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Theorem logccv 20507
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 960 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR+ )
21rpred 10604 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR )
3 simpl2 961 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR+ )
43rpred 10604 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR )
5 simpl3 962 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  <  B )
61rpgt0d 10607 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  0  <  A )
7 ltpnf 10677 . . . . . . . . . . . 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 9087 . . . . . . . . . . . 12  |-  0  e.  RR*
10 pnfxr 10669 . . . . . . . . . . . 12  |-  +oo  e.  RR*
11 iccssioo 10935 . . . . . . . . . . . 12  |-  ( ( ( 0  e.  RR*  /\ 
+oo  e.  RR* )  /\  ( 0  <  A  /\  B  <  +oo )
)  ->  ( A [,] B )  C_  (
0 (,)  +oo ) )
129, 10, 11mpanl12 664 . . . . . . . . . . 11  |-  ( ( 0  <  A  /\  B  <  +oo )  ->  ( A [,] B )  C_  ( 0 (,)  +oo ) )
136, 8, 12syl2anc 643 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A [,] B )  C_  ( 0 (,)  +oo ) )
14 ioorp 10944 . . . . . . . . . 10  |-  ( 0 (,)  +oo )  =  RR+
1513, 14syl6sseq 3354 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A [,] B )  C_  RR+ )
1615sselda 3308 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR+ )
1716relogcld 20471 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  ( log `  x )  e.  RR )
1817renegcld 9420 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  -u ( log `  x )  e.  RR )
19 eqid 2404 . . . . . 6  |-  ( x  e.  ( A [,] B )  |->  -u ( log `  x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )
2018, 19fmptd 5852 . . . . 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 9003 . . . . . 6  |-  RR  C_  CC
22 resabs1 5134 . . . . . . . . 9  |-  ( ( A [,] B ) 
C_  RR+  ->  ( ( log  |`  RR+ )  |`  ( A [,] B ) )  =  ( log  |`  ( A [,] B ) ) )
2315, 22syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  =  ( log  |`  ( A [,] B ) ) )
24 ssid 3327 . . . . . . . . . . 11  |-  CC  C_  CC
25 cncfss 18882 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC ) )
2621, 24, 25mp2an 654 . . . . . . . . . 10  |-  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC )
27 relogcn 20482 . . . . . . . . . 10  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
2826, 27sselii 3305 . . . . . . . . 9  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )
29 rescncf 18880 . . . . . . . . 9  |-  ( ( A [,] B ) 
C_  RR+  ->  ( ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) ) )
3015, 28, 29ee10 1382 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) )
3123, 30eqeltrrd 2479 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
32 fvres 5704 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  (
( log  |`  ( A [,] B ) ) `
 x )  =  ( log `  x
) )
3332negeqd 9256 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  -u (
( log  |`  ( A [,] B ) ) `
 x )  = 
-u ( log `  x
) )
3433mpteq2ia 4251 . . . . . . . . 9  |-  ( x  e.  ( A [,] B )  |->  -u (
( log  |`  ( A [,] B ) ) `
 x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )
3534eqcomi 2408 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  -u ( log `  x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( ( log  |`  ( A [,] B
) ) `  x
) )
3635negfcncf 18902 . . . . . . 7  |-  ( ( log  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC )  ->  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
3731, 36syl 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 ) )
38 cncffvrn 18881 . . . . . 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 ) )
3921, 37, 38sylancr 645 . . . . 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 ) )
4020, 39mpbird 224 . . . 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 ) )
41 ioossre 10928 . . . . . . . 8  |-  ( A (,) B )  C_  RR
42 ltso 9112 . . . . . . . 8  |-  <  Or  RR
43 soss 4481 . . . . . . . 8  |-  ( ( A (,) B ) 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  ( A (,) B ) ) )
4441, 42, 43mp2 9 . . . . . . 7  |-  <  Or  ( A (,) B )
4544a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Or  ( A (,) B
) )
46 ioossicc 10952 . . . . . . . . . . . . . 14  |-  ( A (,) B )  C_  ( A [,] B )
4746, 15syl5ss 3319 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A (,) B )  C_  RR+ )
4847sselda 3308 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  x  e.  RR+ )
4948rprecred 10615 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  (
1  /  x )  e.  RR )
5049renegcld 9420 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  -u (
1  /  x )  e.  RR )
51 eqid 2404 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  =  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )
5250, 51fmptd 5852 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
x  e.  ( A (,) B )  |->  -u ( 1  /  x
) ) : ( A (,) B ) --> RR )
53 frn 5556 . . . . . . . . 9  |-  ( ( x  e.  ( A (,) B )  |->  -u ( 1  /  x
) ) : ( A (,) B ) --> RR  ->  ran  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  C_  RR )
5452, 53syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ran  ( x  e.  ( A (,) B )  |->  -u ( 1  /  x
) )  C_  RR )
55 soss 4481 . . . . . . . 8  |-  ( ran  ( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) )  C_  RR  ->  (  <  Or  RR  ->  <  Or  ran  ( x  e.  ( A (,) B )  |->  -u ( 1  /  x
) ) ) )
5654, 42, 55ee10 1382 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Or 
ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
57 sopo 4480 . . . . . . 7  |-  (  < 
Or  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) )  ->  <  Po  ran  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) )
5856, 57syl 16 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Po 
ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
59 negex 9260 . . . . . . . . 9  |-  -u (
1  /  x )  e.  _V
6059, 51fnmpti 5532 . . . . . . . 8  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  Fn  ( A (,) B )
61 dffn4 5618 . . . . . . . 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 ) ) )
6260, 61mpbi 200 . . . . . . 7  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) : ( A (,) B ) -onto-> ran  ( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) )
6362a1i 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 ) ) )
6447sselda 3308 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  z  e.  ( A (,) B
) )  ->  z  e.  RR+ )
6564adantrl 697 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
z  e.  RR+ )
