Step |
Hyp |
Ref |
Expression |
1 |
|
logdivsqrle.a |
|- ( ph -> A e. RR+ ) |
2 |
|
logdivsqrle.b |
|- ( ph -> B e. RR+ ) |
3 |
|
logdivsqrle.1 |
|- ( ph -> ( exp ` 2 ) <_ A ) |
4 |
|
logdivsqrle.2 |
|- ( ph -> A <_ B ) |
5 |
|
ioorp |
|- ( 0 (,) +oo ) = RR+ |
6 |
5
|
eqcomi |
|- RR+ = ( 0 (,) +oo ) |
7 |
|
simpr |
|- ( ( ph /\ x e. RR+ ) -> x e. RR+ ) |
8 |
7
|
relogcld |
|- ( ( ph /\ x e. RR+ ) -> ( log ` x ) e. RR ) |
9 |
7
|
rpsqrtcld |
|- ( ( ph /\ x e. RR+ ) -> ( sqrt ` x ) e. RR+ ) |
10 |
9
|
rpred |
|- ( ( ph /\ x e. RR+ ) -> ( sqrt ` x ) e. RR ) |
11 |
|
rpsqrtcl |
|- ( x e. RR+ -> ( sqrt ` x ) e. RR+ ) |
12 |
|
rpne0 |
|- ( ( sqrt ` x ) e. RR+ -> ( sqrt ` x ) =/= 0 ) |
13 |
11 12
|
syl |
|- ( x e. RR+ -> ( sqrt ` x ) =/= 0 ) |
14 |
13
|
adantl |
|- ( ( ph /\ x e. RR+ ) -> ( sqrt ` x ) =/= 0 ) |
15 |
8 10 14
|
redivcld |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) / ( sqrt ` x ) ) e. RR ) |
16 |
15
|
fmpttd |
|- ( ph -> ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) : RR+ --> RR ) |
17 |
|
rpcn |
|- ( x e. RR+ -> x e. CC ) |
18 |
17
|
adantl |
|- ( ( ph /\ x e. RR+ ) -> x e. CC ) |
19 |
|
rpne0 |
|- ( x e. RR+ -> x =/= 0 ) |
20 |
19
|
adantl |
|- ( ( ph /\ x e. RR+ ) -> x =/= 0 ) |
21 |
18 20
|
logcld |
|- ( ( ph /\ x e. RR+ ) -> ( log ` x ) e. CC ) |
22 |
18
|
sqrtcld |
|- ( ( ph /\ x e. RR+ ) -> ( sqrt ` x ) e. CC ) |
23 |
21 22 14
|
divrecd |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) / ( sqrt ` x ) ) = ( ( log ` x ) x. ( 1 / ( sqrt ` x ) ) ) ) |
24 |
|
2cnd |
|- ( ph -> 2 e. CC ) |
25 |
24
|
adantr |
|- ( ( ph /\ x e. RR+ ) -> 2 e. CC ) |
26 |
|
2ne0 |
|- 2 =/= 0 |
27 |
26
|
a1i |
|- ( ( ph /\ x e. RR+ ) -> 2 =/= 0 ) |
28 |
25 27
|
reccld |
|- ( ( ph /\ x e. RR+ ) -> ( 1 / 2 ) e. CC ) |
29 |
18 20 28
|
cxpnegd |
|- ( ( ph /\ x e. RR+ ) -> ( x ^c -u ( 1 / 2 ) ) = ( 1 / ( x ^c ( 1 / 2 ) ) ) ) |
30 |
|
cxpsqrt |
|- ( x e. CC -> ( x ^c ( 1 / 2 ) ) = ( sqrt ` x ) ) |
31 |
18 30
|
syl |
|- ( ( ph /\ x e. RR+ ) -> ( x ^c ( 1 / 2 ) ) = ( sqrt ` x ) ) |
32 |
31
|
oveq2d |
|- ( ( ph /\ x e. RR+ ) -> ( 1 / ( x ^c ( 1 / 2 ) ) ) = ( 1 / ( sqrt ` x ) ) ) |
33 |
29 32
|
eqtrd |
|- ( ( ph /\ x e. RR+ ) -> ( x ^c -u ( 1 / 2 ) ) = ( 1 / ( sqrt ` x ) ) ) |
34 |
33
|
oveq2d |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) x. ( x ^c -u ( 1 / 2 ) ) ) = ( ( log ` x ) x. ( 1 / ( sqrt ` x ) ) ) ) |
35 |
23 34
|
eqtr4d |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) / ( sqrt ` x ) ) = ( ( log ` x ) x. ( x ^c -u ( 1 / 2 ) ) ) ) |
36 |
35
|
mpteq2dva |
|- ( ph -> ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) = ( x e. RR+ |-> ( ( log ` x ) x. ( x ^c -u ( 1 / 2 ) ) ) ) ) |
37 |
36
|
oveq2d |
|- ( ph -> ( RR _D ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ) = ( RR _D ( x e. RR+ |-> ( ( log ` x ) x. ( x ^c -u ( 1 / 2 ) ) ) ) ) ) |
38 |
|
reelprrecn |
|- RR e. { RR , CC } |
39 |
38
|
a1i |
|- ( ph -> RR e. { RR , CC } ) |
40 |
7
|
rpreccld |
|- ( ( ph /\ x e. RR+ ) -> ( 1 / x ) e. RR+ ) |
41 |
|
