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Theorem logccv 23687
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 1033 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR+ )
21rpred 11364 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR )
3 simpl2 1034 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR+ )
43rpred 11364 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR )
5 simpl3 1035 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  <  B )
61rpgt0d 11367 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  0  <  A )
7 ltpnf 11445 . . . . . . . . . . . 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 9705 . . . . . . . . . . . 12  |-  0  e.  RR*
10 pnfxr 11435 . . . . . . . . . . . 12  |- +oo  e.  RR*
11 iccssioo 11728 . . . . . . . . . . . 12  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <  A  /\  B  < +oo ) )  -> 
( A [,] B
)  C_  ( 0 (,) +oo ) )
129, 10, 11mpanl12 696 . . . . . . . . . . 11  |-  ( ( 0  <  A  /\  B  < +oo )  ->  ( A [,] B )  C_  ( 0 (,) +oo ) )
136, 8, 12syl2anc 673 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A [,] B )  C_  ( 0 (,) +oo ) )
14 ioorp 11737 . . . . . . . . . 10  |-  ( 0 (,) +oo )  = 
RR+
1513, 14syl6sseq 3464 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A [,] B )  C_  RR+ )
1615sselda 3418 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR+ )
1716relogcld 23651 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  ( log `  x )  e.  RR )
1817renegcld 10067 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  -u ( log `  x )  e.  RR )
19 eqid 2471 . . . . . 6  |-  ( x  e.  ( A [,] B )  |->  -u ( log `  x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )
2018, 19fmptd 6061 . . . . 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 9614 . . . . . 6  |-  RR  C_  CC
2215resabs1d 5140 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  =  ( log  |`  ( A [,] B ) ) )
23 ssid 3437 . . . . . . . . . . 11  |-  CC  C_  CC
24 cncfss 22009 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC ) )
2521, 23, 24mp2an 686 . . . . . . . . . 10  |-  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC )
26 relogcn 23662 . . . . . . . . . 10  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
2725, 26sselii 3415 . . . . . . . . 9  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )
28 rescncf 22007 . . . . . . . . 9  |-  ( ( A [,] B ) 
C_  RR+  ->  ( ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) ) )
2915, 27, 28mpisyl 21 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) )
3022, 29eqeltrrd 2550 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
31 fvres 5893 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  (
( log  |`  ( A [,] B ) ) `
 x )  =  ( log `  x
) )
3231negeqd 9889 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  -u (
( log  |`  ( A [,] B ) ) `
 x )  = 
-u ( log `  x
) )
3332mpteq2ia 4478 . . . . . . . . 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 22029 . . . . . . 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 22008 . . . . . 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 676 . . . . 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 240 . . . 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 11721 . . . . . . . 8  |-  ( A (,) B )  C_  RR
41 ltso 9732 . . . . . . . 8  |-  <  Or  RR
42 soss 4778 . . . . . . . 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 11745 . . . . . . . . . . . . . 14  |-  ( A (,) B )  C_  ( A [,] B )
4645, 15syl5ss 3429 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A (,) B )  C_  RR+ )
4746sselda 3418 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  x  e.  RR+ )
4847rprecred 11375 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  (
1  /  x )  e.  RR )
4948renegcld 10067 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  -u (
1  /  x )  e.  RR )
50 eqid 2471 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  =  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )
5149, 50fmptd 6061 . . . . . . . . 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 5747 . . . . . . . . 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 4778 . . . . . . . 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 21 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Or 
ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
56 sopo 4777 . . . . . . 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 9893 . . . . . . . . 9  |-  -u (
1  /  x )  e.  _V
5958, 50fnmpti 5716 . . . . . . . 8  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  Fn  ( A (,) B )
60 dffn4 5812 . . . . . . . 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 213 . . . . . . 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 3418 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  z  e.  ( A (,) B
) )  ->  z  e.  RR+ )
6463adantrl 730 . . . . . . . . . . 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 11375 . . . . . . . . . 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 3418 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  y  e.  ( A (,) B
) )  ->  y  e.  RR+ )
6766adantrr 731 . . . . . . . . . . 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 11375 . . . . . . . . . 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 10212 . . . . . . . . 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 11382 . . . . . . . . 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 6316 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
1  /  x )  =  ( 1  / 
y ) )
7271negeqd 9889 . . . . . . . . . . . 12  |-  ( x  =  y  ->  -u (
1  /  x )  =  -u ( 1  / 
y ) )
73 negex 9893 . . . . . . . . . . . 12  |-  -u (
1  /  y )  e.  _V
7472, 50, 73fvmpt 5963 . . . . . . . . . . 11  |-  ( y  e.  ( A (,) B )  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  y )  =  -u ( 1  /  y
) )
75 oveq2 6316 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
1  /  x )  =  ( 1  / 
z ) )
7675negeqd 9889 . . . . . . . . . . . 12  |-  ( x  =  z  ->  -u (
1  /  x )  =  -u ( 1  / 
z ) )
77 negex 9893 . . . . . . . . . . . 12  |-  -u (
1  /  z )  e.  _V
7876, 50, 77fvmpt 5963 . . . . . . . . . . 11  |-  ( z  e.  ( A (,) B )  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  z )  =  -u ( 1  /  z
) )
7974, 78breqan12d 4411 . . . . . . . . . 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 473 . . . . . . . . 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 293 . . . . . . . 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 212 . . . . . . 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 2814 . . . . . 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 6237 . . . . . 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 1293 . . . . 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 9649 . . . . . . . 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 23604 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
8988adantl 473 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
9089recnd 9687 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
