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Theorem logccv 22908
Description: The natural logarithm function on the reals is a strictly concave function. (Contributed by Mario Carneiro, 20-Jun-2015.)
Assertion
Ref Expression
logccv  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  ( log `  A ) )  +  ( ( 1  -  T )  x.  ( log `  B
) ) )  < 
( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )

Proof of Theorem logccv
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 999 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR+ )
21rpred 11268 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR )
3 simpl2 1000 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR+ )
43rpred 11268 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR )
5 simpl3 1001 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  <  B )
61rpgt0d 11271 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  0  <  A )
7 ltpnf 11343 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  B  < +oo )
84, 7syl 16 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  < +oo )
9 0xr 9652 . . . . . . . . . . . 12  |-  0  e.  RR*
10 pnfxr 11333 . . . . . . . . . . . 12  |- +oo  e.  RR*
11 iccssioo 11605 . . . . . . . . . . . 12  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <  A  /\  B  < +oo ) )  -> 
( A [,] B
)  C_  ( 0 (,) +oo ) )
129, 10, 11mpanl12 682 . . . . . . . . . . 11  |-  ( ( 0  <  A  /\  B  < +oo )  ->  ( A [,] B )  C_  ( 0 (,) +oo ) )
136, 8, 12syl2anc 661 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A [,] B )  C_  ( 0 (,) +oo ) )
14 ioorp 11614 . . . . . . . . . 10  |-  ( 0 (,) +oo )  = 
RR+
1513, 14syl6sseq 3555 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A [,] B )  C_  RR+ )
1615sselda 3509 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR+ )
1716relogcld 22872 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  ( log `  x )  e.  RR )
1817renegcld 9998 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  -u ( log `  x )  e.  RR )
19 eqid 2467 . . . . . 6  |-  ( x  e.  ( A [,] B )  |->  -u ( log `  x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )
2018, 19fmptd 6056 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
x  e.  ( A [,] B )  |->  -u ( log `  x ) ) : ( A [,] B ) --> RR )
21 ax-resscn 9561 . . . . . 6  |-  RR  C_  CC
2215resabs1d 5309 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  =  ( log  |`  ( A [,] B ) ) )
23 ssid 3528 . . . . . . . . . . 11  |-  CC  C_  CC
24 cncfss 21269 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC ) )
2521, 23, 24mp2an 672 . . . . . . . . . 10  |-  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC )
26 relogcn 22883 . . . . . . . . . 10  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
2725, 26sselii 3506 . . . . . . . . 9  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )
28 rescncf 21267 . . . . . . . . 9  |-  ( ( A [,] B ) 
C_  RR+  ->  ( ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) ) )
2915, 27, 28mpisyl 18 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) )
3022, 29eqeltrrd 2556 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
31 fvres 5886 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  (
( log  |`  ( A [,] B ) ) `
 x )  =  ( log `  x
) )
3231negeqd 9826 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  -u (
( log  |`  ( A [,] B ) ) `
 x )  = 
-u ( log `  x
) )
3332mpteq2ia 4535 . . . . . . . . 9  |-  ( x  e.  ( A [,] B )  |->  -u (
( log  |`  ( A [,] B ) ) `
 x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )
3433eqcomi 2480 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  -u ( log `  x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( ( log  |`  ( A [,] B
) ) `  x
) )
3534negfcncf 21289 . . . . . . 7  |-  ( ( log  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC )  ->  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
3630, 35syl 16 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
x  e.  ( A [,] B )  |->  -u ( log `  x ) )  e.  ( ( A [,] B )
-cn-> CC ) )
37 cncffvrn 21268 . . . . . 6  |-  ( ( RR  C_  CC  /\  (
x  e.  ( A [,] B )  |->  -u ( log `  x ) )  e.  ( ( A [,] B )
-cn-> CC ) )  -> 
( ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )  e.  ( ( A [,] B
) -cn-> RR )  <->  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) ) : ( A [,] B ) --> RR ) )
3821, 36, 37sylancr 663 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) )  e.  ( ( A [,] B
) -cn-> RR )  <->  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) ) : ( A [,] B ) --> RR ) )
3920, 38mpbird 232 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
x  e.  ( A [,] B )  |->  -u ( log `  x ) )  e.  ( ( A [,] B )
-cn-> RR ) )
40 ioossre 11598 . . . . . . . 8  |-  ( A (,) B )  C_  RR
41 ltso 9677 . . . . . . . 8  |-  <  Or  RR