logf1o |
|- log : ( CC \ { 0 } ) -1-1-onto-> ran log |
42 |
|
f1of |
|- ( log : ( CC \ { 0 } ) -1-1-onto-> ran log -> log : ( CC \ { 0 } ) --> ran log ) |
43 |
41 42
|
ax-mp |
|- log : ( CC \ { 0 } ) --> ran log |
44 |
43
|
a1i |
|- ( ph -> log : ( CC \ { 0 } ) --> ran log ) |
45 |
17
|
ssriv |
|- RR+ C_ CC |
46 |
|
0nrp |
|- -. 0 e. RR+ |
47 |
|
ssdifsn |
|- ( RR+ C_ ( CC \ { 0 } ) <-> ( RR+ C_ CC /\ -. 0 e. RR+ ) ) |
48 |
45 46 47
|
mpbir2an |
|- RR+ C_ ( CC \ { 0 } ) |
49 |
48
|
a1i |
|- ( ph -> RR+ C_ ( CC \ { 0 } ) ) |
50 |
44 49
|
feqresmpt |
|- ( ph -> ( log |` RR+ ) = ( x e. RR+ |-> ( log ` x ) ) ) |
51 |
50
|
oveq2d |
|- ( ph -> ( RR _D ( log |` RR+ ) ) = ( RR _D ( x e. RR+ |-> ( log ` x ) ) ) ) |
52 |
|
dvrelog |
|- ( RR _D ( log |` RR+ ) ) = ( x e. RR+ |-> ( 1 / x ) ) |
53 |
51 52
|
eqtr3di |
|- ( ph -> ( RR _D ( x e. RR+ |-> ( log ` x ) ) ) = ( x e. RR+ |-> ( 1 / x ) ) ) |
54 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
55 |
54
|
halfcld |
|- ( ph -> ( 1 / 2 ) e. CC ) |
56 |
55
|
negcld |
|- ( ph -> -u ( 1 / 2 ) e. CC ) |
57 |
56
|
adantr |
|- ( ( ph /\ x e. RR+ ) -> -u ( 1 / 2 ) e. CC ) |
58 |
18 57
|
cxpcld |
|- ( ( ph /\ x e. RR+ ) -> ( x ^c -u ( 1 / 2 ) ) e. CC ) |
59 |
54
|
adantr |
|- ( ( ph /\ x e. RR+ ) -> 1 e. CC ) |
60 |
57 59
|
subcld |
|- ( ( ph /\ x e. RR+ ) -> ( -u ( 1 / 2 ) - 1 ) e. CC ) |
61 |
18 60
|
cxpcld |
|- ( ( ph /\ x e. RR+ ) -> ( x ^c ( -u ( 1 / 2 ) - 1 ) ) e. CC ) |
62 |
57 61
|
mulcld |
|- ( ( ph /\ x e. RR+ ) -> ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) e. CC ) |
63 |
|
dvcxp1 |
|- ( -u ( 1 / 2 ) e. CC -> ( RR _D ( x e. RR+ |-> ( x ^c -u ( 1 / 2 ) ) ) ) = ( x e. RR+ |-> ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) ) |
64 |
56 63
|
syl |
|- ( ph -> ( RR _D ( x e. RR+ |-> ( x ^c -u ( 1 / 2 ) ) ) ) = ( x e. RR+ |-> ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) ) |
65 |
39 21 40 53 58 62 64
|
dvmptmul |
|- ( ph -> ( RR _D ( x e. RR+ |-> ( ( log ` x ) x. ( x ^c -u ( 1 / 2 ) ) ) ) ) = ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) ) |
66 |
37 65
|
eqtrd |
|- ( ph -> ( RR _D ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ) = ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) ) |
67 |
|
ax-resscn |
|- RR C_ CC |
68 |
67
|
a1i |
|- ( ph -> RR C_ CC ) |
69 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
70 |
69
|
addcn |
|- + e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) |
71 |
70
|
a1i |
|- ( ph -> + e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) ) |
72 |
45
|
a1i |
|- ( ph -> RR+ C_ CC ) |
73 |
|
ssid |
|- CC C_ CC |
74 |
73
|
a1i |
|- ( ph -> CC C_ CC ) |
75 |
|
cncfmptc |
|- ( ( 1 e. CC /\ RR+ C_ CC /\ CC C_ CC ) -> ( x e. RR+ |-> 1 ) e. ( RR+ -cn-> CC ) ) |
76 |
54 72 74 75
|
syl3anc |
|- ( ph -> ( x e. RR+ |-> 1 ) e. ( RR+ -cn-> CC ) ) |
77 |
|
difss |
|- ( CC \ { 0 } ) C_ CC |
78 |
|
cncfmptid |
|- ( ( RR+ C_ ( CC \ { 0 } ) /\ ( CC \ { 0 } ) C_ CC ) -> ( x e. RR+ |-> x ) e. ( RR+ -cn-> ( CC \ { 0 } ) ) ) |
79 |
49 77 78
