y ) ) |
93 |
90 92
|
eqtrdi |
|- ( ( A e. RR /\ 0 <_ A ) -> ( log |` RR+ ) = ( y e. RR+ |-> ( log ` y ) ) ) |
94 |
93
|
oveq2d |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( log |` RR+ ) ) = ( RR _D ( y e. RR+ |-> ( log ` y ) ) ) ) |
95 |
|
dvrelog |
|- ( RR _D ( log |` RR+ ) ) = ( y e. RR+ |-> ( 1 / y ) ) |
96 |
94 95
|
eqtr3di |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( y e. RR+ |-> ( log ` y ) ) ) = ( y e. RR+ |-> ( 1 / y ) ) ) |
97 |
|
fveq2 |
|- ( y = ( x + 1 ) -> ( log ` y ) = ( log ` ( x + 1 ) ) ) |
98 |
|
oveq2 |
|- ( y = ( x + 1 ) -> ( 1 / y ) = ( 1 / ( x + 1 ) ) ) |
99 |
53 53 57 61 64 66 86 96 97 98
|
dvmptco |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( x e. RR+ |-> ( log ` ( x + 1 ) ) ) ) = ( x e. RR+ |-> ( ( 1 / ( x + 1 ) ) x. 1 ) ) ) |
100 |
60
|
rpcnd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( 1 / ( x + 1 ) ) e. CC ) |
101 |
100
|
mulid1d |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( ( 1 / ( x + 1 ) ) x. 1 ) = ( 1 / ( x + 1 ) ) ) |
102 |
101
|
mpteq2dva |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. RR+ |-> ( ( 1 / ( x + 1 ) ) x. 1 ) ) = ( x e. RR+ |-> ( 1 / ( x + 1 ) ) ) ) |
103 |
99 102
|
eqtrd |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( x e. RR+ |-> ( log ` ( x + 1 ) ) ) ) = ( x e. RR+ |-> ( 1 / ( x + 1 ) ) ) ) |
104 |
|
ioossicc |
|- ( 0 (,) A ) C_ ( 0 [,] A ) |
105 |
104
|
sseli |
|- ( x e. ( 0 (,) A ) -> x e. ( 0 [,] A ) ) |
106 |
105 7
|
sylan2 |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. ( 0 (,) A ) ) -> x e. RR ) |
107 |
|
eliooord |
|- ( x e. ( 0 (,) A ) -> ( 0 < x /\ x < A ) ) |
108 |
107
|
simpld |
|- ( x e. ( 0 (,) A ) -> 0 < x ) |
109 |
108
|
adantl |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. ( 0 (,) A ) ) -> 0 < x ) |
110 |
106 109
|
elrpd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. ( 0 (,) A ) ) -> x e. RR+ ) |
111 |
110
|
ex |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 (,) A ) -> x e. RR+ ) ) |
112 |
111
|
ssrdv |
|- ( ( A e. RR /\ 0 <_ A ) -> ( 0 (,) A ) C_ RR+ ) |
113 |
|
iooretop |
|- ( 0 (,) A ) e. ( topGen ` ran (,) ) |
114 |
113
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> ( 0 (,) A ) e. ( topGen ` ran (,) ) ) |
115 |
53 59 60 103 112 81 12 114
|
dvmptres |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( x e. ( 0 (,) A ) |-> ( log ` ( x + 1 ) ) ) ) = ( x e. ( 0 (,) A ) |-> ( 1 / ( x + 1 ) ) ) ) |
116 |
|
elrege0 |
|- ( x e. ( 0 [,) +oo ) <-> ( x e. RR /\ 0 <_ x ) ) |
117 |
7 8 116
|
sylanbrc |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. ( 0 [,] A ) ) -> x e. ( 0 [,) +oo ) ) |
118 |
117
|
ex |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) -> x e. ( 0 [,) +oo ) ) ) |
119 |
118
|
ssrdv |
|- ( ( A e. RR /\ 0 <_ A ) -> ( 0 [,] A ) C_ ( 0 [,) +oo ) ) |
120 |
119
|
resabs1d |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt |` ( 0 [,) +oo ) ) |` ( 0 [,] A ) ) = ( sqrt |` ( 0 [,] A ) ) ) |
121 |
|
sqrtf |
|- sqrt : CC --> CC |
122 |
121
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> sqrt : CC --> CC ) |
123 |
122 17
|
feqresmpt |
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt |` ( 0 [,] A ) ) = ( x e. ( 0 [,] A ) |-> ( sqrt ` x ) ) ) |
124 |
120 123
|
