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Theorem logccv 20026
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 958 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR+ )
21rpred 10406 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR )
3 simpl2 959 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR+ )
43rpred 10406 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR )
5 simpl3 960 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  <  B )
61rpgt0d 10409 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  0  <  A )
7 ltpnf 10479 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  B  <  +oo )
84, 7syl 15 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  <  +oo )
9 0xr 8894 . . . . . . . . . . . 12  |-  0  e.  RR*
10 pnfxr 10471 . . . . . . . . . . . 12  |-  +oo  e.  RR*
11 iccssioo 10735 . . . . . . . . . . . 12  |-  ( ( ( 0  e.  RR*  /\ 
+oo  e.  RR* )  /\  ( 0  <  A  /\  B  <  +oo )
)  ->  ( A [,] B )  C_  (
0 (,)  +oo ) )
129, 10, 11mpanl12 663 . . . . . . . . . . 11  |-  ( ( 0  <  A  /\  B  <  +oo )  ->  ( A [,] B )  C_  ( 0 (,)  +oo ) )
136, 8, 12syl2anc 642 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A [,] B )  C_  ( 0 (,)  +oo ) )
14 ioorp 10743 . . . . . . . . . 10  |-  ( 0 (,)  +oo )  =  RR+
1513, 14syl6sseq 3237 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A [,] B )  C_  RR+ )
1615sselda 3193 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR+ )
1716relogcld 19990 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  ( log `  x )  e.  RR )
1817renegcld 9226 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A [,] B
) )  ->  -u ( log `  x )  e.  RR )
19 eqid 2296 . . . . . 6  |-  ( x  e.  ( A [,] B )  |->  -u ( log `  x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )
2018, 19fmptd 5700 . . . . 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 8810 . . . . . 6  |-  RR  C_  CC
22 resabs1 5000 . . . . . . . . 9  |-  ( ( A [,] B ) 
C_  RR+  ->  ( ( log  |`  RR+ )  |`  ( A [,] B ) )  =  ( log  |`  ( A [,] B ) ) )
2315, 22syl 15 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  =  ( log  |`  ( A [,] B ) ) )
24 ssid 3210 . . . . . . . . . . 11  |-  CC  C_  CC
25 cncfss 18419 . . . . . . . . . . 11  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC ) )
2621, 24, 25mp2an 653 . . . . . . . . . 10  |-  ( RR+ -cn-> RR )  C_  ( RR+ -cn-> CC )
27 relogcn 20001 . . . . . . . . . 10  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
2826, 27sselii 3190 . . . . . . . . 9  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )
29 rescncf 18417 . . . . . . . . 9  |-  ( ( A [,] B ) 
C_  RR+  ->  ( ( log  |`  RR+ )  e.  (
RR+ -cn-> CC )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) ) )
3015, 28, 29ee10 1366 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( log  |`  RR+ )  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) )
3123, 30eqeltrrd 2371 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
32 fvres 5558 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  (
( log  |`  ( A [,] B ) ) `
 x )  =  ( log `  x
) )
3332negeqd 9062 . . . . . . . . . 10  |-  ( x  e.  ( A [,] B )  ->  -u (
( log  |`  ( A [,] B ) ) `
 x )  = 
-u ( log `  x
) )
3433mpteq2ia 4118 . . . . . . . . 9  |-  ( x  e.  ( A [,] B )  |->  -u (
( log  |`  ( A [,] B ) ) `
 x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )
3534eqcomi 2300 . . . . . . . 8  |-  ( x  e.  ( A [,] B )  |->  -u ( log `  x ) )  =  ( x  e.  ( A [,] B
)  |->  -u ( ( log  |`  ( A [,] B
) ) `  x
) )
3635negfcncf 18438 . . . . . . 7  |-  ( ( log  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC )  ->  ( x  e.  ( A [,] B
)  |->  -u ( log `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
3731, 36syl 15 . . . . . 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 18418 . . . . . 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 644 . . . . 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 223 . . . 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 10728 . . . . . . . 8  |-  ( A (,) B )  C_  RR
42 ltso 8919 . . . . . . . 8  |-  <  Or  RR
43 soss 4348 . . . . . . . 8  |-  ( ( A (,) B ) 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  ( A (,) B ) ) )
4441, 42, 43mp2 17 . . . . . . 7  |-  <  Or  ( A (,) B )
4544a1i 10 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Or  ( A (,) B
) )
46 ioossicc 10751 . . . . . . . . . . . . . 14  |-  ( A (,) B )  C_  ( A [,] B )
4746, 15syl5ss 3203 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( A (,) B )  C_  RR+ )