|
sylancl |
|- ( ph -> ( x e. RR+ |-> x ) e. ( RR+ -cn-> ( CC \ { 0 } ) ) ) |
80 |
76 79
|
divcncf |
|- ( ph -> ( x e. RR+ |-> ( 1 / x ) ) e. ( RR+ -cn-> CC ) ) |
81 |
|
ax-1 |
|- ( x e. RR+ -> ( x e. RR -> x e. RR+ ) ) |
82 |
17 81
|
jca |
|- ( x e. RR+ -> ( x e. CC /\ ( x e. RR -> x e. RR+ ) ) ) |
83 |
|
eqid |
|- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) ) |
84 |
83
|
ellogdm |
|- ( x e. ( CC \ ( -oo (,] 0 ) ) <-> ( x e. CC /\ ( x e. RR -> x e. RR+ ) ) ) |
85 |
82 84
|
sylibr |
|- ( x e. RR+ -> x e. ( CC \ ( -oo (,] 0 ) ) ) |
86 |
85
|
ssriv |
|- RR+ C_ ( CC \ ( -oo (,] 0 ) ) |
87 |
86
|
a1i |
|- ( ph -> RR+ C_ ( CC \ ( -oo (,] 0 ) ) ) |
88 |
56 87
|
cxpcncf1 |
|- ( ph -> ( x e. RR+ |-> ( x ^c -u ( 1 / 2 ) ) ) e. ( RR+ -cn-> CC ) ) |
89 |
80 88
|
mulcncf |
|- ( ph -> ( x e. RR+ |-> ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) ) e. ( RR+ -cn-> CC ) ) |
90 |
|
cncfmptc |
|- ( ( -u ( 1 / 2 ) e. CC /\ RR+ C_ CC /\ CC C_ CC ) -> ( x e. RR+ |-> -u ( 1 / 2 ) ) e. ( RR+ -cn-> CC ) ) |
91 |
56 72 74 90
|
syl3anc |
|- ( ph -> ( x e. RR+ |-> -u ( 1 / 2 ) ) e. ( RR+ -cn-> CC ) ) |
92 |
56 54
|
subcld |
|- ( ph -> ( -u ( 1 / 2 ) - 1 ) e. CC ) |
93 |
92 87
|
cxpcncf1 |
|- ( ph -> ( x e. RR+ |-> ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) e. ( RR+ -cn-> CC ) ) |
94 |
91 93
|
mulcncf |
|- ( ph -> ( x e. RR+ |-> ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) e. ( RR+ -cn-> CC ) ) |
95 |
|
cncfss |
|- ( ( RR C_ CC /\ CC C_ CC ) -> ( RR+ -cn-> RR ) C_ ( RR+ -cn-> CC ) ) |
96 |
67 73 95
|
mp2an |
|- ( RR+ -cn-> RR ) C_ ( RR+ -cn-> CC ) |
97 |
|
relogcn |
|- ( log |` RR+ ) e. ( RR+ -cn-> RR ) |
98 |
50 97
|
eqeltrrdi |
|- ( ph -> ( x e. RR+ |-> ( log ` x ) ) e. ( RR+ -cn-> RR ) ) |
99 |
96 98
|
sselid |
|- ( ph -> ( x e. RR+ |-> ( log ` x ) ) e. ( RR+ -cn-> CC ) ) |
100 |
94 99
|
mulcncf |
|- ( ph -> ( x e. RR+ |-> ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) e. ( RR+ -cn-> CC ) ) |
101 |
69 71 89 100
|
cncfmpt2f |
|- ( ph -> ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) e. ( RR+ -cn-> CC ) ) |
102 |
|
rpre |
|- ( x e. RR+ -> x e. RR ) |
103 |
102 19
|
rereccld |
|- ( x e. RR+ -> ( 1 / x ) e. RR ) |
104 |
|
rpge0 |
|- ( x e. RR+ -> 0 <_ x ) |
105 |
|
halfre |
|- ( 1 / 2 ) e. RR |
106 |
105
|
renegcli |
|- -u ( 1 / 2 ) e. RR |
107 |
106
|
a1i |
|- ( x e. RR+ -> -u ( 1 / 2 ) e. RR ) |
108 |
102 104 107
|
recxpcld |
|- ( x e. RR+ -> ( x ^c -u ( 1 / 2 ) ) e. RR ) |
109 |
103 108
|
remulcld |
|- ( x e. RR+ -> ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) e. RR ) |
110 |
|
1re |
|- 1 e. RR |
111 |
106 110
|
resubcli |
|- ( -u ( 1 / 2 ) - 1 ) e. RR |
112 |
111
|
a1i |
|- ( x e. RR+ -> ( -u ( 1 / 2 ) - 1 ) e. RR ) |
113 |
102 104 112
|
recxpcld |
|- ( x e. RR+ -> ( x ^c ( -u ( 1 / 2 ) - 1 ) ) e. RR ) |
114 |
107 113
|
remulcld |
|- ( x e. RR+ -> ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) e. RR ) |
115 |
|
relogcl |
|- ( x e. RR+ -> ( log ` x ) e. RR ) |
116 |
114 115
|
remulcld |
