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Theorem efcvx 21798
Description: The exponential function on the reals is a strictly convex function. (Contributed by Mario Carneiro, 20-Jun-2015.)
Assertion
Ref Expression
efcvx  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  <  ( ( T  x.  ( exp `  A ) )  +  ( ( 1  -  T )  x.  ( exp `  B ) ) ) )

Proof of Theorem efcvx
StepHypRef Expression
1 simpl1 984 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  RR )
2 simpl2 985 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  RR )
3 simpl3 986 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  <  B
)
4 reeff1o 21796 . . . . . . 7  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
5 f1of 5629 . . . . . . 7  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
64, 5ax-mp 5 . . . . . 6  |-  ( exp  |`  RR ) : RR --> RR+
7 rpssre 10988 . . . . . 6  |-  RR+  C_  RR
8 fss 5555 . . . . . 6  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
96, 7, 8mp2an 665 . . . . 5  |-  ( exp  |`  RR ) : RR --> RR
10 iccssre 11364 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
111, 2, 10syl2anc 654 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( A [,] B )  C_  RR )
12 fssres2 5567 . . . . 5  |-  ( ( ( exp  |`  RR ) : RR --> RR  /\  ( A [,] B ) 
C_  RR )  -> 
( exp  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
139, 11, 12sylancr 656 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
14 ax-resscn 9326 . . . . 5  |-  RR  C_  CC
1511, 14syl6ss 3356 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( A [,] B )  C_  CC )
16 efcn 21792 . . . . . 6  |-  exp  e.  ( CC -cn-> CC )
17 rescncf 20314 . . . . . 6  |-  ( ( A [,] B ) 
C_  CC  ->  ( exp 
e.  ( CC -cn-> CC )  ->  ( exp  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) ) )
1815, 16, 17mpisyl 18 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
19 cncffvrn 20315 . . . . 5  |-  ( ( RR  C_  CC  /\  ( exp  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )  ->  ( ( exp  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  <-> 
( exp  |`  ( A [,] B ) ) : ( A [,] B ) --> RR ) )
2014, 18, 19sylancr 656 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> RR )  <->  ( exp  |`  ( A [,] B
) ) : ( A [,] B ) --> RR ) )
2113, 20mpbird 232 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
22 reefiso 21797 . . . . . 6  |-  ( exp  |`  RR )  Isom  <  ,  <  ( RR ,  RR+ )
2322a1i 11 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp  |`  RR ) 
Isom  <  ,  <  ( RR ,  RR+ ) )
24 ioossre 11344 . . . . . 6  |-  ( A (,) B )  C_  RR
2524a1i 11 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( A (,) B )  C_  RR )
26 eqidd 2434 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  RR ) " ( A (,) B ) )  =  ( ( exp  |`  RR ) " ( A (,) B ) ) )
27 isores3 6013 . . . . 5  |-  ( ( ( exp  |`  RR ) 
Isom  <  ,  <  ( RR ,  RR+ )  /\  ( A (,) B ) 
C_  RR  /\  (
( exp  |`  RR )
" ( A (,) B ) )  =  ( ( exp  |`  RR )
" ( A (,) B ) ) )  ->  ( ( exp  |`  RR )  |`  ( A (,) B ) ) 
Isom  <  ,  <  (
( A (,) B
) ,  ( ( exp  |`  RR ) " ( A (,) B ) ) ) )
2823, 25, 26, 27syl3anc 1211 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  RR )  |`  ( A (,) B ) ) 
Isom  <  ,  <  (
( A (,) B
) ,  ( ( exp  |`  RR ) " ( A (,) B ) ) ) )
29 ssid 3363 . . . . . . 7  |-  RR  C_  RR
30 fss 5555 . . . . . . . . 9  |-  ( ( ( exp  |`  RR ) : RR --> RR  /\  RR  C_  CC )  -> 
( exp  |`  RR ) : RR --> CC )
319, 14, 30mp2an 665 . . . . . . . 8  |-  ( exp  |`  RR ) : RR --> CC
32 eqid 2433 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3332tgioo2 20221 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3432, 33dvres 21227 . . . . . . . 8  |-  ( ( ( RR  C_  CC  /\  ( exp  |`  RR ) : RR --> CC )  /\  ( RR  C_  RR  /\  ( A [,] B )  C_  RR ) )  ->  ( RR  _D  ( ( exp  |`  RR )  |`  ( A [,] B ) ) )  =  ( ( RR  _D  ( exp  |`  RR ) )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
3514, 31, 34mpanl12 675 . . . . . . 7  |-  ( ( RR  C_  RR  /\  ( A [,] B )  C_  RR )  ->  ( RR 
_D  ( ( exp  |`  RR )  |`  ( A [,] B ) ) )  =  ( ( RR  _D  ( exp  |`  RR ) )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
3629, 11, 35sylancr 656 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( ( exp  |`  RR )  |`  ( A [,] B
) ) )  =  ( ( RR  _D  ( exp  |`  RR )
)  |`  ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) ) ) )