4847sselda 3193 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  x  e.  RR+ )
4948rprecred 10417 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  (
1  /  x )  e.  RR )
5049renegcld 9226 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  ( A (,) B
) )  ->  -u (
1  /  x )  e.  RR )
51 eqid 2296 . . . . . . . . . 10  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  =  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )
5250, 51fmptd 5700 . . . . . . . . 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 5411 . . . . . . . . 9  |-  ( ( x  e.  ( A (,) B )  |->  -u ( 1  /  x
) ) : ( A (,) B ) --> RR  ->  ran  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  C_  RR )
5452, 53syl 15 . . . . . . . 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 4348 . . . . . . . 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 1366 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Or 
ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
57 sopo 4347 . . . . . . 7  |-  (  < 
Or  ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) )  ->  <  Po  ran  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) )
5856, 57syl 15 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  <  Po 
ran  ( x  e.  ( A (,) B
)  |->  -u ( 1  /  x ) ) )
59 negex 9066 . . . . . . . . 9  |-  -u (
1  /  x )  e.  _V
6059, 51fnmpti 5388 . . . . . . . 8  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) )  Fn  ( A (,) B )
61 dffn4 5473 . . . . . . . 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 199 . . . . . . 7  |-  ( x  e.  ( A (,) B )  |->  -u (
1  /  x ) ) : ( A (,) B ) -onto-> ran  ( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) )
6362a1i 10 . . . . . 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 3193 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  z  e.  ( A (,) B
) )  ->  z  e.  RR+ )
6564adantrl 696 . . . . . . . . . . 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 10417 . . . . . . . . . 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 3193 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  y  e.  ( A (,) B
) )  ->  y  e.  RR+ )
6867adantrr 697 . . . . . . . . . . 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 10417 . . . . . . . . . 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 9366 . . . . . . . . 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 10424 . . . . . . . . 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 5882 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
1  /  x )  =  ( 1  / 
y ) )
7372negeqd 9062 . . . . . . . . . . . 12  |-  ( x  =  y  ->  -u (
1  /  x )  =  -u ( 1  / 
y ) )
74 negex 9066 . . . . . . . . . . . 12  |-  -u (
1  /  y )  e.  _V
7573, 51, 74fvmpt 5618 . . . . . . . . . . 11  |-  ( y  e.  ( A (,) B )  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  y )  =  -u ( 1  /  y
) )
76 oveq2 5882 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
1  /  x )  =  ( 1  / 
z ) )
7776negeqd 9062 . . . . . . . . . . . 12  |-  ( x  =  z  ->  -u (
1  /  x )  =  -u ( 1  / 
z ) )
78 negex 9066 . . . . . . . . . . . 12  |-  -u (
1  /  z )  e.  _V
7977, 51, 78fvmpt 5618 . . . . . . . . . . 11  |-  ( z  e.  ( A (,) B )  ->  (
( x  e.  ( A (,) B ) 
|->  -u ( 1  /  x ) ) `  z )  =  -u ( 1  /  z
) )
8075, 79breqan12d 4054 . . . . . . . . . 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 452 . . . . . . . . 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 276 . . . . . . . 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 198 . . . . . . 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 2648 . . . . . 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 5841 . . . . . 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 1183 . . . . 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 8844 . . . . . . . . 9  |-  RR  e.  _V
8887prid1 3747 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
8988a1i 10 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  RR  e.  { RR ,  CC } )
90 relogcl 19948 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
9190adantl 452 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
9291recnd 8877 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
9392negcld 9160 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  -u ( log `  x )  e.  CC )
9459a1i 10 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  -u (
1  /  x )  e.  _V )
95 ovex 5899 . . . . . . . . 9  |-  ( 1  /  x )  e. 
_V
9695a1i 10 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  /\  x  e.  RR+ )  ->  ( 1  /  x )  e. 