|- ( x e. RR+ -> ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) e. RR ) |
117 |
109 116
|
readdcld |
|- ( x e. RR+ -> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) e. RR ) |
118 |
117
|
adantl |
|- ( ( ph /\ x e. RR+ ) -> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) e. RR ) |
119 |
118
|
fmpttd |
|- ( ph -> ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) : RR+ --> RR ) |
120 |
|
cncffvrn |
|- ( ( RR C_ CC /\ ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) e. ( RR+ -cn-> CC ) ) -> ( ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) e. ( RR+ -cn-> RR ) <-> ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) : RR+ --> RR ) ) |
121 |
120
|
biimpar |
|- ( ( ( RR C_ CC /\ ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) e. ( RR+ -cn-> CC ) ) /\ ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) : RR+ --> RR ) -> ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) e. ( RR+ -cn-> RR ) ) |
122 |
68 101 119 121
|
syl21anc |
|- ( ph -> ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) e. ( RR+ -cn-> RR ) ) |
123 |
66 122
|
eqeltrd |
|- ( ph -> ( RR _D ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ) e. ( RR+ -cn-> RR ) ) |
124 |
66
|
fveq1d |
|- ( ph -> ( ( RR _D ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ) ` y ) = ( ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) ` y ) ) |
125 |
124
|
adantr |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( RR _D ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ) ` y ) = ( ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) ` y ) ) |
126 |
59
|
negcld |
|- ( ( ph /\ x e. RR+ ) -> -u 1 e. CC ) |
127 |
|
cxpadd |
|- ( ( ( x e. CC /\ x =/= 0 ) /\ -u ( 1 / 2 ) e. CC /\ -u 1 e. CC ) -> ( x ^c ( -u ( 1 / 2 ) + -u 1 ) ) = ( ( x ^c -u ( 1 / 2 ) ) x. ( x ^c -u 1 ) ) ) |
128 |
18 20 57 126 127
|
syl211anc |
|- ( ( ph /\ x e. RR+ ) -> ( x ^c ( -u ( 1 / 2 ) + -u 1 ) ) = ( ( x ^c -u ( 1 / 2 ) ) x. ( x ^c -u 1 ) ) ) |
129 |
61
|
mulid2d |
|- ( ( ph /\ x e. RR+ ) -> ( 1 x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) = ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) |
130 |
57 59
|
negsubd |
|- ( ( ph /\ x e. RR+ ) -> ( -u ( 1 / 2 ) + -u 1 ) = ( -u ( 1 / 2 ) - 1 ) ) |
131 |
130
|
oveq2d |
|- ( ( ph /\ x e. RR+ ) -> ( x ^c ( -u ( 1 / 2 ) + -u 1 ) ) = ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) |
132 |
129 131
|
eqtr4d |
|- ( ( ph /\ x e. RR+ ) -> ( 1 x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) = ( x ^c ( -u ( 1 / 2 ) + -u 1 ) ) ) |
133 |
45 40
|
sselid |
|- ( ( ph /\ x e. RR+ ) -> ( 1 / x ) e. CC ) |
134 |
133 58
|
mulcomd |
|- ( ( ph /\ x e. RR+ ) -> ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) = ( ( x ^c -u ( 1 / 2 ) ) x. ( 1 / x ) ) ) |
135 |
|
cxpneg |
|- ( ( x e. CC /\ x =/= 0 /\ 1 e. CC ) -> ( x ^c -u 1 ) = ( 1 / ( x ^c 1 ) ) ) |
136 |
18 20 59 135
|
syl3anc |
|- ( ( ph /\ x e. RR+ ) -> ( x ^c -u 1 ) = ( 1 / ( x ^c 1 ) ) ) |
137 |
18
|
cxp1d |
|- ( ( ph /\ x e. RR+ ) -> ( x ^c 1 ) = x ) |
