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Theorem logccv 23512
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 1008 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR+ )
21rpred 11330 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR )
3 simpl2 1009 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR+ )
43rpred 11330 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR )
5 simpl3 1010 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  <  B )
61rpgt0d 11333 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  0  <  A )
7 ltpnf 11411 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  B  < +oo )
84, 7syl 17 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  < +oo )
9 0xr 9676 . . . . . . . . . . . 12  |-  0  e.  RR*
10 pnfxr 11401 . . . . . . . . . . . 12  |- +oo  e.  RR*
11 iccssioo 11692 . . . . . . . . . . . 12  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <  A  /\  B  < +oo ) )  -> 
( A [,] B
)  C_  ( 0 (,) +oo ) )
129, 10, 11mpanl12 686 . . . . . . . . . . 11  |-  ( ( 0  <  A  /\  B  < +oo )  ->  ( A [,] B )  C_  ( 0 (,) +oo ) )
136, 8, 12syl2anc 665 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A [,] B )  C_  ( 0 (,) +oo ) )
14 ioorp 11701 . . . . . . . . . 10  |-  ( 0 (,) +oo )  = 
RR+
1513, 14syl6sseq 3507 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A [,] B )  C_  RR+ )
1615sselda 3461 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR+ )
1716relogcld 23476 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  ( log `  x )  e.  RR )
1817renegcld 10035 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  -u ( log `  x )  e.  RR )
19 eqid 2420 . . . . . 6  |-  ( x  e.  ( A [,] B )  |->  -u ( log `  x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )
2018, 19fmptd 6052 . . . . 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 9585 . . . . . 6  |-  RR  C_  CC
2215resabs1d 5145 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  =  ( log  |`  ( A [,] B ) ) )
23 ssid 3480 . . . . . . . . . . 11  |-  CC  C_  CC
24 cncfss 21853 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC ) )
2521, 23, 24mp2an 676 . . . . . . . . . 10  |-  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC )
26 relogcn 23487 . . . . . . . . . 10  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
2725, 26sselii 3458 . . . . . . . . 9  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )
28 rescncf 21851 . . . . . . . . 9  |-  ( ( A [,] B ) 
C_  RR+  ->  ( ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) ) )
2915, 27, 28mpisyl 22 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) )
3022, 29eqeltrrd 2509 . . . . . . 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 9858 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  -u (
( log  |`  ( A [,] B ) ) `
 x )  = 
-u ( log `  x
) )
3332mpteq2ia 4499 . . . . . . . . 9  |-  ( x  e.  ( A [,] B )  |->  -u (
( log  |`  ( A [,] B ) ) `
 x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )
3433eqcomi 2433 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  -u ( log `  x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( ( log  |`  ( A [,] B
) ) `  x
) )
3534negfcncf 21873 . . . . . . 7  |-  ( ( log  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC )  ->  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
3630, 35syl 17 . . . . . 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 21852 . . . . . 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 667 . . . . 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 235 . . . 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 11685 . . . . . . . 8  |-  ( A (,) B )  C_  RR
41 ltso 9703 . . . . . . . 8  |-  <  Or  RR
42 soss 4784 . . . . . . . 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 11709 . . . . . . . . . . . . . 14  |-  ( A (,) B )  C_  ( A [,] B )
4645, 15syl5ss 3472 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A (,) B )  C_  RR+ )
4746sselda 3461 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  x  e.  RR+ )
4847rprecred 11341 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  (
1  /  x )  e.  RR )
4948renegcld 10035 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  -u (
1  /  x )  e.  RR )
50 eqid 2420 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  =  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )
5149, 50fmptd 6052 . . . . . . . . 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 17 . . . . . . . 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 4784 . . . . . . . 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 22 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Or 
ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
56 sopo 4783 . . . . . . 7  |-  (  < 
Or  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) )  ->  <  Po  ran  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) )
5755, 56syl 17 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Po 
ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
58 negex 9862 . . . . . . . . 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 211 . . . . . . 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 3461 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  z  e.  ( A (,) B
) )  ->  z  e.  RR+ )
6463adantrl 720 . . . . . . . . . . 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 11341 . . . . . . . . . 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 3461 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  y  e.  ( A (,) B
) )  ->  y  e.  RR+ )