eqtrd |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt |` ( 0 [,) +oo ) ) |` ( 0 [,] A ) ) = ( x e. ( 0 [,] A ) |-> ( sqrt ` x ) ) ) |
125 |
|
resqrtcn |
|- ( sqrt |` ( 0 [,) +oo ) ) e. ( ( 0 [,) +oo ) -cn-> RR ) |
126 |
|
rescncf |
|- ( ( 0 [,] A ) C_ ( 0 [,) +oo ) -> ( ( sqrt |` ( 0 [,) +oo ) ) e. ( ( 0 [,) +oo ) -cn-> RR ) -> ( ( sqrt |` ( 0 [,) +oo ) ) |` ( 0 [,] A ) ) e. ( ( 0 [,] A ) -cn-> RR ) ) ) |
127 |
119 125 126
|
mpisyl |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt |` ( 0 [,) +oo ) ) |` ( 0 [,] A ) ) e. ( ( 0 [,] A ) -cn-> RR ) ) |
128 |
124 127
|
eqeltrrd |
|- ( ( A e. RR /\ 0 <_ A ) -> ( x e. ( 0 [,] A ) |-> ( sqrt ` x ) ) e. ( ( 0 [,] A ) -cn-> RR ) ) |
129 |
|
rpcn |
|- ( x e. RR+ -> x e. CC ) |
130 |
129
|
adantl |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> x e. CC ) |
131 |
130
|
sqrtcld |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( sqrt ` x ) e. CC ) |
132 |
|
2rp |
|- 2 e. RR+ |
133 |
|
rpsqrtcl |
|- ( x e. RR+ -> ( sqrt ` x ) e. RR+ ) |
134 |
133
|
adantl |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( sqrt ` x ) e. RR+ ) |
135 |
|
rpmulcl |
|- ( ( 2 e. RR+ /\ ( sqrt ` x ) e. RR+ ) -> ( 2 x. ( sqrt ` x ) ) e. RR+ ) |
136 |
132 134 135
|
sylancr |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( 2 x. ( sqrt ` x ) ) e. RR+ ) |
137 |
136
|
rpreccld |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( 1 / ( 2 x. ( sqrt ` x ) ) ) e. RR+ ) |
138 |
|
dvsqrt |
|- ( RR _D ( x e. RR+ |-> ( sqrt ` x ) ) ) = ( x e. RR+ |-> ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) |
139 |
138
|
a1i |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( x e. RR+ |-> ( sqrt ` x ) ) ) = ( x e. RR+ |-> ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) ) |
140 |
53 131 137 139 112 81 12 114
|
dvmptres |
|- ( ( A e. RR /\ 0 <_ A ) -> ( RR _D ( x e. ( 0 (,) A ) |-> ( sqrt ` x ) ) ) = ( x e. ( 0 (,) A ) |-> ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) ) |
141 |
134
|
rpred |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( sqrt ` x ) e. RR ) |
142 |
|
1re |
|- 1 e. RR |
143 |
|
resubcl |
|- ( ( ( sqrt ` x ) e. RR /\ 1 e. RR ) -> ( ( sqrt ` x ) - 1 ) e. RR ) |
144 |
141 142 143
|
sylancl |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( ( sqrt ` x ) - 1 ) e. RR ) |
145 |
144
|
sqge0d |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> 0 <_ ( ( ( sqrt ` x ) - 1 ) ^ 2 ) ) |
146 |
130
|
sqsqrtd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( ( sqrt ` x ) ^ 2 ) = x ) |
147 |
146
|
oveq1d |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( ( ( sqrt ` x ) ^ 2 ) - ( 2 x. ( sqrt ` x ) ) ) = ( x - ( 2 x. ( sqrt ` x ) ) ) ) |
148 |
147
|
oveq1d |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( ( ( ( sqrt ` x ) ^ 2 ) - ( 2 x. ( sqrt ` x ) ) ) + 1 ) = ( ( x - ( 2 x. ( sqrt ` x ) ) ) + 1 ) ) |
149 |
|
binom2sub1 |
|- ( ( sqrt ` x ) e. CC -> ( ( ( sqrt ` x ) - 1 ) ^ 2 ) = ( ( ( ( sqrt ` x ) ^ 2 ) - ( 2 x. ( sqrt ` x ) ) ) + 1 ) ) |
150 |
131 149
|
syl |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( ( ( sqrt ` x ) - 1 ) ^ 2 ) = ( ( ( ( sqrt ` x ) ^ 2 ) - ( 2 x. ( sqrt ` x ) ) ) + 1 ) ) |
151 |
136
|
rpcnd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( 2 x. ( sqrt ` x ) ) e. CC ) |
