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Theorem efcvx 21889
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 991 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  RR )
2 simpl2 992 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  RR )
3 simpl3 993 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  <  B
)
4 reeff1o 21887 . . . . . . 7  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
5 f1of 5636 . . . . . . 7  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
64, 5ax-mp 5 . . . . . 6  |-  ( exp  |`  RR ) : RR --> RR+
7 rpssre 10993 . . . . . 6  |-  RR+  C_  RR
8 fss 5562 . . . . . 6  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
96, 7, 8mp2an 672 . . . . 5  |-  ( exp  |`  RR ) : RR --> RR
10 iccssre 11369 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
111, 2, 10syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( A [,] B )  C_  RR )
12 fssres2 5574 . . . . 5  |-  ( ( ( exp  |`  RR ) : RR --> RR  /\  ( A [,] B ) 
C_  RR )  -> 
( exp  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
139, 11, 12sylancr 663 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
14 ax-resscn 9331 . . . . 5  |-  RR  C_  CC
1511, 14syl6ss 3363 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( A [,] B )  C_  CC )
16 efcn 21883 . . . . . 6  |-  exp  e.  ( CC -cn-> CC )
17 rescncf 20448 . . . . . 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 20449 . . . . 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 663 . . . 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 21888 . . . . . 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 11349 . . . . . 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 2439 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  RR ) " ( A (,) B ) )  =  ( ( exp  |`  RR ) " ( A (,) B ) ) )
27 isores3 6021 . . . . 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 1218 . . . 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 3370 . . . . . . 7  |-  RR  C_  RR
30 fss 5562 . . . . . . . . 9  |-  ( ( ( exp  |`  RR ) : RR --> RR  /\  RR  C_  CC )  -> 
( exp  |`  RR ) : RR --> CC )
319, 14, 30mp2an 672 . . . . . . . 8  |-  ( exp  |`  RR ) : RR --> CC
32 eqid 2438 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3332tgioo2 20355 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3432, 33dvres 21361 . . . . . . . 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 682 . . . . . . 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 663 . . . . . 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 5134 . . . . . . . 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 6102 . . . . . 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 9366 . . . . . . . . . 10  |-  RR  e.  { RR ,  CC }
41 eff 13359 . . . . . . . . . 10  |-  exp : CC
--> CC
42 ssid 3370 . . . . . . . . . 10  |-  CC  C_  CC
43 dvef 21427 . . . . . . . . . . . . 13  |-  ( CC 
_D  exp )  =  exp
4443dmeqi 5036 . . . . . . . . . . . 12  |-  dom  ( CC  _D  exp )  =  dom  exp
4541fdmi 5559 . . . . . . . . . . . 12  |-  dom  exp  =  CC
4644, 45eqtri 2458 . . . . . . . . . . 11  |-  dom  ( CC  _D  exp )  =  CC
4714, 46sseqtr4i 3384 . . . . . . . . . 10  |-  RR  C_  dom  ( CC  _D  exp )
48 dvres3 21363 . . . . . . . . . 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 673 . . . . . . . . 9  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( ( CC  _D  exp )  |`  RR )
5043reseq1i 5101 . . . . . . . . 9  |-  ( ( CC  _D  exp )  |`  RR )  =  ( exp  |`  RR )
5149, 50eqtri 2458 . . . . . . . 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 20373 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
541, 2, 53syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5552, 54reseq12d 5106 . . . . . 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 2478 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( exp  |`  ( A [,] B ) ) )  =  ( ( exp  |`  RR )  |`  ( A (,) B ) ) )
57 isoeq1 6005 . . . . 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 461 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  ( 0 (,) 1 ) )
61 eqid 2438 . . 3  |-  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) )
621, 2, 3, 21, 59, 60, 61dvcvx 21467 . 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 9332 . . . . . . 7  |-  1  e.  CC
64 ioossre 11349 . . . . . . . . 9  |-  ( 0 (,) 1 )  C_  RR
6564, 60sseldi 3349 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  RR )
6665recnd 9404 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  CC )
67 nncan 9630 . . . . . . 7  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
6863, 66, 67sylancr 663 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( 1  -  ( 1  -  T
) )  =  T )
6968oveq1d 6101 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( 1  -  ( 1  -  T ) )  x.  A )  =  ( T  x.  A ) )
7069oveq1d 6101 . . . 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 11373 . . . . . . 7  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
7271, 60sseldi 3349 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  ( 0 [,] 1 ) )
73 iirev 20476 . . . . . 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 11420 . . . . 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 470 . . . 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 2513 . . 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 5699 . . 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 9425 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  RR* )
812rexrd 9425 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  RR* )
821, 2, 3ltled 9514 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  <_  B
)
83 lbicc2 11393 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
8480, 81, 82, 83syl3anc 1218 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  ( A [,] B ) )
85 fvres 5699 . . . . 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 6102 . . 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 11394 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
8980, 81, 82, 88syl3anc 1218 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  ( A [,] B ) )
90 fvres 5699 . . . . 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 6102 . . 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 6104 . 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 4316 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 965    = wceq 1369    e. wcel 1756    C_ wss 3323   {cpr 3874   class class class wbr 4287   dom cdm 4835   ran crn 4836    |` cres 4837   "cima 4838   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413    Isom wiso 5414  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   RR*cxr 9409    < clt 9410    <_ cle 9411    - cmin 9587   RR+crp 10983   (,)cioo 11292   [,]cicc 11295   expce 13339   TopOpenctopn 14352   topGenctg 14368  ℂfldccnfld 17793   intcnt 18596   -cn->ccncf 20427    _D cdv 21313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-cmp 18965  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317
This theorem is referenced by: (None)
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