37 resabs1 5127 . . . . . . . 8  |-  ( ( A [,] B ) 
C_  RR  ->  ( ( exp  |`  RR )  |`  ( A [,] B
) )  =  ( exp  |`  ( A [,] B ) ) )
3811, 37syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  RR )  |`  ( A [,] B ) )  =  ( exp  |`  ( A [,] B ) ) )
3938oveq2d 6096 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( ( exp  |`  RR )  |`  ( A [,] B
) ) )  =  ( RR  _D  ( exp  |`  ( A [,] B ) ) ) )
40 reelprrecn 9361 . . . . . . . . . 10  |-  RR  e.  { RR ,  CC }
41 eff 13349 . . . . . . . . . 10  |-  exp : CC
--> CC
42 ssid 3363 . . . . . . . . . 10  |-  CC  C_  CC
43 dvef 21293 . . . . . . . . . . . . 13  |-  ( CC 
_D  exp )  =  exp
4443dmeqi 5028 . . . . . . . . . . . 12  |-  dom  ( CC  _D  exp )  =  dom  exp
4541fdmi 5552 . . . . . . . . . . . 12  |-  dom  exp  =  CC
4644, 45eqtri 2453 . . . . . . . . . . 11  |-  dom  ( CC  _D  exp )  =  CC
4714, 46sseqtr4i 3377 . . . . . . . . . 10  |-  RR  C_  dom  ( CC  _D  exp )
48 dvres3 21229 . . . . . . . . . 10  |-  ( ( ( RR  e.  { RR ,  CC }  /\  exp : CC --> CC )  /\  ( CC  C_  CC  /\  RR  C_  dom  ( CC  _D  exp )
) )  ->  ( RR  _D  ( exp  |`  RR ) )  =  ( ( CC  _D  exp )  |`  RR ) )
4940, 41, 42, 47, 48mp4an 666 . . . . . . . . 9  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( ( CC  _D  exp )  |`  RR )
5043reseq1i 5093 . . . . . . . . 9  |-  ( ( CC  _D  exp )  |`  RR )  =  ( exp  |`  RR )
5149, 50eqtri 2453 . . . . . . . 8  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( exp  |`  RR )
5251a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( exp  |`  RR )
)  =  ( exp  |`  RR ) )
53 iccntr 20239 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
541, 2, 53syl2anc 654 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5552, 54reseq12d 5098 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( RR 
_D  ( exp  |`  RR ) )  |`  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )  =  ( ( exp  |`  RR )  |`  ( A (,) B
) ) )
5636, 39, 553eqtr3d 2473 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( exp  |`  ( A [,] B ) ) )  =  ( ( exp  |`  RR )  |`  ( A (,) B ) ) )
57 isoeq1 5997 . . . . 5  |-  ( ( RR  _D  ( exp  |`  ( A [,] B
) ) )  =  ( ( exp  |`  RR )  |`  ( A (,) B
) )  ->  (
( RR  _D  ( exp  |`  ( A [,] B ) ) ) 
Isom  <  ,  <  (
( A (,) B
) ,  ( ( exp  |`  RR ) " ( A (,) B ) ) )  <-> 
( ( exp  |`  RR )  |`  ( A (,) B
) )  Isom  <  ,  <  ( ( A (,) B ) ,  ( ( exp  |`  RR )
" ( A (,) B ) ) ) ) )
5856, 57syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( RR 
_D  ( exp  |`  ( A [,] B ) ) )  Isom  <  ,  <  ( ( A (,) B
) ,  ( ( exp  |`  RR ) " ( A (,) B ) ) )  <-> 
( ( exp  |`  RR )  |`  ( A (,) B
) )  Isom  <  ,  <  ( ( A (,) B ) ,  ( ( exp  |`  RR )
" ( A (,) B ) ) ) ) )
5928, 58mpbird 232 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( exp  |`  ( A [,] B ) ) ) 
Isom  <  ,  <  (
( A (,) B
) ,  ( ( exp  |`  RR ) " ( A (,) B ) ) ) )
60 simpr 458 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  ( 0 (,) 1 ) )
61 eqid 2433 . . 3  |-  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) )
621, 2, 3, 21, 59, 60, 61dvcvx 21333 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  ( A [,] B
) ) `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  <  ( ( T  x.  ( ( exp  |`  ( A [,] B ) ) `  A ) )  +  ( ( 1  -  T )  x.  (
( exp  |`  ( A [,] B ) ) `
 B ) ) ) )
63 ax-1cn 9327 . . . . . . 7  |-  1  e.  CC
64 ioossre 11344 . . . . . . . . 9  |-  ( 0 (,) 1 )  C_  RR
6564, 60sseldi 3342 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  RR )
6665recnd 9399 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  CC )
67 nncan 9625 . . . . . . 7  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
6863, 66, 67sylancr 656 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( 1  -  ( 1  -  T
) )  =  T )
6968oveq1d 6095 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( 1  -  ( 1  -  T ) )  x.  A )  =  ( T  x.  A ) )
7069oveq1d 6095 . . . 4  |-  ( ( ( 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 ) ) )
71 ioossicc 11368 . . . . . . 7  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
7271, 60sseldi 3342 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  ( 0 [,] 1 ) )
73 iirev 20342 . . . . . 6  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
7472, 73syl 16 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( 1  -  T )  e.  ( 0 [,] 1 ) )