9190negcld 9992 . . . . . . 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 6336 . . . . . . . . 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 23661 . . . . . . . . 9  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
96 relogf1o 23595 . . . . . . . . . . . . 13  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
97 f1of 5828 . . . . . . . . . . . . 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 5932 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
100 fvres 5893 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
101100mpteq2ia 4478 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
10299, 101syl6eq 2521 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
103102oveq2d 6324 . . . . . . . . 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 2520 . . . . . . . 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 22999 . . . . . . 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 2471 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
107106tgioo2 21899 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
108 iccntr 21917 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
1092, 4, 108syl2anc 673 . . . . . . 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 22995 . . . . . 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 6228 . . . . . 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 240 . . . 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 468 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  ( 0 (,) 1
) )
115 eqid 2471 . . . 4  |-  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) )
1162, 4, 5, 39, 113, 114, 115dvcvx 23051 . . 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 9615 . . . . . . . 8  |-  1  e.  CC
118 elioore 11691 . . . . . . . . . 10  |-  ( T  e.  ( 0 (,) 1 )  ->  T  e.  RR )
119118adantl 473 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  RR )
120119recnd 9687 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  CC )
121 nncan 9923 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
122117, 120, 121sylancr 676 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  ( 1  -  T ) )  =  T )
123122oveq1d 6323 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  (
1  -  T ) )  x.  A )  =  ( T  x.  A ) )
124123oveq1d 6323 . . . . 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 11745 . . . . . . . 8  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
126125, 114sseldi 3416 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  ( 0 [,] 1
) )
127 iirev 22035 . . . . . . 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 11801 . . . . . 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 1295 . . . . 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 2550 . . . 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 5879 . . . . . 6  |-  ( x  =  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  ->  ( log `  x )  =  ( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
133132negeqd 9889 . . . . 5  |-  ( x  =  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  ->  -u ( log `  x )  = 
-u ( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
134 negex 9893 . . . . 5  |-  -u ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )  e. 
_V
135133, 19, 134fvmpt 5963 . . . 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 11365 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR* )
1383rpxrd 11365 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR* )
1392, 4, 5ltled 9800 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  <_  B )
140 lbicc2 11774 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
141137, 138, 139, 140syl3anc 1292 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  ( A [,] B
) )
142 fveq2 5879 . . . . . . . . . 10  |-  ( x  =  A  ->  ( log `  x )  =  ( log `  A
) )
143142negeqd 9889 . . . . . . . . 9  |-  ( x  =  A  ->  -u ( log `  x )  = 
-u ( log `  A
) )
144 negex 9893 . . . . . . . . 9  |-  -u ( log `  A )  e. 
_V
145143, 19, 144fvmpt 5963 . . . . . . . 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 6324 . . . . . 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 23651 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  A )  e.  RR )
149148recnd 9687 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  A )  e.  CC )
150120, 149mulneg2d 10093 . . . . . 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 2505 . . . . 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 11775 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
153137, 138, 139, 152syl3anc 1292 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  ( A [,] B
) )
154 fveq2 5879 . . . . . . . . . 10  |-  ( x  =  B  ->  ( log `  x )  =  ( log `  B
) )
155154negeqd 9889 . . . . . . . . 9  |-  ( x  =  B  ->  -u ( log `  x )  = 
-u ( log `  B
) )
156 negex 9893 . . . . . . . . 9  |-  -u ( log `  B )  e. 
_V
157155, 19, 156fvmpt 5963 . . . . . . . 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 6324 . . . . . 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 9660 . . . . . . . . 9  |-  1  e.  RR
161 resubcl 9958 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
162160, 119, 161sylancr 676 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  RR )
163162recnd 9687 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  CC )
1643relogcld 23651 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  B )  e.  RR )
165164recnd 9687 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  B )  e.  CC )
166163, 165mulneg2d 10093 . . . . . 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 2505 . . . . 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 6326 . . . 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 9689 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( log `  A ) )  e.  RR )
170169recnd 9687 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( log `  A ) )  e.  CC )
171162, 164remulcld 9689 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( log `  B ) )  e.  RR )
172171recnd 9687 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( log `  B ) )  e.  CC )
173170, 172negdid 10018 . . . 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 2508 . . 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 4425 . 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 9688 . . 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 3419 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  RR+ )
178177relogcld 23651 . . 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 10212 . 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 240 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    C_ wss 3390   {cpr 3961   class class class wbr 4395    |-> cmpt 4454    Po wpo 4758    Or wor 4759   ran crn 4840    |` cres 4841    Fn wfn 5584   -->wf 5585   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   +oocpnf 9690   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   -ucneg 9881    / cdiv 10291   RR+crp 11325   (,)cioo 11660   [,]cicc 11663   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047   intcnt 20109   -cn->ccncf 21986    _D cdv 22897   logclog 23583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585
This theorem is referenced by:  amgmlem  23994
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