138 |
137
|
oveq2d |
|- ( ( ph /\ x e. RR+ ) -> ( 1 / ( x ^c 1 ) ) = ( 1 / x ) ) |
139 |
136 138
|
eqtr2d |
|- ( ( ph /\ x e. RR+ ) -> ( 1 / x ) = ( x ^c -u 1 ) ) |
140 |
139
|
oveq2d |
|- ( ( ph /\ x e. RR+ ) -> ( ( x ^c -u ( 1 / 2 ) ) x. ( 1 / x ) ) = ( ( x ^c -u ( 1 / 2 ) ) x. ( x ^c -u 1 ) ) ) |
141 |
134 140
|
eqtrd |
|- ( ( ph /\ x e. RR+ ) -> ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) = ( ( x ^c -u ( 1 / 2 ) ) x. ( x ^c -u 1 ) ) ) |
142 |
128 132 141
|
3eqtr4rd |
|- ( ( ph /\ x e. RR+ ) -> ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) = ( 1 x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) |
143 |
57 61 21
|
mul32d |
|- ( ( ph /\ x e. RR+ ) -> ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) = ( ( -u ( 1 / 2 ) x. ( log ` x ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) |
144 |
142 143
|
oveq12d |
|- ( ( ph /\ x e. RR+ ) -> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) = ( ( 1 x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) + ( ( -u ( 1 / 2 ) x. ( log ` x ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) ) |
145 |
57 21
|
mulcld |
|- ( ( ph /\ x e. RR+ ) -> ( -u ( 1 / 2 ) x. ( log ` x ) ) e. CC ) |
146 |
59 145 61
|
adddird |
|- ( ( ph /\ x e. RR+ ) -> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) = ( ( 1 x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) + ( ( -u ( 1 / 2 ) x. ( log ` x ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) ) |
147 |
144 146
|
eqtr4d |
|- ( ( ph /\ x e. RR+ ) -> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) = ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) |
148 |
147
|
mpteq2dva |
|- ( ph -> ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) = ( x e. RR+ |-> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) ) |
149 |
148
|
fveq1d |
|- ( ph -> ( ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) ` y ) = ( ( x e. RR+ |-> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) ` y ) ) |
150 |
149
|
adantr |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) ` y ) = ( ( x e. RR+ |-> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) ` y ) ) |
151 |
|
eqidd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( x e. RR+ |-> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) = ( x e. RR+ |-> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) ) |
152 |
|
simpr |
|- ( ( ( ph /\ y e. ( A (,) B ) ) /\ x = y ) -> x = y ) |
153 |
152
|
fveq2d |
|- ( ( ( ph /\ y e. ( A (,) B ) ) /\ x = y ) -> ( log ` x ) = ( log ` y ) ) |
154 |
153
|
oveq2d |
|- ( ( ( ph /\ y e. ( A (,) B ) ) /\ x = y ) -> ( -u ( 1 / 2 ) x. ( log ` x ) ) = ( -u ( 1 / 2 ) x. ( log ` y ) ) ) |
155 |
154
|
oveq2d |
|- ( ( ( ph /\ y e. ( A (,) B ) ) /\ x = y ) -> ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) = ( 1 + ( -u ( 1 / 2 ) x. ( log ` y ) ) ) ) |
156 |
152
|
oveq1d |
|- ( ( ( ph /\ y e. ( A (,) B ) ) /\ x = y ) -> ( x ^c ( -u ( 1 / 2 ) - 1 ) ) = ( y ^c ( -u ( 1 / 2 ) - 1 ) ) ) |
157 |
155 156
|
oveq12d |