152 |
130 61 151
|
addsubd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( ( x + 1 ) - ( 2 x. ( sqrt ` x ) ) ) = ( ( x - ( 2 x. ( sqrt ` x ) ) ) + 1 ) ) |
153 |
148 150 152
|
3eqtr4d |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( ( ( sqrt ` x ) - 1 ) ^ 2 ) = ( ( x + 1 ) - ( 2 x. ( sqrt ` x ) ) ) ) |
154 |
145 153
|
breqtrd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> 0 <_ ( ( x + 1 ) - ( 2 x. ( sqrt ` x ) ) ) ) |
155 |
57
|
rpred |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( x + 1 ) e. RR ) |
156 |
136
|
rpred |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( 2 x. ( sqrt ` x ) ) e. RR ) |
157 |
155 156
|
subge0d |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( 0 <_ ( ( x + 1 ) - ( 2 x. ( sqrt ` x ) ) ) <-> ( 2 x. ( sqrt ` x ) ) <_ ( x + 1 ) ) ) |
158 |
154 157
|
mpbid |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( 2 x. ( sqrt ` x ) ) <_ ( x + 1 ) ) |
159 |
136 57
|
lerecd |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( ( 2 x. ( sqrt ` x ) ) <_ ( x + 1 ) <-> ( 1 / ( x + 1 ) ) <_ ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) ) |
160 |
158 159
|
mpbid |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. RR+ ) -> ( 1 / ( x + 1 ) ) <_ ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) |
161 |
110 160
|
syldan |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ x e. ( 0 (,) A ) ) -> ( 1 / ( x + 1 ) ) <_ ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) |
162 |
|
rexr |
|- ( A e. RR -> A e. RR* ) |
163 |
|
0xr |
|- 0 e. RR* |
164 |
|
lbicc2 |
|- ( ( 0 e. RR* /\ A e. RR* /\ 0 <_ A ) -> 0 e. ( 0 [,] A ) ) |
165 |
163 164
|
mp3an1 |
|- ( ( A e. RR* /\ 0 <_ A ) -> 0 e. ( 0 [,] A ) ) |
166 |
162 165
|
sylan |
|- ( ( A e. RR /\ 0 <_ A ) -> 0 e. ( 0 [,] A ) ) |
167 |
|
ubicc2 |
|- ( ( 0 e. RR* /\ A e. RR* /\ 0 <_ A ) -> A e. ( 0 [,] A ) ) |
168 |
163 167
|
mp3an1 |
|- ( ( A e. RR* /\ 0 <_ A ) -> A e. ( 0 [,] A ) ) |
169 |
162 168
|
sylan |
|- ( ( A e. RR /\ 0 <_ A ) -> A e. ( 0 [,] A ) ) |
170 |
|
simpr |
|- ( ( A e. RR /\ 0 <_ A ) -> 0 <_ A ) |
171 |
|
fv0p1e1 |
|- ( x = 0 -> ( log ` ( x + 1 ) ) = ( log ` 1 ) ) |
172 |
|
log1 |
|- ( log ` 1 ) = 0 |
173 |
171 172
|
eqtrdi |
|- ( x = 0 -> ( log ` ( x + 1 ) ) = 0 ) |
174 |
|
fveq2 |
|- ( x = 0 -> ( sqrt ` x ) = ( sqrt ` 0 ) ) |
175 |
|
sqrt0 |
|- ( sqrt ` 0 ) = 0 |
176 |
174 175
|
eqtrdi |
|- ( x = 0 -> ( sqrt ` x ) = 0 ) |
177 |
|
fvoveq1 |
|- ( x = A -> ( log ` ( x + 1 ) ) = ( log ` ( A + 1 ) ) ) |
178 |
|
fveq2 |
|- ( x = A -> ( sqrt ` x ) = ( sqrt ` A ) ) |
179 |
2 3 51 115 128 140 161 166 169 170 173 176 177 178
|
dvle |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( log ` ( A + 1 ) ) - 0 ) <_ ( ( sqrt ` A ) - 0 ) ) |
180 |
|
ge0p1rp |
|- ( ( A e. RR /\ 0 <_ A ) -> ( A + 1 ) e. RR+ ) |
181 |
180
|
relogcld |
|- ( ( A e. RR /\ 0 <_ A ) -> ( log ` ( A + 1 ) ) e. RR ) |
182 |
|
resqrtcl |
|- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` A ) e. RR ) |
183 |
181 182 2
|
lesub1d |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( log ` ( A + 1 ) ) <_ ( sqrt ` A ) <-> ( ( log ` ( A + 1 ) ) - 0 ) <_ ( ( sqrt ` A ) - 0 ) ) ) |
184 |
179 183
|
mpbird |
|- ( ( A e. RR /\ 0 <_ A ) -> ( log ` ( A + 1 ) ) <_ ( sqrt ` A ) ) |