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Theorem efcvx 22040
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 22038 . . . . . . 7  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
5 f1of 5742 . . . . . . 7  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
64, 5ax-mp 5 . . . . . 6  |-  ( exp  |`  RR ) : RR --> RR+
7 rpssre 11105 . . . . . 6  |-  RR+  C_  RR
8 fss 5668 . . . . . 6  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
96, 7, 8mp2an 672 . . . . 5  |-  ( exp  |`  RR ) : RR --> RR
10 iccssre 11481 . . . . . 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 5680 . . . . 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 9443 . . . . 5  |-  RR  C_  CC
1511, 14syl6ss 3469 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( A [,] B )  C_  CC )
16 efcn 22034 . . . . . 6  |-  exp  e.  ( CC -cn-> CC )
17 rescncf 20598 . . . . . 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 20599 . . . . 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 22039 . . . . . 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 11461 . . . . . 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 2452 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  RR ) " ( A (,) B ) )  =  ( ( exp  |`  RR ) " ( A (,) B ) ) )
27 isores3 6128 . . . . 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 1219 . . . 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 3476 . . . . . . 7  |-  RR  C_  RR
30 fss 5668 . . . . . . . . 9  |-  ( ( ( exp  |`  RR ) : RR --> RR  /\  RR  C_  CC )  -> 
( exp  |`  RR ) : RR --> CC )
319, 14, 30mp2an 672 . . . . . . . 8  |-  ( exp  |`  RR ) : RR --> CC
32 eqid 2451 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3332tgioo2 20505 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3432, 33dvres 21512 . . . . . . . 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 5240 . . . . . . . 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 6209 . . . . . 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 9478 . . . . . . . . . 10  |-  RR  e.  { RR ,  CC }
41 eff 13478 . . . . . . . . . 10  |-  exp : CC
--> CC
42 ssid 3476 . . . . . . . . . 10  |-  CC  C_  CC
43 dvef 21578 . . . . . . . . . . . . 13  |-  ( CC 
_D  exp )  =  exp
4443dmeqi 5142 . . . . . . . . . . . 12  |-  dom  ( CC  _D  exp )  =  dom  exp
4541fdmi 5665 . . . . . . . . . . . 12  |-  dom  exp  =  CC
4644, 45eqtri 2480 . . . . . . . . . . 11  |-  dom  ( CC  _D  exp )  =  CC
4714, 46sseqtr4i 3490 . . . . . . . . . 10  |-  RR  C_  dom  ( CC  _D  exp )
48 dvres3 21514 . . . . . . . . . 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 5207 . . . . . . . . 9  |-  ( ( CC  _D  exp )  |`  RR )  =  ( exp  |`  RR )
5149, 50eqtri 2480 . . . . . . . 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 20523 . . . . . . . 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 5212 . . . . . 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 2500 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( exp  |`  ( A [,] B ) ) )  =  ( ( exp  |`  RR )  |`  ( A (,) B ) ) )
57 isoeq1 6112 . . . . 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 2451 . . 3  |-  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) )
621, 2, 3, 21, 59, 60, 61dvcvx 21618 . 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 9444 . . . . . . 7  |-  1  e.  CC
64 ioossre 11461 . . . . . . . . 9  |-  ( 0 (,) 1 )  C_  RR
6564, 60sseldi 3455 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  RR )
6665recnd 9516 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  CC )
67 nncan 9742 . . . . . . 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 6208 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( 1  -  ( 1  -  T ) )  x.  A )  =  ( T  x.  A ) )
7069oveq1d 6208 . . . 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 11485 . . . . . . 7  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
7271, 60sseldi 3455 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  ( 0 [,] 1 ) )
73 iirev 20626 . . . . . 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 11538 . . . . 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 2540 . . 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 5806 . . 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 9537 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  RR* )
812rexrd 9537 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  RR* )
821, 2, 3ltled 9626 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  <_  B
)
83 lbicc2 11511 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
8480, 81, 82, 83syl3anc 1219 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  ( A [,] B ) )
85 fvres 5806 . . . . 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 6209 . . 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 11512 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
8980, 81, 82, 88syl3anc 1219 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  ( A [,] B ) )
90 fvres 5806 . . . . 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 6209 . . 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 6211 . 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 4422 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 1370    e. wcel 1758    C_ wss 3429   {cpr 3980   class class class wbr 4393   dom cdm 4941   ran crn 4942    |` cres 4943   "cima 4944   -->wf 5515   -1-1-onto->wf1o 5518   ` cfv 5519    Isom wiso 5520  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387    + caddc 9389    x. cmul 9391   RR*cxr 9521    < clt 9522    <_ cle 9523    - cmin 9699   RR+crp 11095   (,)cioo 11404   [,]cicc 11407   expce 13458   TopOpenctopn 14471   topGenctg 14487  ℂfldccnfld 17936   intcnt 18746   -cn->ccncf 20577    _D cdv 21464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-fac 12162  df-bc 12189  df-hash 12214  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-sum 13275  df-ef 13464  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-lp 18865  df-perf 18866  df-cn 18956  df-cnp 18957  df-haus 19044  df-cmp 19115  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-xms 20020  df-ms 20021  df-tms 20022  df-cncf 20579  df-limc 21467  df-dv 21468
This theorem is referenced by: (None)
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