|- ( ( ( ph /\ y e. ( A (,) B ) ) /\ x = y ) -> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) = ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` y ) ) ) x. ( y ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) |
158 |
|
ioossicc |
|- ( A (,) B ) C_ ( A [,] B ) |
159 |
158
|
a1i |
|- ( ph -> ( A (,) B ) C_ ( A [,] B ) ) |
160 |
6 1 2
|
fct2relem |
|- ( ph -> ( A [,] B ) C_ RR+ ) |
161 |
159 160
|
sstrd |
|- ( ph -> ( A (,) B ) C_ RR+ ) |
162 |
161
|
sselda |
|- ( ( ph /\ y e. ( A (,) B ) ) -> y e. RR+ ) |
163 |
|
ovexd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` y ) ) ) x. ( y ^c ( -u ( 1 / 2 ) - 1 ) ) ) e. _V ) |
164 |
151 157 162 163
|
fvmptd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( x e. RR+ |-> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) ` y ) = ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` y ) ) ) x. ( y ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) |
165 |
110
|
a1i |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 1 e. RR ) |
166 |
106
|
a1i |
|- ( ( ph /\ y e. ( A (,) B ) ) -> -u ( 1 / 2 ) e. RR ) |
167 |
162
|
relogcld |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( log ` y ) e. RR ) |
168 |
166 167
|
remulcld |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( -u ( 1 / 2 ) x. ( log ` y ) ) e. RR ) |
169 |
165 168
|
readdcld |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( 1 + ( -u ( 1 / 2 ) x. ( log ` y ) ) ) e. RR ) |
170 |
|
0red |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 0 e. RR ) |
171 |
|
rpcxpcl |
|- ( ( y e. RR+ /\ ( -u ( 1 / 2 ) - 1 ) e. RR ) -> ( y ^c ( -u ( 1 / 2 ) - 1 ) ) e. RR+ ) |
172 |
162 111 171
|
sylancl |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( y ^c ( -u ( 1 / 2 ) - 1 ) ) e. RR+ ) |
173 |
172
|
rpred |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( y ^c ( -u ( 1 / 2 ) - 1 ) ) e. RR ) |
174 |
172
|
rpge0d |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 0 <_ ( y ^c ( -u ( 1 / 2 ) - 1 ) ) ) |
175 |
|
2cn |
|- 2 e. CC |
176 |
175
|
mulid2i |
|- ( 1 x. 2 ) = 2 |
177 |
|
2re |
|- 2 e. RR |
178 |
177
|
a1i |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 2 e. RR ) |
179 |
178
|
reefcld |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( exp ` 2 ) e. RR ) |
180 |
1
|
rpred |
|- ( ph -> A e. RR ) |
181 |
180
|
adantr |
|- ( ( ph /\ y e. ( A (,) B ) ) -> A e. RR ) |
182 |
162
|
rpred |
|- ( ( ph /\ y e. ( A (,) B ) ) -> y e. RR ) |
183 |
3
|
adantr |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( exp ` 2 ) <_ A ) |
184 |
|
eliooord |
|- ( y e. ( A (,) B ) -> ( A < y /\ y < B ) ) |
185 |
184
|
simpld |
|- ( y e. ( A (,) B ) -> A < y ) |
186 |
185
|
adantl |
|- ( ( ph /\ y e. ( A (,) B ) ) -> A < y ) |
187 |
181 182 186
|
ltled |
|- ( ( ph /\ y e. ( A (,) B ) ) -> A <_ y ) |
188 |
179 181 182 183 187
|
letrd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( exp ` 2 ) <_ y ) |
189 |
|
reeflog |
|- ( y e. RR+ -> ( exp ` ( log ` y ) ) = y ) |
190 |
162 189
|