42 soss 4824 . . . . . . . 8  |-  ( ( A (,) B ) 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  ( A (,) B ) ) )
4340, 41, 42mp2 9 . . . . . . 7  |-  <  Or  ( A (,) B )
4443a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Or  ( A (,) B
) )
45 ioossicc 11622 . . . . . . . . . . . . . 14  |-  ( A (,) B )  C_  ( A [,] B )
4645, 15syl5ss 3520 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A (,) B )  C_  RR+ )
4746sselda 3509 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  x  e.  RR+ )
4847rprecred 11279 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  (
1  /  x )  e.  RR )
4948renegcld 9998 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  -u (
1  /  x )  e.  RR )
50 eqid 2467 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  =  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )
5149, 50fmptd 6056 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
x  e.  ( A (,) B )  |->  -u ( 1  /  x
) ) : ( A (,) B ) --> RR )
52 frn 5743 . . . . . . . . 9  |-  ( ( x  e.  ( A (,) B )  |->  -u ( 1  /  x
) ) : ( A (,) B ) --> RR  ->  ran  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  C_  RR )
5351, 52syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ran  ( x  e.  ( A (,) B )  |->  -u ( 1  /  x
) )  C_  RR )
54 soss 4824 . . . . . . . 8  |-  ( ran  ( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) )  C_  RR  ->  (  <  Or  RR  ->  <  Or  ran  ( x  e.  ( A (,) B )  |->  -u ( 1  /  x
) ) ) )
5553, 41, 54mpisyl 18 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Or 
ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
56 sopo 4823 . . . . . . 7  |-  (  < 
Or  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) )  ->  <  Po  ran  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) )
5755, 56syl 16 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Po 
ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
58 negex 9830 . . . . . . . . 9  |-  -u (
1  /  x )  e.  _V
5958, 50fnmpti 5715 . . . . . . . 8  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  Fn  ( A (,) B )
60 dffn4 5807 . . . . . . . 8  |-  ( ( x  e.  ( A (,) B )  |->  -u ( 1  /  x
) )  Fn  ( A (,) B )  <->  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) : ( A (,) B
) -onto-> ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
6159, 60mpbi 208 . . . . . . 7  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) : ( A (,) B ) -onto-> ran  ( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) )
6261a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
x  e.  ( A (,) B )  |->  -u ( 1  /  x
) ) : ( A (,) B )
-onto->
ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
6346sselda 3509 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  z  e.  ( A (,) B
) )  ->  z  e.  RR+ )
6463adantrl 715 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
z  e.  RR+ )
6564rprecred 11279 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
( 1  /  z
)  e.  RR )
6646sselda 3509 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  y  e.  ( A (,) B
) )  ->  y  e.  RR+ )
6766adantrr 716 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
y  e.  RR+ )
6867rprecred 11279 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
( 1  /  y
)  e.  RR )
6965, 68ltnegd 10142 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
( ( 1  / 
z )  <  (
1  /  y )  <->  -u ( 1  /  y
)  <  -u ( 1  /  z ) ) )
7067, 64ltrecd 11286 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
( y  <  z  <->  ( 1  /  z )  <  ( 1  / 
y ) ) )
71 oveq2 6303 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
1  /  x )  =  ( 1  / 
y ) )
7271negeqd 9826 . . . . . . . . . . . 12  |-  ( x  =  y  ->  -u (
1  /  x )  =  -u ( 1  / 
y ) )
73 negex 9830 . . . . . . . . . . . 12  |-  -u (
1  /  y )  e.  _V
7472, 50, 73fvmpt 5957 . . . . . . . . . . 11  |-  ( y  e.  ( A (,) B )  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  y )  =  -u ( 1  /  y
) )
75 oveq2 6303 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
1  /  x )  =  ( 1  / 
z ) )
7675negeqd 9826 . . . . . . . . . . . 12  |-  ( x  =  z  ->  -u (
1  /  x )  =  -u ( 1  / 
z ) )
77 negex 9830 . . . . . . . . . . . 12  |-  -u (
1  /  z )  e.  _V
7876, 50, 77fvmpt 5957 . . . . . . . . . . 11  |-  ( z  e.  ( A (,) B )  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  z )  =  -u ( 1  /  z
) )
7974, 78breqan12d 4468 . . . . . . . . . 10  |-  ( ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B ) )  -> 
( ( ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) `  y )  <  ( ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) `  z )  <->  -u ( 1  /  y
)  <  -u ( 1  /  z ) ) )
8079adantl 466 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
( ( ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) `  y )  <  ( ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) `  z )  <->  -u ( 1  /  y
)  <  -u ( 1  /  z ) ) )