6766adantrr 721 . . . . . . . . . . 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 11341 . . . . . . . . . 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 10180 . . . . . . . . 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 11348 . . . . . . . . 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 6304 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
1  /  x )  =  ( 1  / 
y ) )
7271negeqd 9858 . . . . . . . . . . . 12  |-  ( x  =  y  ->  -u (
1  /  x )  =  -u ( 1  / 
y ) )
73 negex 9862 . . . . . . . . . . . 12  |-  -u (
1  /  y )  e.  _V
7472, 50, 73fvmpt 5955 . . . . . . . . . . 11  |-  ( y  e.  ( A (,) B )  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  y )  =  -u ( 1  /  y
) )
75 oveq2 6304 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
1  /  x )  =  ( 1  / 
z ) )
7675negeqd 9858 . . . . . . . . . . . 12  |-  ( x  =  z  ->  -u (
1  /  x )  =  -u ( 1  / 
z ) )
77 negex 9862 . . . . . . . . . . . 12  |-  -u (
1  /  z )  e.  _V
7876, 50, 77fvmpt 5955 . . . . . . . . . . 11  |-  ( z  e.  ( A (,) B )  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  z )  =  -u ( 1  /  z
) )
7974, 78breqan12d 4432 . . . . . . . . . 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 467 . . . . . . . . 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 288 . . . . . . . 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 210 . . . . . . 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 2844 . . . . . 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 6225 . . . . . 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 1265 . . . . 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 9620 . . . . . . . 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 23429 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
8988adantl 467 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
9089recnd 9658 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
9190negcld 9962 . . . . . . 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 6324 . . . . . . . . 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 23486 . . . . . . . . 9  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
96 relogf1o 23420 . . . . . . . . . . . . 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 13 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ ) : RR+ --> RR )
9998feqmptd 5925 . . . . . . . . . . 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 4499 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
10299, 101syl6eq 2477 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
103102oveq2d 6312 . . . . . . . . 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 2476 . . . . . . . 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 22827 . . . . . . 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 2420 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
107106tgioo2 21758 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
108 iccntr 21776 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
1092, 4, 108syl2anc 665 . . . . . . 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 22823 . . . . . 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 6216 . . . . . 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 17 . . . . 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 235 . . . 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 462 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  ( 0 (,) 1
) )
115 eqid 2420 . . . 4  |-  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) )
1162, 4, 5, 39, 113, 114, 115dvcvx 22879 . . 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 9586 . . . . . . . 8  |-  1  e.  CC
118 elioore 11655 . . . . . . . . . 10  |-  ( T  e.  ( 0 (,) 1 )  ->  T  e.  RR )
119118adantl 467 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  RR )
120119recnd 9658 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  CC )
121 nncan 9892 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
122117, 120, 121sylancr 667 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  ( 1  -  T ) )  =  T )
123122oveq1d 6311 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  (
1  -  T ) )  x.  A )  =  ( T  x.  A ) )
124123oveq1d 6311 . . . . 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 11709 . . . . . . . 8  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
126125, 114sseldi 3459 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  ( 0 [,] 1
) )
127 iirev 21879 . . . . . . 7  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
128126, 127syl 17 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
129 lincmb01cmp 11762 . . . . . 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 1267 . . . . 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 2509 . . . 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 9858 . . . . 5  |-  ( x  =  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  ->  -u ( log `  x )  = 
-u ( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
134 negex 9862 . . . . 5  |-  -u ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )  e. 
_V
135133, 19, 134fvmpt 5955 . . . 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 17 . . 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 11331 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR* )
1383rpxrd 11331 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR* )
1392, 4, 5ltled 9772 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  <_  B )
140 lbicc2 11735 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
141137, 138, 139, 140syl3anc 1264 . . . . . . . 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 9858 . . . . . . . . 9  |-  ( x  =  A  ->  -u ( log `  x )  = 
-u ( log `  A
) )
144 negex 9862 . . . . . . . . 9  |-  -u ( log `  A )  e. 