syl |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( exp ` ( log ` y ) ) = y ) |
191 |
188 190
|
breqtrrd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( exp ` 2 ) <_ ( exp ` ( log ` y ) ) ) |
192 |
|
efle |
|- ( ( 2 e. RR /\ ( log ` y ) e. RR ) -> ( 2 <_ ( log ` y ) <-> ( exp ` 2 ) <_ ( exp ` ( log ` y ) ) ) ) |
193 |
177 167 192
|
sylancr |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( 2 <_ ( log ` y ) <-> ( exp ` 2 ) <_ ( exp ` ( log ` y ) ) ) ) |
194 |
191 193
|
mpbird |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 2 <_ ( log ` y ) ) |
195 |
176 194
|
eqbrtrid |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( 1 x. 2 ) <_ ( log ` y ) ) |
196 |
|
2rp |
|- 2 e. RR+ |
197 |
196
|
a1i |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 2 e. RR+ ) |
198 |
165 167 197
|
lemuldivd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( 1 x. 2 ) <_ ( log ` y ) <-> 1 <_ ( ( log ` y ) / 2 ) ) ) |
199 |
195 198
|
mpbid |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 1 <_ ( ( log ` y ) / 2 ) ) |
200 |
67 167
|
sselid |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( log ` y ) e. CC ) |
201 |
24
|
adantr |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 2 e. CC ) |
202 |
26
|
a1i |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 2 =/= 0 ) |
203 |
200 201 202
|
divrec2d |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( log ` y ) / 2 ) = ( ( 1 / 2 ) x. ( log ` y ) ) ) |
204 |
199 203
|
breqtrd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 1 <_ ( ( 1 / 2 ) x. ( log ` y ) ) ) |
205 |
55
|
adantr |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( 1 / 2 ) e. CC ) |
206 |
205 200
|
mulneg1d |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( -u ( 1 / 2 ) x. ( log ` y ) ) = -u ( ( 1 / 2 ) x. ( log ` y ) ) ) |
207 |
206
|
oveq2d |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( 0 - ( -u ( 1 / 2 ) x. ( log ` y ) ) ) = ( 0 - -u ( ( 1 / 2 ) x. ( log ` y ) ) ) ) |
208 |
67 170
|
sselid |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 0 e. CC ) |
209 |
205 200
|
mulcld |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( 1 / 2 ) x. ( log ` y ) ) e. CC ) |
210 |
208 209
|
subnegd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( 0 - -u ( ( 1 / 2 ) x. ( log ` y ) ) ) = ( 0 + ( ( 1 / 2 ) x. ( log ` y ) ) ) ) |
211 |
209
|
addid2d |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( 0 + ( ( 1 / 2 ) x. ( log ` y ) ) ) = ( ( 1 / 2 ) x. ( log ` y ) ) ) |
212 |
207 210 211
|
3eqtrd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( 0 - ( -u ( 1 / 2 ) x. ( log ` y ) ) ) = ( ( 1 / 2 ) x. ( log ` y ) ) ) |
213 |
204 212
|
breqtrrd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> 1 <_ ( 0 - ( -u ( 1 / 2 ) x. ( log ` y ) ) ) ) |
214 |
|
leaddsub |
|- ( ( 1 e. RR /\ ( -u ( 1 / 2 ) x. ( log ` y ) ) e. RR /\ 0 e. RR ) -> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` y ) ) ) <_ 0 <-> 1 <_ ( 0 - ( -u ( 1 / 2 ) x. ( log ` y ) ) ) ) ) |
215 |
165 168 170 214
|