8169, 70, 803bitr4d 285 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
( y  <  z  <->  ( ( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  y )  <  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  z ) ) )
8281biimpd 207 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
( y  <  z  ->  ( ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) `  y )  <  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  z ) ) )
8382ralrimivva 2888 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A. y  e.  ( A (,) B
) A. z  e.  ( A (,) B
) ( y  < 
z  ->  ( (
x  e.  ( A (,) B )  |->  -u ( 1  /  x
) ) `  y
)  <  ( (
x  e.  ( A (,) B )  |->  -u ( 1  /  x
) ) `  z
) ) )
84 soisoi 6223 . . . . . 6  |-  ( ( (  <  Or  ( A (,) B )  /\  <  Po  ran  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) )  /\  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) : ( A (,) B
) -onto-> ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) )  /\  A. y  e.  ( A (,) B ) A. z  e.  ( A (,) B ) ( y  <  z  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  y )  <  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  z ) ) ) )  ->  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) )  Isom  <  ,  <  ( ( A (,) B ) ,  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) ) )
8544, 57, 62, 83, 84syl22anc 1229 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
x  e.  ( A (,) B )  |->  -u ( 1  /  x
) )  Isom  <  ,  <  ( ( A (,) B ) ,  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) ) )
86 reelprrecn 9596 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
8786a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  RR  e.  { RR ,  CC } )
88 relogcl 22827 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
8988adantl 466 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
9089recnd 9634 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
9190negcld 9929 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  -u ( log `  x )  e.  CC )
9258a1i 11 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  -u (
1  /  x )  e.  _V )
93 ovex 6320 . . . . . . . . 9  |-  ( 1  /  x )  e. 
_V
9493a1i 11 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( 1  /  x )  e. 
_V )
95 dvrelog 22882 . . . . . . . . 9  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
96 relogf1o 22818 . . . . . . . . . . . . 13  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
97 f1of 5822 . . . . . . . . . . . . 13  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
9896, 97mp1i 12 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ ) : RR+ --> RR )
9998feqmptd 5927 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
100 fvres 5886 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
101100mpteq2ia 4535 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
10299, 101syl6eq 2524 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
103102oveq2d 6311 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( RR  _D  ( log  |`  RR+ )
)  =  ( RR 
_D  ( x  e.  RR+  |->  ( log `  x
) ) ) )
10495, 103syl5reqr 2523 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( RR  _D  ( x  e.  RR+  |->  ( log `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) ) )
10587, 90, 94, 104dvmptneg 22235 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( RR  _D  ( x  e.  RR+  |->  -u ( log `  x
) ) )  =  ( x  e.  RR+  |->  -u ( 1  /  x
) ) )
106 eqid 2467 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
107106tgioo2 21174 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
108 iccntr 21192 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
1092, 4, 108syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
11087, 91, 92, 105, 15, 107, 106, 109dvmptres2 22231 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( RR  _D  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) ) )  =  ( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) )
111 isoeq1 6214 . . . . . 6  |-  ( ( RR  _D  ( x  e.  ( A [,] B )  |->  -u ( log `  x ) ) )  =  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  ->  ( ( RR  _D  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) ) )  Isom  <  ,  <  ( ( A (,) B ) ,  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )  <-> 
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) )  Isom  <  ,  <  ( ( A (,) B ) ,  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) ) ) )
112110, 111syl 16 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( RR  _D  (
x  e.  ( A [,] B )  |->  -u ( log `  x ) ) )  Isom  <  ,  <  ( ( A (,) B ) ,  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )  <-> 
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) )  Isom  <  ,  <  ( ( A (,) B ) ,  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) ) ) )