_V )
97 dvrelog 20000 . . . . . . . . 9  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
98 relogf1o 19940 . . . . . . . . . . . . 13  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
99 f1of 5488 . . . . . . . . . . . . 13  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
10098, 99mp1i 11 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ ) : RR+ --> RR )
101100feqmptd 5591 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
102 fvres 5558 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
103102mpteq2ia 4118 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
104101, 103syl6eq 2344 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
105104oveq2d 5890 . . . . . . . . 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 2343 . . . . . . . 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 19331 . . . . . . 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 2296 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
109108tgioo2 18325 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
110 iccntr 18342 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
1112, 4, 110syl2anc 642 . . . . . . 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 19327 . . . . . 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 5832 . . . . . 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 15 . . . . 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 223 . . . 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 447 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  ( 0 (,) 1
) )
117 eqid 2296 . . . 4  |-  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) )
1182, 4, 5, 40, 115, 116, 117dvcvx 19383 . . 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 8811 . . . . . . . 8  |-  1  e.  CC
120 elioore 10702 . . . . . . . . . 10  |-  ( T  e.  ( 0 (,) 1 )  ->  T  e.  RR )
121120adantl 452 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  RR )
122121recnd 8877 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  CC )
123 nncan 9092 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
124119, 122, 123sylancr 644 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  ( 1  -  T ) )  =  T )
125124oveq1d 5889 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  (
1  -  T ) )  x.  A )  =  ( T  x.  A ) )
126125oveq1d 5889 . . . . 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 10751 . . . . . . . 8  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
128127, 116sseldi 3191 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  T  e.  ( 0 [,] 1
) )
129 iirev 18443 . . . . . . 7  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
130128, 129syl 15 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
131 lincmb01cmp 10793 . . . . . 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 1185 . . . . 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 2371 . . . 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 5541 . . . . . 6  |-  ( x  =  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  ->  ( log `  x )  =  ( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
135134negeqd 9062 . . . . 5  |-  ( x  =  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  ->  -u ( log `  x )  = 
-u ( log `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
136 negex 9066 . . . . 5  |-  -u ( log `  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )  e. 
_V
137135, 19, 136fvmpt 5618 . . . 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 15 . . 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 10407 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  RR* )
1403rpxrd 10407 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  RR* )
1412, 4, 5ltled 8983 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  <_  B )
142 lbicc2 10768 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
143139, 140, 141, 142syl3anc 1182 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  A  e.  ( A [,] B
) )
144 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  A  ->  ( log `  x )  =  ( log `  A
) )
145144negeqd 9062 . . . . . . . . 9  |-  ( x  =  A  ->  -u ( log `  x )  = 
-u ( log `  A
) )
146 negex 9066 . . . . . . . . 9  |-  -u ( log `  A )  e. 
_V
147145, 19, 146fvmpt 5618 . . . . . . . 8  |-  ( A  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  A
)  =  -u ( log `  A ) )
148143, 147syl 15 . . . . . . 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 5890 . . . . . 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 19990 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  A )  e.  RR )
151150recnd 8877 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  A )  e.  CC )
152122, 151mulneg2d 9249 . . . . . 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 2328 . . . . 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 10769 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
155139, 140, 141, 154syl3anc 1182 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  B  e.  ( A [,] B
) )
156 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  B  ->  ( log `  x )  =  ( log `  B
) )
157156negeqd 9062 . . . . . . . . 9  |-  ( x  =  B  ->  -u ( log `  x )  = 
-u ( log `  B
) )
158 negex 9066 . . . . . . . . 9  |-  -u ( log `  B )  e. 
_V
159157, 19, 158fvmpt 5618 . . . . . . . 8  |-  ( B  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  -u ( log `  x
) ) `  B
)  =  -u ( log `  B ) )
160155, 159syl 15 . . . . . . 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 5890 . . . . . 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 8853 . . . . . . . . 9  |-  1  e.  RR
163 resubcl 9127 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
164162, 121, 163sylancr 644 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  RR )
165164recnd 8877 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
1  -  T )  e.  CC )
1663relogcld 19990 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  B )  e.  RR )
167166recnd 8877 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( log `  B )  e.  CC )
168165, 167mulneg2d 9249 . . . . . 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 2328 . . . . 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 5892 . . . 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 8879 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( log `  A ) )  e.  RR )
172171recnd 8877 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  ( T  x.  ( log `  A ) )  e.  CC )
173164, 166remulcld 8879 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( log `  B ) )  e.  RR )
174173recnd 8877 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( 1  -  T
)  x.  ( log `  B ) )  e.  CC )
175172, 174negdid 9186 . . . 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 2331 . . 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 4068 . 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 8878 . . 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 3194 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+  /\  A  <  B )  /\  T  e.  ( 0 (,) 1
) )  ->  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  RR+ )
180179relogcld 19990 . . 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 9366 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   {cpr 3654   class class class wbr 4039    e. cmpt 4093    Po wpo 4328    Or wor 4329   ran crn 4706    |` cres 4707    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   RR+crp 10370   (,)cioo 10672   [,]cicc 10675   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393   intcnt 16770   -cn->ccncf 18396    _D cdv 19229   logclog 19928
This theorem is referenced by:  amgmlem  20300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930
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