_V
145143, 19, 144fvmpt 5955 . . . . . . . 8  |-  ( A  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  A
)  =  -u ( log `  A ) )
146141, 145syl 17 . . . . . . 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 6312 . . . . . 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 23476 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  A )  e.  RR )
149148recnd 9658 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  A )  e.  CC )
150120, 149mulneg2d 10061 . . . . . 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 2461 . . . . 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 11736 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
153137, 138, 139, 152syl3anc 1264 . . . . . . . 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 9858 . . . . . . . . 9  |-  ( x  =  B  ->  -u ( log `  x )  = 
-u ( log `  B
) )
156 negex 9862 . . . . . . . . 9  |-  -u ( log `  B )  e. 
_V
157155, 19, 156fvmpt 5955 . . . . . . . 8  |-  ( B  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  B
)  =  -u ( log `  B ) )
158153, 157syl 17 . . . . . . 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 6312 . . . . . 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 9631 . . . . . . . . 9  |-  1  e.  RR
161 resubcl 9927 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
162160, 119, 161sylancr 667 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  RR )
163162recnd 9658 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  CC )
1643relogcld 23476 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  B )  e.  RR )
165164recnd 9658 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  B )  e.  CC )
166163, 165mulneg2d 10061 . . . . . 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 2461 . . . . 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 6314 . . . 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 9660 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( log `  A ) )  e.  RR )
170169recnd 9658 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( log `  A ) )  e.  CC )
171162, 164remulcld 9660 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( log `  B ) )  e.  RR )
172171recnd 9658 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( log `  B ) )  e.  CC )
173170, 172negdid 9988 . . . 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 2464 . . 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 4446 . 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 9659 . . 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 3462 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  RR+ )
178177relogcld 23476 . . 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 10180 . 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 235 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078    C_ wss 3433   {cpr 3995   class class class wbr 4417    |-> cmpt 4475    Po wpo 4764    Or wor 4765   ran crn 4846    |` cres 4847    Fn wfn 5587   -->wf 5588   -onto->wfo 5590   -1-1-onto->wf1o 5591   ` cfv 5592    Isom wiso 5593  (class class class)co 6296   CCcc 9526   RRcr 9527   0cc0 9528   1c1 9529    + caddc 9531    x. cmul 9533   +oocpnf 9661   RR*cxr 9663    < clt 9664    <_ cle 9665    - cmin 9849   -ucneg 9850    / cdiv 10258   RR+crp 11291   (,)cioo 11624   [,]cicc 11627   TopOpenctopn 15280   topGenctg 15296  ℂfldccnfld 18911   intcnt 19969   -cn->ccncf 21830    _D cdv 22725   logclog 23408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-ioo 11628  df-ioc 11629  df-ico 11630  df-icc 11631  df-fz 11772  df-fzo 11903  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-fac 12446  df-bc 12474  df-hash 12502  df-shft 13098  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-limsup 13493  df-clim 13519  df-rlim 13520  df-sum 13720  df-ef 14088  df-sin 14090  df-cos 14091  df-pi 14093  df-struct 15083  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-ress 15088  df-plusg 15163  df-mulr 15164  df-starv 15165  df-sca 15166  df-vsca 15167  df-ip 15168  df-tset 15169  df-ple 15170  df-ds 15172  df-unif 15173  df-hom 15174  df-cco 15175  df-rest 15281  df-topn 15282  df-0g 15300  df-gsum 15301  df-topgen 15302  df-pt 15303  df-prds 15306  df-xrs 15360  df-qtop 15365  df-imas 15366  df-xps 15368  df-mre 15444  df-mrc 15445  df-acs 15447  df-mgm 16440  df-sgrp 16479  df-mnd 16489  df-submnd 16535  df-mulg 16628  df-cntz 16923  df-cmn 17373  df-psmet 18903  df-xmet 18904  df-met 18905  df-bl 18906  df-mopn 18907  df-fbas 18908  df-fg 18909  df-cnfld 18912  df-top 19858  df-bases 19859  df-topon 19860  df-topsp 19861  df-cld 19971  df-ntr 19972  df-cls 19973  df-nei 20051  df-lp 20089  df-perf 20090  df-cn 20180  df-cnp 20181  df-haus 20268  df-cmp 20339  df-tx 20514  df-hmeo 20707  df-fil 20798  df-fm 20890  df-flim 20891  df-flf 20892  df-xms 21272  df-ms 21273  df-tms 21274  df-cncf 21832  df-limc 22728  df-dv 22729  df-log 23410
This theorem is referenced by:  amgmlem  23819
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