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Theorem efcvx 22578
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 999 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  RR )
2 simpl2 1000 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  RR )
3 simpl3 1001 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  <  B
)
4 reeff1o 22576 . . . . . . 7  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
5 f1of 5814 . . . . . . 7  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
64, 5ax-mp 5 . . . . . 6  |-  ( exp  |`  RR ) : RR --> RR+
7 rpssre 11226 . . . . . 6  |-  RR+  C_  RR
8 fss 5737 . . . . . 6  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
96, 7, 8mp2an 672 . . . . 5  |-  ( exp  |`  RR ) : RR --> RR
10 iccssre 11602 . . . . . 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 5751 . . . . 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 9545 . . . . 5  |-  RR  C_  CC
1511, 14syl6ss 3516 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( A [,] B )  C_  CC )
16 efcn 22572 . . . . . 6  |-  exp  e.  ( CC -cn-> CC )
17 rescncf 21136 . . . . . 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 21137 . . . . 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 22577 . . . . . 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 11582 . . . . . 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 2468 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  RR ) " ( A (,) B ) )  =  ( ( exp  |`  RR ) " ( A (,) B ) ) )
27 isores3 6217 . . . . 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 1228 . . . 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 3523 . . . . . . 7  |-  RR  C_  RR
30 fss 5737 . . . . . . . . 9  |-  ( ( ( exp  |`  RR ) : RR --> RR  /\  RR  C_  CC )  -> 
( exp  |`  RR ) : RR --> CC )
319, 14, 30mp2an 672 . . . . . . . 8  |-  ( exp  |`  RR ) : RR --> CC
32 eqid 2467 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3332tgioo2 21043 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3432, 33dvres 22050 . . . . . . . 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 5300 . . . . . . . 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 6298 . . . . . 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 9580 . . . . . . . . . 10  |-  RR  e.  { RR ,  CC }
41 eff 13675 . . . . . . . . . 10  |-  exp : CC
--> CC
42 ssid 3523 . . . . . . . . . 10  |-  CC  C_  CC
43 dvef 22116 . . . . . . . . . . . . 13  |-  ( CC 
_D  exp )  =  exp
4443dmeqi 5202 . . . . . . . . . . . 12  |-  dom  ( CC  _D  exp )  =  dom  exp
4541fdmi 5734 . . . . . . . . . . . 12  |-  dom  exp  =  CC
4644, 45eqtri 2496 . . . . . . . . . . 11  |-  dom  ( CC  _D  exp )  =  CC
4714, 46sseqtr4i 3537 . . . . . . . . . 10  |-  RR  C_  dom  ( CC  _D  exp )
48 dvres3 22052 . . . . . . . . . 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 5267 . . . . . . . . 9  |-  ( ( CC  _D  exp )  |`  RR )  =  ( exp  |`  RR )
5149, 50eqtri 2496 . . . . . . . 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 21061 . . . . . . . 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 5272 . . . . . 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 2516 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( exp  |`  ( A [,] B ) ) )  =  ( ( exp  |`  RR )  |`  ( A (,) B ) ) )
57 isoeq1 6201 . . . . 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 2467 . . 3  |-  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) )
621, 2, 3, 21, 59, 60, 61dvcvx 22156 . 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 9546 . . . . . . 7  |-  1  e.  CC
64 ioossre 11582 . . . . . . . . 9  |-  ( 0 (,) 1 )  C_  RR
6564, 60sseldi 3502 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  RR )
6665recnd 9618 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  CC )
67 nncan 9844 . . . . . . 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 6297 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( 1  -  ( 1  -  T ) )  x.  A )  =  ( T  x.  A ) )
7069oveq1d 6297 . . . 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 11606 . . . . . . 7  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
7271, 60sseldi 3502 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  ( 0 [,] 1 ) )
73 iirev 21164 . . . . . 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 11659 . . . . 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 2556 . . 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 5878 . . 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 9639 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  RR* )
812rexrd 9639 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  RR* )
821, 2, 3ltled 9728 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  <_  B
)
83 lbicc2 11632 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
8480, 81, 82, 83syl3anc 1228 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  ( A [,] B ) )
85 fvres 5878 . . . . 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 6298 . . 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 11633 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
8980, 81, 82, 88syl3anc 1228 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  ( A [,] B ) )
90 fvres 5878 . . . . 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 6298 . . 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 6300 . 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 4476 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 973    = wceq 1379    e. wcel 1767    C_ wss 3476   {cpr 4029   class class class wbr 4447   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   -->wf 5582   -1-1-onto->wf1o 5585   ` cfv 5586    Isom wiso 5587  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493   RR*cxr 9623    < clt 9624    <_ cle 9625    - cmin 9801   RR+crp 11216   (,)cioo 11525   [,]cicc 11528   expce 13655   TopOpenctopn 14673   topGenctg 14689  ℂfldccnfld 18191   intcnt 19284   -cn->ccncf 21115    _D cdv 22002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-cmp 19653  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-limc 22005  df-dv 22006
This theorem is referenced by: (None)
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