6665rprecred 10615 . . . . . . . . . 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 )
6747sselda 3308 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  y  e.  ( A (,) B
) )  ->  y  e.  RR+ )
6867adantrr 698 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  ( y  e.  ( A (,) B )  /\  z  e.  ( A (,) B
) ) )  -> 
y  e.  RR+ )
6968rprecred 10615 . . . . . . . . . 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 )
7066, 69ltnegd 9560 . . . . . . . . 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 ) ) )
7168, 65ltrecd 10622 . . . . . . . . 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 ) ) )
72 oveq2 6048 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
1  /  x )  =  ( 1  / 
y ) )
7372negeqd 9256 . . . . . . . . . . . 12  |-  ( x  =  y  ->  -u (
1  /  x )  =  -u ( 1  / 
y ) )
74 negex 9260 . . . . . . . . . . . 12  |-  -u (
1  /  y )  e.  _V
7573, 51, 74fvmpt 5765 . . . . . . . . . . 11  |-  ( y  e.  ( A (,) B )  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  y )  =  -u ( 1  /  y
) )
76 oveq2 6048 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
1  /  x )  =  ( 1  / 
z ) )
7776negeqd 9256 . . . . . . . . . . . 12  |-  ( x  =  z  ->  -u (
1  /  x )  =  -u ( 1  / 
z ) )
78 negex 9260 . . . . . . . . . . . 12  |-  -u (
1  /  z )  e.  _V
7977, 51, 78fvmpt 5765 . . . . . . . . . . 11  |-  ( z  e.  ( A (,) B )  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  z )  =  -u ( 1  /  z
) )
8075, 79breqan12d 4187 . . . . . . . . . 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 ) ) )
8180adantl 453 . . . . . . . . 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 ) ) )
8270, 71, 813bitr4d 277 . . . . . . . 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 ) ) )
8382biimpd 199 . . . . . . 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 ) ) )
8483ralrimivva 2758 . . . . . 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
) ) )
85 soisoi 6007 . . . . . 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 ) ) ) )
8645, 58, 63, 84, 85syl22anc 1185 . . . . 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 ) ) ) )
87 reex 9037 . . . . . . . . 9  |-  RR  e.  _V
8887prid1 3872 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
8988a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  RR  e.  { RR ,  CC } )
90 relogcl 20426 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
9190adantl 453 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
9291recnd 9070 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
9392negcld 9354 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  -u ( log `  x )  e.  CC )
9459a1i 11 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  -u (
1  /  x )  e.  _V )
95 ovex 6065 . . . . . . . . 9  |-  ( 1  /  x )  e. 
_V
9695a1i 11 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( 1  /  x )  e. 
_V )
97 dvrelog 20481 . . . . . . . . 9  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
98 relogf1o 20417 . . . . . . . . . . . . 13  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
99 f1of 5633 . . . . . . . . . . . . 13  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
10098, 99mp1i 12 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ ) : RR+ --> RR )
101100feqmptd 5738 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
102 fvres 5704 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
103102mpteq2ia 4251 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
104101, 103syl6eq 2452 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
105104oveq2d 6056 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( RR  _D  ( log  |`  RR+ )
)  =  ( RR 
_D  ( x  e.  RR+  |->  ( log `  x
) ) ) )
10697, 105syl5reqr 2451 . . . . . . . 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
) ) )
10789, 92, 96, 106dvmptneg 19805 . . . . . . 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
) ) )
108 eqid 2404 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
109108tgioo2 18787 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
110 iccntr 18805 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
1112, 4, 110syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
11289, 93, 94, 107, 15, 109, 108, 111dvmptres2 19801 . . . . . 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 ) ) )
113 isoeq1 5998 . . . . . 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 ) ) ) ) )
114112, 113syl 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 ) ) ) ) )
11586, 114mpbird 224 . . . 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 ) ) ) )
116 simpr 448 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  ( 0 (,) 1
) )
117 eqid 2404 . . . 4  |-  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) )
1182, 4, 5, 40, 115, 116, 117dvcvx 19857 . . 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 ) ) ) )
119 ax-1cn 9004 . . . . . . . 8  |-  1  e.  CC
120 elioore 10902 . . . . . . . . . 10  |-  ( T  e.  ( 0 (,) 1 )  ->  T  e.  RR )
121120adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  RR )
122121recnd 9070 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  CC )
123 nncan 9286 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
124119, 122, 123sylancr 645 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  ( 1  -  T ) )  =  T )
125124oveq1d 6055 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  (
1  -  T ) )  x.  A )  =  ( T  x.  A ) )
126125oveq1d 6055 . . . . 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
) ) )
127 ioossicc 10952 . . . . . . . 8  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
128127, 116sseldi 3306 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  ( 0 [,] 1
) )
129 iirev 18907 . . . . . . 7  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
130128, 129syl 16 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
131 lincmb01cmp 10994 . . . . . 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 ) )
1322, 4, 5, 130, 131syl31anc 1187 . . . . 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 ) )
133126, 132eqeltrrd 2479 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  ( A [,] B ) )
134 fveq2 5687 . . . . . 6  |-  ( x  =  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  ->  ( log `  x )  =  ( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
135134negeqd 9256 . . . . 5  |-  ( x  =  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  ->  -u ( log `  x )  = 
-u ( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
136 negex 9260 . . . . 5  |-  -u ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )  e. 