11385, 112mpbird 232 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( RR  _D  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) ) )  Isom  <  ,  <  ( ( A (,) B ) ,  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) ) )
114 simpr 461 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  ( 0 (,) 1
) )
115 eqid 2467 . . . 4  |-  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) )
1162, 4, 5, 39, 113, 114, 115dvcvx 22287 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  <  ( ( T  x.  ( ( x  e.  ( A [,] B )  |->  -u ( log `  x ) ) `  A ) )  +  ( ( 1  -  T )  x.  ( ( x  e.  ( A [,] B )  |->  -u ( log `  x ) ) `
 B ) ) ) )
117 ax-1cn 9562 . . . . . . . 8  |-  1  e.  CC
118 elioore 11571 . . . . . . . . . 10  |-  ( T  e.  ( 0 (,) 1 )  ->  T  e.  RR )
119118adantl 466 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  RR )
120119recnd 9634 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  CC )
121 nncan 9860 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
122117, 120, 121sylancr 663 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  ( 1  -  T ) )  =  T )
123122oveq1d 6310 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  (
1  -  T ) )  x.  A )  =  ( T  x.  A ) )
124123oveq1d 6310 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( ( 1  -  ( 1  -  T
) )  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )
125 ioossicc 11622 . . . . . . . 8  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
126125, 114sseldi 3507 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  ( 0 [,] 1
) )
127 iirev 21295 . . . . . . 7  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
128126, 127syl 16 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
129 lincmb01cmp 11675 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( 1  -  T
)  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  ( 1  -  T ) )  x.  A )  +  ( ( 1  -  T )  x.  B
) )  e.  ( A [,] B ) )
1302, 4, 5, 128, 129syl31anc 1231 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( ( 1  -  ( 1  -  T
) )  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  ( A [,] B ) )
131124, 130eqeltrrd 2556 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  ( A [,] B ) )
132 fveq2 5872 . . . . . 6  |-  ( x  =  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  ->  ( log `  x )  =  ( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
133132negeqd 9826 . . . . 5  |-  ( x  =  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  ->  -u ( log `  x )  = 
-u ( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
134 negex 9830 . . . . 5  |-  -u ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )  e. 
_V
135133, 19, 134fvmpt 5957 . . . 4  |-  ( ( ( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  =  -u ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) ) )
136131, 135syl 16 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  =  -u ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) ) )
1371rpxrd 11269 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR* )
1383rpxrd 11269 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR* )
1392, 4, 5ltled 9744 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  <_  B )
140 lbicc2 11648 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
141137, 138, 139, 140syl3anc 1228 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  ( A [,] B
) )
142 fveq2 5872 . . . . . . . . . 10  |-  ( x  =  A  ->  ( log `  x )  =  ( log `  A
) )
143142negeqd 9826 . . . . . . . . 9  |-  ( x  =  A  ->  -u ( log `  x )  = 
-u ( log `  A
) )
144 negex 9830 . . . . . . . . 9  |-  -u ( log `  A )  e. 
_V
145143, 19, 144fvmpt 5957 . . . . . . . 8  |-  ( A  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  A
)  =  -u ( log `  A ) )
146141, 145syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  A
)  =  -u ( log `  A ) )
147146oveq2d 6311 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( (
x  e.  ( A [,] B )  |->  -u ( log `  x ) ) `  A ) )  =  ( T  x.  -u ( log `  A
) ) )
1481relogcld 22872 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  A )  e.  RR )
149148recnd 9634 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  A )  e.  CC )
150120, 149mulneg2d 10022 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  -u ( log `  A ) )  = 
-u ( T  x.  ( log `  A ) ) )
151147, 150eqtrd 2508 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( (
x  e.  ( A [,] B )  |->  -u ( log `  x ) ) `  A ) )  =  -u ( T  x.  ( log `  A ) ) )
152 ubicc2 11649 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
153137, 138, 139, 152syl3anc 1228 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  ( A [,] B
) )
154 fveq2 5872 . . . . . . . . . 10  |-  ( x  =  B  ->  ( log `  x )  =  ( log `  B
) )
155154negeqd 9826 . . . . . . . . 9  |-  ( x  =  B  ->  -u ( log `  x )  = 
-u ( log `  B
) )
156 negex 9830 . . . . . . . . 9  |-  -u ( log `  B )  e. 