75 lincmb01cmp 11414 . . . . 5  |-  ( ( ( 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 ) )
7674, 75syldan 467 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( ( 1  -  ( 1  -  T ) )  x.  A )  +  ( ( 1  -  T )  x.  B
) )  e.  ( A [,] B ) )
7770, 76eqeltrrd 2508 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  e.  ( A [,] B ) )
78 fvres 5692 . . 3  |-  ( ( ( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  ( A [,] B )  ->  (
( exp  |`  ( A [,] B ) ) `
 ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )  =  ( exp `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
7977, 78syl 16 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  ( A [,] B
) ) `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  =  ( exp `  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) ) ) )
801rexrd 9420 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  RR* )
812rexrd 9420 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  RR* )
821, 2, 3ltled 9509 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  <_  B
)
83 lbicc2 11387 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
8480, 81, 82, 83syl3anc 1211 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  ( A [,] B ) )
85 fvres 5692 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  (
( exp  |`  ( A [,] B ) ) `
 A )  =  ( exp `  A
) )
8684, 85syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  ( A [,] B
) ) `  A
)  =  ( exp `  A ) )
8786oveq2d 6096 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( T  x.  ( ( exp  |`  ( A [,] B ) ) `
 A ) )  =  ( T  x.  ( exp `  A ) ) )
88 ubicc2 11388 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
8980, 81, 82, 88syl3anc 1211 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  ( A [,] B ) )
90 fvres 5692 . . . . 5  |-  ( B  e.  ( A [,] B )  ->  (
( exp  |`  ( A [,] B ) ) `
 B )  =  ( exp `  B
) )
9189, 90syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  ( A [,] B
) ) `  B
)  =  ( exp `  B ) )
9291oveq2d 6096 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( 1  -  T )  x.  ( ( exp  |`  ( A [,] B ) ) `
 B ) )  =  ( ( 1  -  T )  x.  ( exp `  B
) ) )
9387, 92oveq12d 6098 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( T  x.  ( ( exp  |`  ( A [,] B
) ) `  A
) )  +  ( ( 1  -  T
)  x.  ( ( exp  |`  ( A [,] B ) ) `  B ) ) )  =  ( ( T  x.  ( exp `  A
) )  +  ( ( 1  -  T
)  x.  ( exp `  B ) ) ) )
9462, 79, 933brtr3d 4309 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  <  ( ( T  x.  ( exp `  A ) )  +  ( ( 1  -  T )  x.  ( exp `  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755    C_ wss 3316   {cpr 3867   class class class wbr 4280   dom cdm 4827   ran crn 4828    |` cres 4829   "cima 4830   -->wf 5402   -1-1-onto->wf1o 5405   ` cfv 5406    Isom wiso 5407  (class class class)co 6080   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270    + caddc 9272    x. cmul 9274   RR*cxr 9404    < clt 9405    <_ cle 9406    - cmin 9582   RR+crp 10978   (,)cioo 11287   [,]cicc 11290   expce 13329   TopOpenctopn 14342   topGenctg 14358  ℂfldccnfld 17661   intcnt 18462   -cn->ccncf 20293    _D cdv 21179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ioo 11291  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-fl 11625  df-seq 11790  df-exp 11849  df-fac 12035  df-bc 12062  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-sum 13147  df-ef 13335  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-starv 14235  df-sca 14236  df-vsca 14237  df-ip 14238  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-hom 14244  df-cco 14245  df-rest 14343  df-topn 14344  df-0g 14362  df-gsum 14363  df-topgen 14364  df-pt 14365  df-prds 14368  df-xrs 14422  df-qtop 14427  df-imas 14428  df-xps 14430  df-mre 14506  df-mrc 14507  df-acs 14509  df-mnd 15397  df-submnd 15447  df-mulg 15527  df-cntz 15814  df-cmn 16258  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-fbas 17657  df-fg 17658  df-cnfld 17662  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-cld 18464  df-ntr 18465  df-cls 18466  df-nei 18543  df-lp 18581  df-perf 18582  df-cn 18672  df-cnp 18673  df-haus 18760  df-cmp 18831  df-tx 18976  df-hmeo 19169  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-xms 19736  df-ms 19737  df-tms 19738  df-cncf 20295  df-limc 21182  df-dv 21183
This theorem is referenced by: (None)
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