_V
137135, 19, 136fvmpt 5765 . . . 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
) ) ) )
138133, 137syl 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
) ) ) )
1391rpxrd 10605 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR* )
1403rpxrd 10605 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR* )
1412, 4, 5ltled 9177 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  <_  B )
142 lbicc2 10969 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
143139, 140, 141, 142syl3anc 1184 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  ( A [,] B
) )
144 fveq2 5687 . . . . . . . . . 10  |-  ( x  =  A  ->  ( log `  x )  =  ( log `  A
) )
145144negeqd 9256 . . . . . . . . 9  |-  ( x  =  A  ->  -u ( log `  x )  = 
-u ( log `  A
) )
146 negex 9260 . . . . . . . . 9  |-  -u ( log `  A )  e. 
_V
147145, 19, 146fvmpt 5765 . . . . . . . 8  |-  ( A  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  A
)  =  -u ( log `  A ) )
148143, 147syl 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 ) )
149148oveq2d 6056 . . . . . 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
) ) )
1501relogcld 20471 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  A )  e.  RR )
151150recnd 9070 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  A )  e.  CC )
152122, 151mulneg2d 9443 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  -u ( log `  A ) )  = 
-u ( T  x.  ( log `  A ) ) )
153149, 152eqtrd 2436 . . . . 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 ) ) )
154 ubicc2 10970 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
155139, 140, 141, 154syl3anc 1184 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  ( A [,] B
) )
156 fveq2 5687 . . . . . . . . . 10  |-  ( x  =  B  ->  ( log `  x )  =  ( log `  B
) )
157156negeqd 9256 . . . . . . . . 9  |-  ( x  =  B  ->  -u ( log `  x )  = 
-u ( log `  B
) )
158 negex 9260 . . . . . . . . 9  |-  -u ( log `  B )  e. 
_V
159157, 19, 158fvmpt 5765 . . . . . . . 8  |-  ( B  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  B
)  =  -u ( log `  B ) )
160155, 159syl 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 ) )
161160oveq2d 6056 . . . . . 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
) ) )
162 1re 9046 . . . . . . . . 9  |-  1  e.  RR
163 resubcl 9321 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
164162, 121, 163sylancr 645 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  RR )
165164recnd 9070 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  CC )
1663relogcld 20471 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  B )  e.  RR )
167166recnd 9070 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  B )  e.  CC )
168165, 167mulneg2d 9443 . . . . . 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
) ) )
169161, 168eqtrd 2436 . . . . 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 ) ) )
170153, 169oveq12d 6058 . . . 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 ) ) ) )
171121, 150remulcld 9072 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( log `  A ) )  e.  RR )
172171recnd 9070 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( log `  A ) )  e.  CC )
173164, 166remulcld 9072 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( log `  B ) )  e.  RR )
174173recnd 9070 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( log `  B ) )  e.  CC )
175172, 174negdid 9380 . . . 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 ) ) ) )
176170, 175eqtr4d 2439 . . 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
) ) ) )
177118, 138, 1763brtr3d 4201 . 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
) ) ) )
178171, 173readdcld 9071 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  ( log `  A ) )  +  ( ( 1  -  T )  x.  ( log `  B
) ) )  e.  RR )
17915, 133sseldd 3309 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  RR+ )
180179relogcld 20471 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )  e.  RR )
181178, 180ltnegd 9560 . 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
) ) ) ) )
182177, 181mpbird 224 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   {cpr 3775   class class class wbr 4172    e. cmpt 4226    Po wpo 4461    Or wor 4462   ran crn 4838    |` cres 4839    Fn wfn 5408   -->wf 5409   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413    Isom wiso 5414  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   RR+crp 10568   (,)cioo 10872   [,]cicc 10875   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658   intcnt 17036   -cn->ccncf 18859    _D cdv 19703   logclog 20405
This theorem is referenced by:  amgmlem  20781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407
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