_V
157155, 19, 156fvmpt 5957 . . . . . . . 8  |-  ( B  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  B
)  =  -u ( log `  B ) )
158153, 157syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  B
)  =  -u ( log `  B ) )
159158oveq2d 6311 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( ( x  e.  ( A [,] B )  |->  -u ( log `  x ) ) `  B ) )  =  ( ( 1  -  T )  x.  -u ( log `  B
) ) )
160 1re 9607 . . . . . . . . 9  |-  1  e.  RR
161 resubcl 9895 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
162160, 119, 161sylancr 663 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  RR )
163162recnd 9634 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  CC )
1643relogcld 22872 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  B )  e.  RR )
165164recnd 9634 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  B )  e.  CC )
166163, 165mulneg2d 10022 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  -u ( log `  B ) )  =  -u ( ( 1  -  T )  x.  ( log `  B
) ) )
167159, 166eqtrd 2508 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( ( x  e.  ( A [,] B )  |->  -u ( log `  x ) ) `  B ) )  =  -u (
( 1  -  T
)  x.  ( log `  B ) ) )
168151, 167oveq12d 6313 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  A
) )  +  ( ( 1  -  T
)  x.  ( ( x  e.  ( A [,] B )  |->  -u ( log `  x ) ) `  B ) ) )  =  (
-u ( T  x.  ( log `  A ) )  +  -u (
( 1  -  T
)  x.  ( log `  B ) ) ) )
169119, 148remulcld 9636 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( log `  A ) )  e.  RR )
170169recnd 9634 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( log `  A ) )  e.  CC )
171162, 164remulcld 9636 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( log `  B ) )  e.  RR )
172171recnd 9634 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( log `  B ) )  e.  CC )
173170, 172negdid 9955 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  -u (
( T  x.  ( log `  A ) )  +  ( ( 1  -  T )  x.  ( log `  B
) ) )  =  ( -u ( T  x.  ( log `  A
) )  +  -u ( ( 1  -  T )  x.  ( log `  B ) ) ) )
174168, 173eqtr4d 2511 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  A
) )  +  ( ( 1  -  T
)  x.  ( ( x  e.  ( A [,] B )  |->  -u ( log `  x ) ) `  B ) ) )  =  -u ( ( T  x.  ( log `  A ) )  +  ( ( 1  -  T )  x.  ( log `  B
) ) ) )
175116, 136, 1743brtr3d 4482 . 2  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  -u ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )  <  -u ( ( T  x.  ( log `  A ) )  +  ( ( 1  -  T )  x.  ( log `  B
) ) ) )
176169, 171readdcld 9635 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  ( log `  A ) )  +  ( ( 1  -  T )  x.  ( log `  B
) ) )  e.  RR )
17715, 131sseldd 3510 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  RR+ )
178177relogcld 22872 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )  e.  RR )
179176, 178ltnegd 10142 . 2  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( ( T  x.  ( log `  A ) )  +  ( ( 1  -  T )  x.  ( log `  B
) ) )  < 
( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  <->  -u ( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  <  -u (
( T  x.  ( log `  A ) )  +  ( ( 1  -  T )  x.  ( log `  B
) ) ) ) )
180175, 179mpbird 232 1  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  ( log `  A ) )  +  ( ( 1  -  T )  x.  ( log `  B
) ) )  < 
( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    C_ wss 3481   {cpr 4035   class class class wbr 4453    |-> cmpt 4511    Po wpo 4804    Or wor 4805   ran crn 5006    |` cres 5007    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509   +oocpnf 9637   RR*cxr 9639    < clt 9640    <_ cle 9641    - cmin 9817   -ucneg 9818    / cdiv 10218   RR+crp 11232   (,)cioo 11541   [,]cicc 11544   TopOpenctopn 14693   topGenctg 14709  ℂfldccnfld 18288   intcnt 19384   -cn->ccncf 21246    _D cdv 22133   logclog 22806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684  df-pi 13686  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-rest 14694  df-topn 14695  df-0g 14713  df-gsum 14714  df-topgen 14715  df-pt 14716  df-prds 14719  df-xrs 14773  df-qtop 14778  df-imas 14779  df-xps 14781  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-submnd 15839  df-mulg 15931  df-cntz 16226  df-cmn 16671  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-lp 19503  df-perf 19504  df-cn 19594  df-cnp 19595  df-haus 19682  df-cmp 19753  df-tx 19929  df-hmeo 20122  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-xms 20689  df-ms 20690  df-tms 20691  df-cncf 21248  df-limc 22136  df-dv 22137  df-log 22808
This theorem is referenced by:  amgmlem  23183
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