syl3anc |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` y ) ) ) <_ 0 <-> 1 <_ ( 0 - ( -u ( 1 / 2 ) x. ( log ` y ) ) ) ) ) |
216 |
213 215
|
mpbird |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( 1 + ( -u ( 1 / 2 ) x. ( log ` y ) ) ) <_ 0 ) |
217 |
169 170 173 174 216
|
lemul1ad |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` y ) ) ) x. ( y ^c ( -u ( 1 / 2 ) - 1 ) ) ) <_ ( 0 x. ( y ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) |
218 |
45 172
|
sselid |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( y ^c ( -u ( 1 / 2 ) - 1 ) ) e. CC ) |
219 |
218
|
mul02d |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( 0 x. ( y ^c ( -u ( 1 / 2 ) - 1 ) ) ) = 0 ) |
220 |
217 219
|
breqtrd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` y ) ) ) x. ( y ^c ( -u ( 1 / 2 ) - 1 ) ) ) <_ 0 ) |
221 |
164 220
|
eqbrtrd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( x e. RR+ |-> ( ( 1 + ( -u ( 1 / 2 ) x. ( log ` x ) ) ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) ) ` y ) <_ 0 ) |
222 |
150 221
|
eqbrtrd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( x e. RR+ |-> ( ( ( 1 / x ) x. ( x ^c -u ( 1 / 2 ) ) ) + ( ( -u ( 1 / 2 ) x. ( x ^c ( -u ( 1 / 2 ) - 1 ) ) ) x. ( log ` x ) ) ) ) ` y ) <_ 0 ) |
223 |
125 222
|
eqbrtrd |
|- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( RR _D ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ) ` y ) <_ 0 ) |
224 |
6 1 2 16 123 4 223
|
fdvnegge |
|- ( ph -> ( ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ` B ) <_ ( ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ` A ) ) |
225 |
|
eqidd |
|- ( ph -> ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) = ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ) |
226 |
|
simpr |
|- ( ( ph /\ x = B ) -> x = B ) |
227 |
226
|
fveq2d |
|- ( ( ph /\ x = B ) -> ( log ` x ) = ( log ` B ) ) |
228 |
226
|
fveq2d |
|- ( ( ph /\ x = B ) -> ( sqrt ` x ) = ( sqrt ` B ) ) |
229 |
227 228
|
oveq12d |
|- ( ( ph /\ x = B ) -> ( ( log ` x ) / ( sqrt ` x ) ) = ( ( log ` B ) / ( sqrt ` B ) ) ) |
230 |
|
ovex |
|- ( ( log ` B ) / ( sqrt ` B ) ) e. _V |
231 |
230
|
a1i |
|- ( ph -> ( ( log ` B ) / ( sqrt ` B ) ) e. _V ) |
232 |
225 229 2 231
|
fvmptd |
|- ( ph -> ( ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ` B ) = ( ( log ` B ) / ( sqrt ` B ) ) ) |
233 |
|
simpr |
|- ( ( ph /\ x = A ) -> x = A ) |
234 |
233
|
fveq2d |
|- ( ( ph /\ x = A ) -> ( log ` x ) = ( log ` A ) ) |
235 |
233
|
fveq2d |
|- ( ( ph /\ x = A ) -> ( sqrt ` x ) = ( sqrt ` A ) ) |
236 |
234 235
|
oveq12d |
|- ( ( ph /\ x = A ) -> ( ( log ` x ) / ( sqrt ` x ) ) = ( ( log ` A ) / ( sqrt ` A ) ) ) |
237 |
|
ovex |
|- ( ( log ` A ) / ( sqrt ` A ) ) e. _V |
238 |
237
|
a1i |
|- ( ph -> ( ( log ` A ) / ( sqrt ` A ) ) e. _V ) |
239 |
225 236 1 238
|
fvmptd |
|- ( ph -> ( ( x e. RR+ |-> ( ( log ` x ) / ( sqrt ` x ) ) ) ` A ) = ( ( log ` A ) / ( sqrt ` A ) ) ) |
240 |
224 232 239
|
3brtr3d |
|- ( ph -> ( ( log ` B ) / ( sqrt ` B ) ) <_ ( ( log ` A ) / ( sqrt ` A ) ) ) |